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Baldur: A one-dimensional plasma transport code
Computer Physics Communications 49 (1988) 275—398 North-Holland, Amsterdam
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BALDUR: A ONE-DIMENSIONAL PLASMA TRANSPORT CODE C.E. SINGER D.E. POST 1, D.R. MIKKELSEN ‘,M.H. REDI 1, A. McKENNEY 2 A. SILVERMAN F.G.P. SEIDL P.H. RUTHERFORD 1, R.J. HAWRYLUK 1, W.D. LANGER1, L. FOOTE ~‘, D.B. HEIFETZ ~, W.A. HOULBERG ~, M.H. HUGHES6 R.V. JENSEN 1, G. LISTER7 and J. OGDEN 8 ~,
~,
~,
‘Plasma Physics Laboratory, Princeton University, Princeton, NJ 08545, USA 2New York University, New York, NY 10012, USA ~ Corporation, Los Angeles, CA 90007, USA 4Los Alamos National Laboratory, Los Alamos, NM 87545, USA ~ Ridge National Laboratory, Oak Ridge, TN 37830, USA ~Grumman Aerospace, Princeton, NJ 08540, USA 7Max-Planck-Institut fur Plasmaphysik, Garching bei München, Fed Rep. Germany 85ch~,lof Engineering and Applied Science, Princeton University, Princeton, NJ 08544, USA Received 10 September 1984; in revised form 1 July 1986; in final form 18 September 1987
A version of the BALDUR plasma transport code which calculates the evolution of plasma parameters is documented. This version uses an MHD equilibrium which can be approximated by concentric circular flux surfaces. Transport of up to six species of ionized particles, of electron and ion energy, and of poloidal magnetic field is computed. A wide variety of source terms are calculated including those due to neutral gas, fusion and auxiliary heating. The code is primarily designed for modelling tokamak plasmas.
PROGRAM SUMMARY Tine ofprogram: BALDUR version BALDP47M
Keywords: plasma physics, thermonuclear fusion, tokainak, one-dimensional, transport
Catalogue number: ABBS Program obtainable from: CPC Program Library, Queen’s Urnversity of Belfast, N. Ireland (see application form in this issue) Computer: CRAY-i; Installation: Magnetic Fusion Energy Computer Center (MFECC), Lawrence Livermore Laboratory, Livermore, California Operating system: CRAY Timesharing System
Nature ofphysical problem The purpose of the version of the BALDUR code documented here is to calculate the evolution of plasma parameters in an MHD equilibrium which can be approximated by concentric circular flux surfaces. Transport of up to six species of ionized particles, of electron and ion energy, and of poloidal magnetic field is computed. A wide variety of source terms are calculated including those due to neutral gas, fusion and auxiliary heating. The code is primarily designed for modeling tokamak plasmas but could be adapted to other toroidal confinement systems.
Programming language used: FORTRAN High speed storage required: 721000 (octal) Number of bits in a word: 64 Peripheral used: disk No. of lines in combined program and test deck: 45271 (decimal)
Method of solution The plasma and the poloidal magnetic field encircling the plasma column are described by up to nine dependent variables, including up to six particle densities, ion energy, electron energy and poloidal magnetic field. These variables are functions of time t and radius r. The code was designed for a circular poloidal cross section but can be applied to constant effipticity ellipses with neutral beam injection.
OO1O-4655/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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The basic equations form a set of parabolic initial value equations with nonlinear coefficients and source terms. The equations are differenced with a conservative Crank—Nicholson scheme with an adjustable degree of implicitness [1]. Time centering of the source terms and transport coefficients is accomplished using a predictor—corrector scheme [21or by extrapolation methods [3]. Convective terms are handled with a variety of up-stream and down-stream weighting schemes [41. Monte Carlo techniques are used to compute (1) a-particle heating, (2) neutral gas transport [5] and (3) the ionization of neutral beams [6]. The evolution of the fast ion distribution, resulting from neutral beam injection, is computed as a function of energy and pitch angle using a simple multigroup Fokker—Planck scheme. Restrictions on the complexity of the problem BALDUR has been applied to a wide variety of tokamak simulations, and contains a variety of checks which terminate the computation if it goes outside the range of validity of the model. The most commonly encountered of these gives a message that the thermal velocity of the ions exceeds the neutral beam injection energy. Thermal collapse of the plasma edge will often drive the time step below a prescribed minimum (cf. section 6). However, care must be taken to provide physically reasonable initial and boundary conditions and to examine carefully the assumptions used in modeling a given device, especially when operating outside the range of parameters covered by published applications. For example, line radiation losses assume coronal equilibrium, and users must use the output to support their own calculations to assess the validity of this assumption. A number of common input errors are checked at the beginning of each computation.
Typical running time Execution time is typically but not exclusively between three and ten minutes on the MFECC CRAY, depending on the complexity and length of the discharge simulation, and the sophistication of the physical models taken from among the various choices available in BALDUR.
Unusual features of the program The majority of BALDUR is written in OLYMPUS [7] Fortran. NAMELIST is used for specifying input parameters. A choice among a variety of graphical and pnnted outputs is provided. The code also accesses US Magnetic Energy Fusion Computer Center Library subroutines such as their random number generator, which must be used to exactly reproduce the test case given below.
References [1] RD. Richtmyer and K.W. Morton, Difference Methods for Initial Value Problems (Interscience—Wiley, New York, 1967). [2] J. Dahiquist and A. Bjorck, Numerical Methods (Prentice Hall, Englewood Cliffs, New Jersey, 1974). [3] J. Chang, Lawrence Livermore Laboratory Report UCID15992 (1972). [4] J. Chang and G. Cooper, J. Comput. Phys. 6 (1970) 1. [5] M. Hughes and D. Post, J. Comput. Phys. 28 (1978) 43. [6] C.G. Lister, D.E. Post and R. Goldston, Third Symp. on Plasma Heating in Toroidal Devices, Varenna, Italy (1976). [7] J.P. Christiansen and Ky. Roberts, Comput. Phys. Cornmun. 7 (1974) 237, 245.
CONTENTS 3. Introduction 2. Diffusion equations 2.1. Diagonal anomalous diffusivities 2.2. Off-diagonal anomalous coefficients 2.3. Ripple diffusivities 2.4. Neoclassical transport 2.4.1. Simplified neoclassical diffusivities 2.4.2. Nearly exact neoclassical particle fluxes 2.5. Summed diffusivities and limits on diffusivities 2.5.1. Summation 25.2. Linear interpolation for scrape-off limits 2.6. Final diffusivities 2.7. Ware pinches 2.8. Resistivity 2.9. Sources 2.9.1. Sources due to neutral hydrogen isotopes 2.9.2. Neutral beam injection and fast-ion thermalization
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2.9.2.1. Neutral beam injection 2.9.2.2. Fast-ion thermalization 2.9.3. Thermonuclear fusion power and fusion-product heating 2.9.3.1. D—T fusion 2.9.3.2. Catalyzed D—D fusion 2.9.4. Radiative losses 2.9.5. Scrape-off losses 2.9.6. Lower hybrid and arbitrarily specified auxiliary heating 2.9.7. Electron cyclotron resonance heating (ECRH) 2.9.8. Ohmic heating 2.9.9. Collisional energy interchange 2.9.10. Cold helium source 2.9.11. Pellet fueling 2.9.12. Recombination 2.9.13. Neutral impurity influxes 2.10. Minimum densities and temperatures 2.11. Compression
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3. Boundary conditions and density control 3.1. Boundary conditions 3.2. Density control 4. Initial conditions 5. Solutions of the diffusion equations 5.1. Space and time mesh 5.1.1. Space grid 5.1.2. Time grid 5.1.3. Variables used in code 5.2. Differencing 0CN 5.2.1. Crank—Nicholson parameter 5.2.2. Poloidal field equation 5.2.3. Particle and energy diffusion equations 5.2.4. Predictor—corrector and extrapolation 5.2.5. Numerical form for sources 5.2.6. Convection 5.3. Compression 6. Time-step control
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7. Structure of the code and test run 7.1. Structure of the FORTRAN program 7.2. COMMON variables 7.3. Atomic data file 7.4. Ripple amplitude input array 7.5. Plotting 7.6. NAMELIST input description and test case sample run 8. Code version nomenclature Appendices A. Definition of matrices B. Formulae C. COMMON blocks D. Input variables E. Subroutines F. Sample run G. Units
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LONG WRITE-UP 1. Introduction BALDUR is a one-dimensional (1-d) transport code designed to simulate a wide variety of plasma conditions in tokamaks. It solves equations similar to those outlined by Düchs, Post and Rutherford [8], but includes many additional physics effects and more flexibility in programming. BALDUR concentrates on a detailed treatment of neutral hydrogen transport, auxiliary heating, multispecies effects (including an extensive atomic physics package), orbits of alpha particles, plasma compression, ripple transport and edge processes such as “scrape-off’ losses. BALDUR is therefore complimentary to other similar transport codes which have more extensive treatments of other effects such as magnetic topology, MHD fluctuations, off-diagonal terms in the transport coefficients (BALDUR has a limited number of off-diagonal terms), etc. [12—16,45—49]. The BALDUR model has been used to aid interpretation of experimental data in smaller tokamaks and for evaluating designs of larger fusion devices. In this paper, we first outline the equations solved in BALDUR, then describe the numerical methods used, the structure of the program, the form of the input and output, and the results of a standard test case. The present work is limited to an overview of the code and its structure and a complete list of the input variables. More detailed information about the details of the code and its implementation on the US Magnetic Fusion Energy Computer Center are available from the authors of the present paper [17].
2. Diffusion equations This is the first of three sections of this paper which give a mathematical formulation of the physical problem solved in BALDUR. The formulae presented give a description of the physics used, except for corrections which are included merely to avoid singularities in the innermost computational zones. For clarity, a backward-pointing arrow in the form (4— eq(s). *) following a symbol or formula denotes that the equation(s) indicated by “*“ define that symbol or formula. We use Gaussian units except where indicated by appropriate subscripts. In the actual computations, inputs are given in the most commonly used engineering and scientific units and converted to a computationally convenient set of internal units (cf. appendix G).
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The diffusion equations solved in BALDUR are —--~-(rI~)+S
~
0,
8E
1
aB0
~2
a
a=1,2,te,h,
(2a)
j=i,e,
(2b)
a ~ a(rB9) ~
a~
a (2c)
~C~(TlJbeam),
where ~a is the number density of species a, E1 is the energy density of thermal ions or of electrons, and B~is the magnetic field along the (poloidal) direction encircling the plasma column. T~[*— eq. (2h)], q1 [4— eq. (2h)], S~[÷— eqs. (2.9a—b)] and Q~[*— eqs. 2.9c—d)] are the fluxes and sources defined below. i~ [— eq. (2.8c)] is the parallel resistivity, and ~beam [*— eq. 2.9.2.2e)] is the net current density driven by circulating fast ions. Consistent with the neglect of finite pressure effects on the magnetic geometry, the bootstrap current is also neglected. A version of the code which includes these effects has been developed by Bateman [15]. Theimpurities subscriptswith 1 and 2 refer to hydrogen andnotation the subscripts t° and h refer to 3He, 4He or heavier a given average atomic isotopes mass. The é,h refers to any of the four impurity species allowed in BALDUR version BALDP47M-PO9A. (In some places, we replace the subscripts I or 2 by D or T to indicate specific hydrogen isotopes.) The electron density is = ~B +
faKZ)aFn~n,
~
(2d)
a=1,21,h
where n 8 [~- eq. (2.9.2.2j)] is the density of fast hydrogen isotope ions due to neutral beams and (D.995) is a user supplied factor used to simulate hydrogen dilution. The contribution to ~ and Ee of electrons associated with fast a-particle is typically small and is ignored.2)~,J. WeThe use average the symbol KZ)O for ( )~ is the the meanover charge of species anda given denotespecies the mean by flux (Z surface. KZ)a and KZ2) average all charge statesa of on asquare given charge magnetic 0 are determined as follows: (~—eq.
KZ) a
2\c~r0~
~
a=
\Za~ /0 Z~, /Z\c0~~
‘
\/Z
Ia
‘
N = 2 ATOMC NATOMC=l,
(2d’)
where NATOMC (.— eq. D. 1023) is an input variable. K Z )coronal and K Z2 )coronal are the average values in coronal equilibrium, which are functions of electron temperature determined as described elsewhere [17]. This model is designed primarily for fully-ionized or high-Z impurities and should be supplemented by coronal nonequilibrium calculations when radiation from carbon or oxygen is significant. A computational package which does this is under development by Wunderlich and Lackner [161. The total thermal ion density (2e) a
defines a locally common thermal ion temperature
T~= (2E 1)/(3n1).
(2f)
The local electron temperature is T~=(2Ee)/(3fle),
where
tie
~
eq. (2d)] is defined above.
(2g)
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Table 1 Symbols not defined in text Symbol defined by Dependent variables in diffusion equations n,,, a =1,2 [~- eq. (2a)] n~,a — e’,h [4— eq. (2a)] E~,j e,i [~-eq. (2b)] B 0 [~- eq. (2c)] Independent variables r
3) density of hydrogen isotope a (cm density of impurity species a (cm3) energy density of electrons or thermal ions (erg/cm3) poloidal magnetic field (G)
minor radius (cm) eq. D.2276) time (s)
t
(‘—
Plasma geometry r,or
(‘— (4--
R
meaning (units)
(4—
C120, appendix D) inner boundary of plasma scrapeoff (cm) eqs. D.2047—D.2066) outer boundary of plasma (cm) eqs. D.2027—D.2046) major radius of magnetic axis (cm)
Temperatures and electron density [(.— eq. (20]
temperature of thermal ions (erg) electron temperature 3) (erg) [(b-eq. (2d)] electron density (cm
T
[(~- eq. (2g)]
Constants e
eq. eq. (~— eq. (4— eq. (+- eq. (i—
c
(f—
m,,
Standard tokamak variables Bz q = rBz/RB 0 C = r/R
(-
B.12) B.8) B.19) B.21) B.23)
electron or proton charge (statcoul) speed of light (cm/s) electron mass (g) proton mass (g) 3.1415926536
eq. D.1461) toroidal magnetic field at r = 0. (G)
(~eq. B.62) (~—eq. B.76)
safety factor inverse aspect ratio
Table 2 Mesh variables N 2 N+1 N+2
(,~ ~ (,~ + ~7)/r~ —
0.5[(r]~i)2
(rI~)2]/r2
2r~~/[(r)~.i)2 _(rb)2] r~~1 R n+1 t~±
(‘~‘),,÷~ = t,,~1— t,, —
e
r,,..~
NZONES LCENTER LEDGE MZONES XZONI(j) XBOUNI(j) DXBOUI(j) DXZONI(j) DX2I(j) DX2INV(j) RMINS (centimeters) RMINI (internal units) RMAJS (centimeters) RMAJI (internal units) NSTEP TAI (internal units) TBI (internal units) DTI (internal units) DTOLDI (internal units) ThETA=THETAI
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We now describe the fluxes T~and q~,and the sources Sa and in eqs. (2a) and (2b), and the terms in the magnetic diffusion eq. (2c). BALDUR draws from an extensive literature on the derivation of 1-d transport processes in tokamak plasmas. We therefore restrict ourselves to a survey of the formulae used and the appropriate references. The particle and energy fluxes for the most general case treated BALDUR are =
F~’°~ + F~ +
F~°’ — n
1t,1,
F~=F~”~ + F~+ F,~ —
—
2~ + F~°’ — flhVh,
Fh
~
qe=q~+~+O_q~,
r2 = F~’~+ F~+ F~ol =
(2h)
F~ + F~
q~=q~lex+qz+q~oI_q~f,
where
an D
Fttex
11
D12
D11
Dlh
0
0
D2~
D22
D21
D2h
0
0
D11
D12
D11
Deh
0
0
an2 F7X
(2i)
Dhl
Dh2
qfIex
Dei1~
De2T~ De~KZ)i1~ DehKZ)hl~
~eXe
0 0
aTe
qflex
D11T
D12T1
0
n~1
-~
Dhe
0
Dhh
D11T1
DIhTI
and the fluxes dependent on gradients of mean charge and fast ion density are: 2~
n1D1
pZ
rz =
q~ Z
n
1D?~
0
r~z
0
tl2L~~2~
ti2L.2h
n~D~
n~D~
0
nhDM
nhDl~Jl
0
ar _____
.
(2i’)
ar
fl~I~De~nh1~D~ XB~ pr~Z TnZ 0
q1 ~I~i~~I( ‘ih 1~-’Ih The particular form chosen for eqs. (2h) and (2i) is the result of the historical evolution of the code as progressively more extensive transport models were included. The models include a flexible combination of anomalous and neoclassical transport to be described next, contributions due to gradients in
+~
j =
$=1,2,e,h,i; except
(2j)
ts=$=i,
e,i
(2k)
of terms which can include anomalous diffusivities D~°m[— eq. (2.la)] and ~ eq. (2.lb)], thermal ion energy diffusivity x~’~ [~— eq. (2.3a)] due to toroidal magnetic field ripple (cf. section 2.3), and an [~—
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approximate treatment of neoclassical diffusivities D~°[4— eqs. (2.4.la—d)], and x7°[~— eqs. (2.4.le—f), and some of (C1,..., C5®)]. The elements of an input array are described in appendix D. This flexible approach to transport is useful in view of the current lack of an experimentally verified “first principles” transport model. We shall now describe each contribution to these diffusivities and then show the precise formulae [~— eqs. (2.5.la—c)] we use to prescribe the combinations, eqs. (2.6a—d). 2.1. Diagonal anomalous diffusivities The contribution from anomalous transport (i.e., transport due to plasma turbulence) to the diagonal elements of the matrix of transport coefficient in eq. (2i) is given by m =D~ + D~mfh0mp10c~~+D~, ~ = 1,2,~,h, (2.la) .D:o =
X~ +
+
j
~
=
i,e.
(2.lb)
Here we have separated anomalous transport into empirical models used to fit experiment, models motivated by simple theories of turbulence and semiempirical transport models. Minor modifications to the theories used by Düchs, Post and Rutherford [8], and a variety of ad hoc prescriptions to fit various experiments are described in appendix D. The use of ad hoc empirical prescriptions has largely been superseded by applications documented elsewhere [9,18,19] using the above semiempirical formulae, so the older empirical formulae are not described in detail here. The semiempirical diagonal diffusivities are =
a
~
,,
=
1,2,i,h,
(2.lc)
k
x7m~’~= ~x*flx~.s n
(2.ld)
j=i,e.
k
The scaling exponents eDflk and efflk are specified by NAMELIST input arrays (i— eqs. D.2537—2904). In the present version of BALDUR, the allowed dimensionless scaling variables are X~,,k= KZ)a, A~,K, B/Be, r/r~,~
ne/n, p/p,
A~,Zett, I Xne I,
A~, X
IXh~,(T
,
I
T~/T~,
q/q~,IA~H,1~~
1/7~); a=1,2,t~’,h,i,e,
(2.le)
for k = 1,. ,20, respectively. The exponent g = v’ (1 + 1/C295). Here Aa is the atomic mass of species a [for a = 1,2,ë~h;we set A, =Ae = =e = 1 to suppress nonphysical contributions to eqs. (2.lc) and (2.ld), p=n~7+njTj, AI=Eal2ehnaAa/nj, e=r/R, K (~eq. D.237) is the constant ellipticity specified by NAMELIST input, and B~,r4, e,~,= r~/R~,n*, p,~,,T~and q~,are constants (4— eqs. D.2531—2915). The so-called profile factors are defined as follows: . .
i)tnei =r/Lne,
IX~i=r/L~, IA~eI=C293+(t~/Lre), ~
(2.lf)
where L~ =
Xa/I
axe/ar I
(2.lg)
is the scale height of variable Xa. Binomial averaging over five radial zones avoids numerical problems with these profile factors. The profile factor f,. (e.g., used to adjust edge transport) is given by (1, =
1 + (C~ — 1)[(r/a) ~C~,
r/aC291, —
C291]/(C292
—
C291),
C29~< r/a C292, r/a> C292.
(2.lg’)
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The circular cylinder approximation used to determine the safety factor neglects a significant singularity in the shear for tokamaks with a separatrix, so we add a correction to obtain for the shear
IXqi
QMHDI
i q—
q ar
i+
QMHD2
+c296,
(2.lh)
where q1~~ = q(r~~) is the “cylindrical q” as calculated in BALDUR at the separatrix. The diffusivities D,7a (4—eqs. (D.2881—2904), x~(*— eqs. D.2904—2912) and QMHD1 and QMHD2 (4—eqs. D.2913—2914) are given by NAMELIST input. To allow an option to keep semiempirical formulae from exceeding Bohm diffusion especially in the plasma periphery, D:~m1emP1~~~t and are limited to a maximum of C51DB (4— 2.lj) when C51 >0, where D’~= cTe/l6eBz.
(2.li)
The factors C2~—C296are typically used to prevent singularities (e.g., C293, C294, C295), or to omit terms such as the anomalous pinch in the scrapeoff (e.g., C2~—C292). 2.2. Off-diagonal anomalous coefficients The off-diagonal anomalous coefficients in eq. (2i) include only so-called thermal convection processes [7]:
D anom = C~D ea
pv
31
D anom = C’ D ~V
a=l,2,e,h,
(2.2a)
00
where ~ and ~ (4— eqs. D.2529—2530) are input constants. In particular, energy transport due to density gradients in the presence of trapped ion or electron modes [8] is ignored. 2.3. Ripple diffusivities Toroidal field ripple affects only the ion thermal diffusivity in BALDUR:
:~i:’
~
~
(2.3a)
where the Ripple-Trapping, Ripple-Plateau and Banana-Drift contributions are 2T~ [Gn(8fjX~IOO,C, =
32.880[(cTj)/(eRBz)]
=
(c1~5(~~)9 + Cl~(6JanabIe)8)DR,
X~’ (C~(~ec~)8+~
4l~~/2)3/2
+ GZ(~va~abIeI9=O,C,~=~/2)3/2],
(2.3b) (2.3c) (2.3d)
where DR ~3(~r/2) v1p91]/R, (2.3e) where iY, and ~ are the ion thermal velocity and poloidal gyroradius. In addition to the coefficients C141 and C1~,there is an input switch C1~to allow a choice of various combinations of X~T, x~and X~D. A proper choice of ripple transport model is particularly important for simulation where the form of the banana-drift coefficient X?D given here may be inappropriate [20]. A variety of methods are available for
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specifying the ripple amplitude, 6 [17]. The G factors in eq. (2.3b) are computed either as described by Goldston and Tower [21], or by Uckan et al. [22] depending on the input switch C149.
2.4. Neoclassical transport Two options are available for computing neoclassical particle transport coefficients. Most published applications of BALDUR use a treatment which follows a summary due to Rutherford [23] (similar to that of Hinton and Hazeltine [24] but with a more complete treatment of impurity transport). There are also three options for computing neoclassical ion thermal diffusivities. Should a more complete and exact treatment of neoclassical particle diffusion be desired, there is an option to include a nearly exact neoclassical treatment of particle fluxes [25,26] which is appropriate in BALDUR for cases with fully-stripped ions. 2.4.1. Simplified neoclassical diffusivities The diagonal neoclassical diffusivities for particle transport in eq. (2i) are: 2XeA~1(Te+ T,)/(7~3~/2B~)], a = 1,2,e,h, = [D~a + CHneq and
D,~ 0=
( (
~
D~n~~)+(~
b=1,2
~
D~n,,) +
b=1,2
(
D~~aKZ2)bnb),
b=C,h ~
(2.4.la)
(2.4.lb)
2) D~
0n~,
a
=
where DI~b= D~b=
/(CHIq2AjA~,/2)/(B~rI1/2),a = é,h; b (0,
f(C1Ajq2,L~2)/(B~r~1/2), a,b=i,h;
~
=
1,2;
C110 C110>0,
C1100,
(2.4.lc)
(2.4.ld)
(0, C110>0. Primarily to allow comparisons with codes that assume fixed isotope ratios, we have included an option which scales like neoclassical coefficients and tends to force similar profiles for both hydrogen isotopes: D~= fC356(1 ~ (0,
C1100, C110>0.
The remaining diffusivities, D~’a[4— eq. (2.4.lj)] and The diagonal thermal diffusivities are
{
X~
=
CEqSA n1~2[i B~T~
~
=
X~°” + X~°~”,
+
0.1(1.6 ++ 2Zeff)pe*] 0.73(1.6 2Z ) ~3/2
DIOb
[4—
+
1.13
(2.4.ld’)
(eq. (2.4.1k)], are defined below. + O.5Ze~~ +
0.59 + ZCff 0.55Z
(2.4.le) (2.4.lf)
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x~’ C~81=0, C12
X?OI1~I=
~
C281
=
1,
~
C281
=
2,
B~T~ (1+o.74e3/2v~)[1+1.o3(v~)1/2+o.31v~I 3”2v~+ [i + l.03(v~)1”2 + 0.31v~](1.76645e3v~)}, (24 if’) (O.661/2C~I) (1 + 0.74 where x~’ (~— C 281) and X~H are the ion thermal conductivities given by Bolton and Ware [27], and Chang and Hinton [28]modified for Zeti # 1 as described in the appendix of ref. [18]. Here HH =
for for
qs={~2
1
qI,
(2.4.lg)
q<1,
1”2(ec)2/3, C~ 1”2/e, AH = ~a1,2Aafa/~a=l.2fla, C CE and CH = 8(2rrm0) 1= C(2AHmP) 1 = C111 = 1”2, A~ = [Z~~ 2.67+ Zeff)]/[3.4(l.l3 + Zeit)], and P~ab= (A CH(mC/mP) 1( 0A b)/(Aa + A,,). Zett is the usual effective charge, A. and A0 the Coulomb logarithms, and p0” and v j!~ the electron and mean hydrogen collisionality. x?°~ is defined below. The off-diagonal diffusivities in this simplified neoclassical treatment are: particle density-gradient interactions involving at least one impurity, anomalous hydrogen mixing which scales like neoclassical transport, the pinch or screening of particle fluxes due to the ion temperature gradient, and thermal convection. There are also neoclassical contributions due to gradients in mean charge. The off-diagonal coefficients affecting particle fluxes due to density gradients are ~D~’aKZ)bfla, a = 1,2; —D~j~’(Z~0n0,a=i,h;
D~° ab =
D,~°=D,~°=
bb=l,2, = ~,h,
)aK)~a’
~DHana,
D~=
a, b
~
(2.4.Ih)
(a=t~’,b=h)
and
(a=h, b=e’),
(a=1, b=2)
and
(a=2, b=1),
=
(2.4.li)
i~,h.
(2.4.1k)
Particle transport due to an electron temperature gradient is not included. Since the particle transport is dominated by anomalous processes in most applications, this is thought not to be a significant omission. The particle transport coefficients due to an ion temperature gradient has only neoclassical contributions. Thus,
a!
D
ai
=
D”
0°=
—
~ nbDH0(KZ)h+KZ~b/2), ~b=t’,h nbD(1 + KZ)a/2) + ~ nbDia(KZ)b
—
a=1,2,
KZ) 0KZ),,),
a
(2.4.i()
=
b=1,2 The particle transport coefficients due to a gradient in mean impurity charge are D,~= —D~’0n,,, a=1,2; b=e,h,
(2.4.lm)
D,~=—D~KZ)ana, a,b=i!’,h.
(2.4.ln)
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285
The so-called thermal convection coefficients are defined below in section 2.6. To define these for the ions, we need the following definitions: D~°=1.5~ =
(D~~a
1.51
(D~’a”— D~b
D~=~1.5[
a=1,2,
(2.4.1o~
(D~
=
b=t’,h
~
D~4’bnb+
b=1,2
~
D~
(2.4.lq)
b=t,h
The part of x~° which depends on particle diffusion coefficients is Cony = DT!e~ X~ pv ai na, (‘~
(~)4 .-~.
1
r
a= 1,2,t,h
where ~
(~—
eq. D.2530) is an input constant.
2.4.2. Nearly exact neoclassical particle fluxes Here we outline the option which calculates all neoclassical contributions to particle diffusion in a thermal plasma with up to four fully-stripped ion species [25,26]. The only approximation made for this case is an interpolation between the banana and plateau regimes. For partially stripped impurities, we make the additional approximation that each isotope can be represented by a single species with charge
~~‘=
a
CI12I?”+C1III~
for C1100) C110>0\ for
a=1,2 t~’h
(2.4.2a)
the classical, Banana-Plateau and Pfirsch—Schlllter contributions. The present version of BALDUR assumes the “strong temperatures equilibration” approximation [25] in applying eq. (2.4.2a); the validity of this approximation for a particular simulation should be checked before using this model. 2.5. Summed diffusivities and limits on diffusivities 2.5.1. Summation First, the terms in the diffusivity matrix of eq. (2i) are summed. The diagonal terms in the neoclassical contribution can also be multiplied by constant factors. Thus, m+ D~°, a, fi = 1,2,e,h,e,i, (2.5.la) = .D:;° = X~m+ Cjx:00, (2.5.lb) =
~anorn +
X~~0
+
(2.5.lc)
~
where C 9,
Ca,Cj= C10, C11, C12,
=
=
a=1,2, a=I,h, j—e,
(2.5.ld)
j=i,
D:r[1 + (Ca — C-5~°, j
=
i,e.
1)6ap],
a$ = 1,2,t’,h,
(2.5.le) (2.5.lf)
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The Kronecker delta symbol 6a~ is used to compact the notation, and we have already defined the diffusivities D~°m[4— eqs. (2.la), (2.2a)], Da7’ [~— eqs. (2.4.la), (2.4.lh), (2.4.li), (2.4.U’), (2.4.10) and (2.4.lp)], x~°m [4— eq. (2.lb)], X~PPk[4— eq. (2.3a)], X~° [4— eq. (2.4.le)] and x~° [+— eq (2.4.lf)]. .
2.5.2. Linear interpolation to scrape-off limits Next, a constant empirical limit can be set on the diagonal particle and electron energy diffusivities in the scrape-off region r> r~~ 1 1a, a=l,2, DaaC53+D0
C
1~aa=
54
a = (,h, r> ~
+ D,~a,
C120> 0, (2.5.2a)
~eC55,
where D~0is defined by eq. (2.4.lb). Using this model sets similar limits on the respective contributions of D11 and Daa of eq. (2.2a) to the electron and ion thermal convection. Continuity in Daa and Xe i5 maintained by linear interpolation to the values (computed as above) at a specified fitting point = C5,, rscr:
=D:~mI~C
1~aa
56~ + [(r
(C~= C53 Xe
for
a
C56r1~1)/(r1~. — Cs,,r~r)] (c~— D,~mI0~550
1,2; C~= C54 for a
=
X~~~mir=Cs6r, +
—
[(r
—
C56rsCr)/(rscr —
=
e,h); a
=
~s,,rscr)I (q~~m
i,2,e’,h, —
)
+
C56r~~~ < r <
‘scr’
(2.5.2b)
~eir=cs6r,~,) + D~a, (2.5.2c)
It is also possible to override other computations of diagonal scrape-off diffusivities by specifying C83 giving D00 = CS3DB + 8
Xi
=
D~0 for r> ‘scr’ a = i,2,e,h, C83 for r> rscr, j = e,i.
>
0, C120> 0,
>
0,
(2.5.2d) (2.5.2e)
C811.5D
Thus for r < C mand ~e = X”m~These limits can be helpful when the user has an a we have D,,~ = D,~ priori empirical56i0~, estimate of the magnitude of the diffusivities in the plasma periphery. 2.6. Final diffusivities To summarize, the diffusivities are sums which can include anomalous, ripple and neoclassical contributions. In addition to the scrape-off limits on Daa and Xe given in eqs. (2.5.2b)—(2.5.2e), a maximum can be placed on the diagonal diffusivities. Stated symbolically, the final diffusivities are Daa
=
min(C
a
43,D00),
=
l,2,é’,h; r
< ‘scr’
(a=i,2; T>TScr~ C830 ~a=l,h; ~ C830 m+D~°~ a#$; a,/3= i,2,t’°,h,e,i, —
Daa=flMn(C43,Daa),
and C530, and C540,
(2.6a) (2.6b)
Da~=D~°
1.5D
11, Dea=
i.5[DIl+(1_l.5(rfl/rT)DT1},
—
i.5D11,
—
1.5[D11
+
(i
—
1.5(~/r~))DTuj,
a=1,2,i,h, C910, a=1,2,e,h, C91>0, a a
= =
l,h, C91 0, i,2,~,h, c91 >0,
(2.6b)
(2.6.b
)
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(C520, ~C >0
287
(2.6c)
52—
r< Xemin(C
{r>r~r,Cg3O
44,Xe), XB
where
=
DTI
Xe
and
C550,
1.5D11,
—
(2.6d) (2.6d’)
is a transport coefficient based on the theory of trapped ion instabilities [7,33].
2.7. Warepinches The total pinch flux for particles of species a is I
~
+ navware, H .
I a nv=c . .. a a ~psem,emplneal
a a
=
=
1 ,2 , I,h,
where rem1emp!~aI =
na~ ~ ,,
fl
(2.7b)
~
k
where v1~’ (4— eqs. D.2857—2880) and eVflk (4— eqs. D.2537—2616) are input constants, and Xak is the dimensionless scaling variables defined in section 2.1. The inward energy fluxes accompanying these particle pinches are qve
=
C~’VTe
~
na
(2.7c)
a=1,2,e,h
q”
=
nava + qware
CJ~~TI ~
(2.7d)
a1,2,t,h
where C~(i— eq. D.2529) and ~ (i— eq. D. 2530) are input constants. The values of the Ware pinch velocities in eq. (2.7a) which describe the flow of hydrogen isotopes due to the presence of a toroidal electric field E,, are given by a formula [29] very similar to that of Hirshman and Sigmar [10], U~iare=
C19L13(Zeti, vC*,)(cl,hJ/B4).
(2.7e)
Here L13 (4—eq. B.115) and L23 (4—eq. B.116) are dimensionless functions with values often of order unity, and flh (~- eq. 2.8a) and J (4—eq. B.63) are defined elsewhere. The analogous impurity pinch is neglected [10,29]. The Ware electron energy pinch is of the following form: ~
~ Ware
T ~
nav~are
a=1,2 flaV~jare
if
C3650,
if
C365 > 0,
a1,2
where V~jare =
C365C19L23 (Z~, ~ )( c’qhj,/B0).
(2.7f)
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The Ware ion energy pinch is i~’i ~r L~pylj
n0vHWare
qware=
0
~ç 1
‘-365 —
if
C365>0.
If the beam-driven current in included as described below, then E. i~Jand the present treatment of the Ware pinch is incorrect. A more complete treatment of the neoclassical effect of neutral injection on particle transport [31,32] has not been included. 2.8. Resistivity The plasma resistivity used to calculate the ohmic heating
[4—
eq. (2.9.8a)] is [17,25]
(2.8a)
C,j/(neTeAeF),
flh
where F—
C,~ fTrap
—
if if
C,~1>0, C~1 0,
(28b)
(see appendix B for C~(4— eq. B.46), i-~ (+—eq. B.73), AE (*—eq. B.65) and fTrap (4—eq. B.81)). The resistivity used in the poloidal magnetic field diffusion equation (2.c) is =
C399~,
(2.8c)
unless: (1) C478 is used to limit the current density in the central region when q( r) < 1 (this simple ‘sawtooth model’ is intended to be used as a substitute for — and not in conjunction with — the more elaborate sawtooth model described below), or (2) a scrape-off region is included in the simulation. (Note that in these cases the equilibrium toroidal electric field will not be constant as a function of minor radius and the B~01conservation check may not be as small as usual.) These simplified models should be avoided in cases where the associated poloidal field energy is significant. If C478> 0, the resistivity is modified to limit the current density inside the q = 1 radius: = C3~i1/(i
+ 6[i
(r/rl)21
—
where r1 is the outermost q
=
},
r
1 radius, ~
< =
r1, ~h(r1) and 6
(2.8d) =
min(2, C478).
If a scrape-off region is included in the simulation the resistivity is modified in the scrape-off region [19] r>r1~~,
(2.8e)
where = S
31Jh, ih1lim~ (10
i~ti i~i ~max, <~s,max,
(2.8f)
where ~1im =
(7~/LeJ)ln(A),
[3, A=\(~,max+~)/(J~,max_~),
=
(eL/4)(ne
—
(2.8g) Jt<0, ~
n~)(1/’rii+ 1/~~).
J~[4— eq. (2.9.8b)] is the toroidal current density, and L, ~ and ~ are defined in section 2.9.5.
(2.8h) (2.8i)
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An option is available to flatten temperature and density profiles periodically inside a sawtooth mixing region defined by helical flux conservation (.i— C~2—C~9).
2.9. Sources The sources or sinks in the diffusion eqs. (2a—b) are computed with a variety of algorithms. These algorithms (denoted by a mnemonic superscript) include neutrals (N), neutral beam injection (B), fusion (F), radiative losses (RAD), scrape-off losses (S), lower hybrid heating (LH), electron cyclotron resonance heating (EC), ohmic heating (OH), collisional ion—electron energy interchange (IE), a volume source of cold helium from the wall or limiter (HE), recombination (REC) and neutral impurity influxes (IMP). In particular, Sa=S~+S+SaF+S+Sa~, Sa Qe
=
S+ S~+ SaHE+
=
Q~+ Q~+
S~MP,
a
a=1,2, =
(2.9a)
i,h,
(2.9b)
Q~+ Q~ + Q~+ Q~H+ Q~+ Q~+ ~
+
Q~+ Q~,
(2.9c)
~
(2.9d)
For each source we now give a brief description of the nature of the model (including any references to more complete descriptions), formulae for the contributions of each model to the sources in the diffusion equations, and a list of the options available for implementation of the model. 2.9.1. Sources due to neutral hydrogen isotopes The neutral gas source terms are computed using a Monte Carlo algorithm [5]. The calculation involves computing the trajectories of a specified number of sample particles (.— eq. D.1010), and then using these trajectories to calculate the source rates to be used by the energy and particle balance equations. Up to three types of neutral sources are computed for each hydrogen species: edge sources due to the recycling of plasma ions and neutrals, edge sources due to gas puffing and volume sources due to charge exchange between injected beam neutrals and plasma ions, and due to radiative recombination of plasma ions and electrons. Thus a total of up to six neutral source types may be included for a run with two hydrogen species. The volume sources due to recombination and the edge sources due to gas puffing may be omitted (4— eq. D.1030, D. 1710—1729 and D.1808—1847, cf. also sections 2.9.12, 3.1 and 3.2). Because the Monte Carlo computations are expensive, they are only done every ten time steps or so (o— eq. D.lOlOa). Separate computations are done for each source type, using a dimensionless neutral source rate of 1.0 for each type. The neutral temperature and density profiles and the ion and energy source terms computed based on these normalized neutral source rates are then linearly rescaled to physical units each time step. The source terms computed are (cf. refs. [17,34])
5:4=
~ Yk{ ~ nenake k=1,N5 a=1,2
+
~
[flb~~a0k<0t~akb>p
+ (1
—
8ab)(fl°akflbCx — ‘2ak<(7t)bka>cx)]}~
b =1,2 =
—
~ k1,N5
Yk{
~ a=1,2
[t1et~k<0t1>e
+
~
h1b1~kPJ
b=~1,2
E~01~}
a
=
1,2,
(2.9.la) (2.9.lb)
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and Q,N =
~
Yk{
k=1,N
~
+
a=1.2
5
~
+
{fle’~kKGu’)e
~
nb~kKa0akh)pj~
b=1.2
flbnak(aVakb)cx(Ea8TI)+(beams)}.
(2.9.ic)
a=1,2 b=1,2
Here Yk is the scaling factor for neutral source type k [17]. N8 is the number of source types, n~8is the unscaled density of species a neutrals originating from neutral source type k, and 6~,, is the Kronecker delta. ~~ v )e is the rate coefficient for electron impact ionization, and K ev0 kb )~,and K “ Va kb )~are the rate coefficients for proton impact ionization and charge exchange, respectively, of species a neutrals originating from source type k interacting with particles of species b. Proton ionization may be omitted via an input switch (~eq. D.2528). The input parameter E~00~1(4—eq. D.1804) representing the average energy lost to the plasma per ionization is typically set to 40 eV. E,~8 is the temperature of species a neutral originating from source type k. E0°is the local temperature of species a neutrals, computed as Ea~
~ k=1.N~
yknakEak/
/
~
‘fk’~ak~
(2.9.ld)
k=1,N5
The density of neutral species a is (2.9.le) k
1, N5
We should note that the Monte Carlo algorithm in the present version of BALDUR includes the approximation that neutrals are launched after every charge-exchange event from a Maxwellian distribution characterized by the local thermal ion temperature T1. Under some circumstances, this can cause noticeable error in the ion energy balance [35], a fact which should be kept in mind when interpreting the results from simulations where charge exchange is an important energy transfer mechanism. The total neutral hydrogen density (2.9.lf)
is also used in equations for slowing down of fast ions and in computing losses due to parallel flow in the scrape-off region, as described below. 2.9.2. Neutral beam injection and fast-ion thermalization The neutral beam heating model calculates the neutral beam deposition and the thermalization of the beam-injected fast ions. Monte Carlo methods are used to model the penetration of a finite-size beam of neutral hydrogen atoms (H, D or T) into a torus with concentric elliptical magnetic surfaces. When these neutral atoms lose an electron in a collision with a background plasma particle, they become part of the suprathermal fast-ion population in the plasma and an electron is added to the plasma electron population [4— eq. (2d)]. Coulomb collisions with background plasma ions slow the fast ions and heat the plasma. When fast ions have slowed to the average energy of the thermal plasma ions, they appear as a source of plasma ions in the ion particle diffusion equation. Coulomb collisions also change the pitch angle of the fast ion; i.e., v11/v is changed. Charge-exchange reactions between fast ions and the thermal neutral gas which is present remove particles from the fast ion population. The beam—target fusion reaction rate [43] between fast ions and background D and T ions are computed. Background ions are assumed to be thermal with zero average velocity and beam—beam reactions are neglected. Velocity space effects of major radius compression of the plasma are modeled by relocating the bins in v11/v space and redistributing the
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291
fast-ions among new energy groups in a manner which conserves the magnetic moments and canonical angular momenta of the fast ions. 2.9.2.1. Neutral beam injection. The Monte Carlo code FREYA is used to model neutral beam penetration in a torus [6]. The results are: (1) a radial array of the volume source-rate of thermal neutrals S,~BCX, created by charge-exchange reactions between the incoming neutral beam atoms and the thermal plasma ions; and (2) the source of fast ions (given as a function of radius, ~s= v11/v, and energy) created by ionization of the incoming neutral beam atoms. The Monte Carlo sample neutral-beam particles are initialized with an energy of W,, (4— eqs. D.1159—1168), W,,/2 or W0/3 at a randomly chosen location on the rectangular (or circular) face of the neutral beam’s ion source The sample particle’s initial velocity includes components for focussing in the horizontal and vertical directions and randomly chosen components which model Gaussian beam divergence in both the vertical and horizontal directions. The neutral beam injection is then oriented in the tokamak coordinate system by a series of two translations and four rotations. The sample particles are then tracked in a straight line until they: (1) do not pass through the beam port at the vacuum vessel (in this case the particle is initialized again and there is no power loss), (2) are ionized in the plasma at a location determined by Monte Carlo techniques, or (3) strike the interior of the vacuum vessel after passing through the plasma region without being ionized. If the sample particle is ionized by a charge-exchange collision with a plasma ion it contributes to the volume neutral-atom source array which is used by the neutral gas transport code. When any kind of ionization of the sample particle takes place, the value of p. = v1/v is calculated and the source array for the fast-ion distribution is incremented. 2.9.2.2. Fast-ion thermalization. The fast ion population is represented by a nonthermal distribution which is a function of radial zone, energy E, and the cosine of pitch angle, ~z= v11/v. The equation for the time evolution of this distribution is a version of the Fokker—Planck equation which includes the effects of average drag and pitch-angle diffusion caused by collisions with the background plasma, and the loss of fast ions caused by D—T fusion reactions with the background plasma ions and by charge-exchange reactions with the neutral gas. The output of the fast-ion model is calculated by integrating over the distribution function to obtain the sinks and sources to both energy and particles which are used by other parts of BALDUR. The equation for the time evolution on the fast ion distribution, f,,~(p.,E, r) (i— eq. B.147) is
—r~~-
a
=S:4BI_(r~X+VDT)faB+
~
a
ala 2)~~_ , (1~~.p.
where the average-drag energy loss rate is —W=2H 5(v~E+y~1/f~).
(2.9.2.2b)
The source term SBI (f— eq. B.125) is calculated by the neutral beam injection code (see section 2.9.2.1). ~ (~- eq. B.163) is the charge-exchange loss rate, and ~DT (f— eq. B.167) is the fusion reaction rate, v~ and ~ (4— eqs. B.160 and B.161) are the energy loss rate coefficients for collisions with plasma electrons and ions, respectively: y5~(4— eq. B. 159) is the pitch angle scattering rate coefficient. The charge-exchange loss rate, ~ includes the “fudge reactor” H1 (4— eq. D.1229) to correct for the neglect of reionization of the escaping neutral fast ions. The slowing down rate, W, includes the factor H5 (4— eq. D.1233). The effects of major radius compression are modeled by redistributing the fast-ion distribution in E and p. space as dictated by conservation of magnetic moment and canonical angular momentum. Under the conditions of adiabatic compression of the major radius from R1 to R2, 2E the perpendicular energy of a fast ion scales as E~—‘ CE~ and the parallel energy scales —* C 11, where C = R1/R2 is the
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compression ratio. The radial zone of the fast ions is unaffected by compression, the absolute radius of each zone scales with the change in the plasma column minor radius, i.e., r c~a cr C°5. The density of fast ions scales inversely as the zone volume, i.e., cr C2. The beam-driven current, ~beam’ in eq. (2c) includes terms for toroidal effects on ion orbits, pitch-angle scattering of ions into trapped orbits and the neoclassical electron return current. The cylindrical approximation to the circulating fast-ion current, 1~eam (~—eq. B.178) is ~beam
Jf
=
‘f(p., E, r)p.[2E/(A,,m~)~1/2 du,
(2.9.2.2c)
where Emax (4— eq. B.144) and Emin (+— eq. B.145) define the fast ion energy range, and A,, (~—eqs. D.1069—1078) is the atomic mass of the beam injected fast ions. Toroidal corrections can be applied to 1~eam to produce ~i~eam ~r.)’~eam(1
—
H
1(m)),
(2.9.2.2d)
3Zeft~)~TT/(2K
where
1(m)=
K(m),
H
c~,
H
4>0,00, 1m,
K ir/2,
m=2t,/[(1+f)p.~,];
H40;
where (— eq. B.179) is the aspect ratio r/R0, and K(m) (f— eq. B.183) is the complete elliptic integral of the second kind. p.1, = Rtang/(r + R0) is an estimate of the average initial value of v1/v of the injected fast ions where Rtang (+~. B.180) is the tangency radius of the neutral beamline. The user supplied factor H3 (f— eq. D.i231) controls the approximate corrections for pitch-angle scattering of fast ions onto bananatrapped orbits. The user sets H4 (+—eq. D.i232) to include (H4> 0) or exclude (H4 0) a correction for the toroidal variation of v~1. The total beam-driven current, ~beam (*— eq. B.184) includes the electron return current (complete with a trapped-electron correction): ~beam
=
CjJ~eam(1—
(1/Zett)E1
—
((1.55
+ 0.85/Zeft)\/~
—
(0.20 +
1.55/Zett))/(1
+ H2pe*
)1 },
(2.9.2
.2e)
where e = r/R0, C1 is a user prescribed factor (— eq. D.1068), and the user prescribed factor H2 (— eq. D.1230) controls the (typically small) effect of electron collisionality, i.~” (— eq. B.75). Eq. (2.9.2.2e) represents a fit to the results of Start and Cordey [37]; the term H2 i~’~is a generalization of their results to finite Ve*. The net hydrogen particle source of thermalized fast ions, h~th (~— eq. B.150) is given by the rate at which fast ions join the thermal ion population minus the beam—target fusion rate: 4a,th =
6aa’f1 d~i[
—
14/(Emin)1 f~’~(p., E,,,,,,, r)
—
6a?~DT’
(2.9.2.2f)
‘
where a’ is the injected hydrogenic species, and ? = T if the fast ions are deuterium, or ? = D if the fast ions are tritium. The beam—target fusion term is the rate at which beam—target fusion reactions remove thermal hydrogen ions from the background plasma: ~DT
=
j
dEfn~aDT(E)fa(~, E,
r)[2E/(A,,mp)I1~’2 dp.,
(2.9.2.2g)
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293
where 0DT (~—eq. B.164) is the cross section for D—T reactions between a fast ion and cold background ions. The net hydrogen ion particle source-rate in eq. (2.9a), ç’B...
-B
.B
— ~NB
nacx, is the sum of the source of thermalized fast-ions, ~~th [4— eq. (2.9.2.21’)], the source of thermal hydrogen ions due to fast-ion charge-exchange reactions with thermal neutral hydrogen (4— eq. B.169), —
nath + nacx
L,,,~,1_1
=
E,
r) dp.,
(2.9.2.2i)
and the loss of hydrogen ions which charge exchange with injected neutral beam atoms, ~ (see section 2.9.2.1 and B.126). We note that the electron density ne (- eq. 2d) responds instantly to changes in the fast ion density (4—eq. B.175) nB =
J
dEffa(I.L, E, r)
dp..
(2.9.2.2j)
Thus, beam injection may cause a brief decline in T~when cold electrons from the incoming beam neutrals dilute the available electron energy before the beam ions have appreciably heated the electrons. The beam heating of electrons, Q~(4— eq. B.151) in eq. (2.9c), and beam heating of ions, (4— eq. B.152) in eq. (2.9d), are given by integrating each energy loss rate [eq.(2.9.2.2b)] over the fast ion distribution:
Qr
Q~=
=
H5J~”2y~EdEf f~(p.,E, r) dp., HSJ Em~(2’,sj/v~)
dEf f,~(p.,E, r) dp. + fdp. ~
(2.9.2.2k) —
J~’(E,p~n)1 f(p.,
~
r). (2.9.2.21)
When deuterium beams are injected into a plasma containing deuterium, the neutron production rate for the reaction 3He+n(2.5 MeV), (2.9.2.2m) D+D—. due to beam—target reactions between fast ions and background ions is calculated for comparison with experimental diagnostics. These reactions do not affect the plasma particle or energy sources or the fast ion distribution. The beam—target neutron production rates is (4— eq. B.173), ‘~DDn = ‘E,~~~1-l
where
GDDn
(2.9.2.2n)
nDGDDn(E)[2E/(Abmp)lfa,
(~ eq. B.172) is the cross section for the reaction in eq.
(2.9.2.2m).
2.9.3. Thermonuclear fusion power and fusion-product heating There are several features which are common to the various thermonuclear fusion models available in BALDUR. These are the calculation of “sink” rates of reacting fuel species, source rates of charged fusion products that have thermalized, and the heating of plasma ions and electrons by charged fusion products. The compression heating of suprathermal fusion products is not modeled, and the distribution in scaled
294
G.E. Singer ci al.
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minor radius, r/a, is unchanged by compression; the compression modeling simply moves the fast ions with the plasma as it compresses in minor and major radius. If NTYPE = 1, the density of alphas in each zone is unaffected by compression; i.e., the number of alphas is not conserved. 2.9.3.1. D— T fusion.
D
+
The fusion reaction rate for
T —*4He (3.52 MeV)
+
n (14.1 Mev)
(2.9.3.la)
is calculated for reactions between tritium and deuterium ions which are in either the neutral beam injected fast-ion population or the thermal population [37]. The beam—target reaction rate is calculated in the fast ion code [see eq. (2.9.2.2g)] and the thermonuclear reaction rate is calculated in the D—T alpha thermalization code. The sum of these reaction rates is the rate at which 3.5 MeV fusion-product alpha particles are produced in the plasma and enter the suprathermal population. The prompt-loss fraction is calculated by determining the fraction of alpha particles which are created on unconfined trajectories and promptly collide with a toroidal surface of major radius RCWALS (4— eq. D.1438) and minor radius RDWALS (— eq. D.1439). Coulomb collisions with plasma ions and electrons slow down the confined alphas over a finite time and heat the plasma. When the alphas have thermalized, they enter the source rate of 4He impurity ions. The sink rate of thermal D and T due to the D—T fusion reaction, S (~—eq. B.222) is given by S,,F=h~T=nDnT(av)DT, a=D,T, where an analytic approximation to reaction is given by 1~DDn=
KGV)DT
(2.9.3.lb) (~—
eq. B.2i9) is used. The reaction rate for the D(D,n)3He
0.5n~/av)DDfl,
(2.9.3.lc)
and KOV~DDfl (~— eq. B.220) is the D—D fusion rate coefficient. This latter reaction rate is used only to calculate the neutron production for diagnostic purposes and is not used to calculate a sink of D, a source of 3He or plasma heating. The different versions of the D—T plasma heating model are based on the same alpha-particle energy-loss model. Coulomb collisions with plasma electrons and ions remove energy from suprathermal alphas at the rate _E,y=fs0[CaeXe3/(1+0.75Xe3)+Cai1/Va,
(2.9.3.ld)
and Caj ~ eqs. B.227 and B.229) are the coefficients of the electron and ion heating terms, is the ratio of the speed of the alpha particle, Va~ and the thermal electron speed Vthe~ f~’5 (~— eq. D.2527) is a factor which varies the slowing down rate. In the limit of small the fraction of the initial energy which is deposited in the ions over the course of the thermalization process is where
Cae
= V,,,/V~
Xe,
2 c~_udu h~(x~)=—~i 3 x~’0
1+u
= (2/x~){[arctanE(2x~-1)/V~I+1T/6I/~_ln[(1
+x~)2/(1
-x~+xflj/6}, (2.9.3.le)
where x~is the ratio of the initial energy, En 0 = 3.52 MeV, and the critical energy E~(4— eq. B.230) is x~= Eao/Ec. By setting the input variable NTYPE = 3 (4— eq. D.1440), the user selects the most elaborate nonlocal D—T plasma heating model in BALDUR. The suprathermal alpha distribution is presented by a group of sample alpha particles which are selected by Monte Carlo techniques from the confined alpha population.
given
by
CE. Singer et at
/ One-dimensional plasma transport code
295
Each sample particle’s trajectory in the poloidal plane is calculated, the plasma heating is calculated along the trajectory, and the collisional perturbations of the trajectory are also included. The model with NTYPE = 2 is very similar to the above; the only difference is that collisional perturbations of the sample trajectories are not included, and the alpha energy is deposited along the initial trajectory. A very fast local plasma heating model is selected with NTYPE = 1. The confined-alpha production contributed to a local density of fast alphas and the slowing down process removes alphas from this population. More specifically, the time evolution of the fast alpha density, n a (*— eq. B.233), is governed by
dna/dt= —na/;a+ñDT, where
;a
~DT
(2.9.3.11)
eq. B.232) is the thermalization time for alphas and
(4—
(n~,T+ ~lDT)(1— L(r))
=
(2.9.3.lg)
is the confined-alpha source rate, and L(r) rates [see eqs. (2.9c—d)] are given by
eq. B.226) is the prompt-loss fraction. The plasma heating
(~—
Q~’=naEa(1—hj(xc))/;a,
(2.9.3.lh)
Q~’= naEahj(xc)/;a,
(2.9.3.li)
(see eqs. B.236—237) and the particle source of thermalized 4He SaF~~la/Tsa’
[4—
eqs. B.235 and (2.9b)] is
a=4He.
(2.9.3.lj)
2.9.3.2. Catalyzed D—D fusion. The catalyzed D—D reactions are D +D
—~
T(1.01 MeV)
+
D + D —*3He(0.82 MeV) D
+
T
D
+ 3He
p(3.03
+
MeV),
n(2.45 MeV),
He(3.52 MeV) + n(14.06 MeV),
—~‘~
—~
He(3.67 MeV) + p(14.67 MeV).
(2.9.3.2a) (2.9.3.2b) (2.9.3.2c) (2.9.3.2d)
This model is called “catalyzed” D—D fusion because we assume that tritium produced in the first reaction is quickly and completely consumed in the third reaction because of the much larger D—T fusion cross section. The particle sinks and sources are calculated for the chief ion species, D and 3He, and the incidental species H and 4He, if they are present in the simulation. All charged fusion products are assumed to be confined in the zone of their birth, and in the model they all slow down instantly and deposit their initial energy in the plasma electrons and ions. The reaction rates are (*— eqs. B.262—264) =
0.5n~Kav)DDP,
(2.9.3.2e)
1~DDn=
0.5n~
(2.9.3.2f)
~DDp
11D’He =
nDnl~KcJv)D3H~.
(2.9.3.2g)
The particle sources of H, D, 3He and 4He in eqs. (2.9a) and (2.9b) (see eqs. B.268—270) are: S,~”=+(~DDp+~D3He), a=H,
S,’= —(2ñDDP+2ñDDfl+ñD3HC),
(2.9.3.2h) a=D,
(2.9.3.2i)
296
G.E. Singer ci at. t4D3He+flDDn~
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One-dimensional plasma transport code
3~j a— rae, 4r1
oF . . ~a =flDDP+nD3He,
a— rae.
.
The plasma heating due to the six charged reactions products is (see eqs.
B.271—273)
6 Q~’=
~ñ,E 01(1—h(x1)),
(2.9.3.21)
i-I
6 (2.9.3.2m)
~ñ1E0,h(X,),
QF=
i= I
4He), E where ñ, is the reaction rate for the ith reaction product (ñDDP is the rate for 3.52 MeV 0, is the initial energy given in eqs. (2.9.3.2a—d), h(x) is given by eq. (2.9.3.le), and x7 is the ratio of initial energy to the critical energy, E~,(4—eq. B.274): x,~= EOI/EC,. 2.9.4. Radiative losses Hydrogenic bremsstrah.lung is always included as a loss in the electron energy balance unless NATOMC = 0. Options also exist to include impurity bremsstrahlung, line radiation and synchrotron radiation. (Additional losses due to ionization of neutral isotopes of hydrogen, helium or impurities are described in sections 2.9.1, 2.9.10 and 2.9.13, respectively.) The following formula for the radiative source terms illustrates how these options are2R(T controlled: \ —6 ( j. \ Qhydrogen(y — 1,NATOMC~ eff ~ eJ 2,NATOMC~e\~1 ~ ~RAD — —~ (z — 1~ 2J
e
coronal (
‘~2,NATOMC~e
~a
a
e
a=e’,h
2.5
X 10_25Cioo(1 +
-~4~)(i +
____
1
~t~b0tk~~ev
(2.9.4a)
Note that setting the input switch NA TOMC = 1 (4— eq. D.1023) also forces impurities to be fully ionized, and therefore also directly affects neoclassical and other transport coefficients. C1~is a (usually factor 3tot = ~ small) 7~+ n~T correcting the synchrotron volume emission for reabsorption and reflection. Here / 1= (2/3)n~]/(B~/8rr).RI~~em is the bremsstrahlung radiation, and ~ is the coronal radiation for species a and is given by tabular input (ci. section 7.3). 2.9.5. Scrape-off losses Losses due to flow of plasma along field lines to a material plate in the scrape-off region are approximated by the order-of-magnitude estimates [39] S,~=—C~°,
Q~= =
L=
=
—
where C~=~4’
~
~nJ1/(TH + ;~)— 61~q~’/L, j = e,i,
for for
(2.9.5b) (2.9.5c)
L/(Mv~),
I( C127 C128,
(2.9.5a)
r < min(r~~,r~5i1), r,~i
( (Lv~)/(21/2v5) ~ 0,
T11,
C123 ~123
0, ~ 0.
(2.9.5e)
>
(2.9.5f)
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/ One-dimensional plasma transport code
297
In the appendices we define the mean ion mass ~ (~— eq. B.309), the inner boundary of the scrape-off region ~ as determined by the input parameter C120, the inner boundary of a “second scrapeoff” r~ras determined by the input parameter C121, and the charge-exchange frequency v.~ (4— eq. B.316). The connection length L [‘f— eq. (2.9.5e)] is typically set to half the average length of a magnetic field line between intersections of a material boundary. The parallel Mach number, M, can be determined by a ‘two-chamber” model (~— C381), which describes the continuity of particles, momentum and energy between a dense, sonic flow at the material boundary and a subsonic flow in the BALDUR scrapeoff [39]. The resulting Mach number is with
:~=~,‘
7..{C384~ (cf. ref.
(2.9.5g)
[52]).
T
ad
M=—~
1/2
~
(2.9.5h)
,
~-
div
where = 1
Adjv
=
ndIV—
—
(1
— C381)(1
—
exp( —
(2.9.5i)
C38o/AdI~)),
vO/(ndIV
(2.9.5j)
neTe/2Tdjv,
(2.9.5k)
=
ex~(~AOFJ ln~(k_11iivev)).
(2.9.51)
A~ are the electron impact ionization coefficients of Freeman and Jones [40], v0 = (2 E0/m0 )1/2 (*— eq. C385) is the neutral energy, m0 is the average neutral mass, and Td1~is the solution of 1”2TdIV —
5ne(~v)
2(7e + yI)ndIV(2TthV/~i~)
E0
1/2 a
q~’ =
41~+
C382(1
—
(2.9.5m) q~’=
min[C386n0v4h0(7~ — T41~),K~(7~, — TdIV)/LII],
where Vthe = (2[(1 — C387)7~+ C3g7Tav]/me }1/2 with C387 = to approximate a so-called thermal barrier.
(2.9.5n) 0
for conventional heat flux limit and C387
5~2 = (12.5
—
9.34Z
~/2)(3m1/2)/[4(21T)h/2Ztre4AI
= 1
(2.9.5o)
~
where A 0 is the Coulomb logarithm and Tay = (1 — C388)1 + C388Td1V. Note that the model of charge-exchange friction used here [eqs. (2.95a—b)] differs slightly from ref. [39]. The value Ye = 2.9 [4— eq. (2.95g)] is chosen to approximate the effects of a plasma sheath at the material boundary assuming no secondary electron emission. 2.9.6. Lower hybrid and arbitrarily specified auxiliary heating In the lower hybrid (LH) heating model, a spectrum of radio frequency waves is launched from the edge of the plasma and propagates inward along a radial line perpendicular to the cylindrical cross section of the plasma. Absorption by linear electron Landau damping heats electrons until a lower hybrid resonance point rLH is reached. At TLH, the remaining power is deposited in the ions. If there is no lower hybrid
298
G.E. Singer
ci
at. / One-dimensional plasma transport code
resonance point in the plasma, electron Landau damping continues as the waves propagate across the magnetic axis and outward along a radial line [41]. The implementation of the LH model in BALDUR allows options to neglect ion heating or electron Landau damping, or to substitute in place of LH heating an arbitrary specified heating source for ions and electrons, as described in appendix D (~— eqs. D.322—D.379, D.381—D.438, D.489—D.492). 2.9.7. Electron cyclotron resonance heating (ECRH) The standard ECRH model adds a specified power p~C(4—eqs. D.238—D.257) to the electrons and pEC (~— eqs. D.258—D.277) to the thermal ions. The power is deposited in a torus with a specified rectangular cross section (— eqs. D.318—D.321). The power added to each radial zone is proportional to the electron density multiplied by the volume of the zone lying within the given box. Thus, QEC =
n~ dv,
PECnfb(r)/J
j =
(2.9.7a)
e,i,
VECRH
where f~( r) is the volume of that portion of the heating region lying in a given radial zone and VECRH is the volume of the entire heating region. Versions of BALDUR are also available which contain ray-tracing models of ECRH [42]and detailed modelling of ICRH [33]. 2.9.8. Ohmic heating Electron heating by current dissipation is (2.9.8a)
QoHnhJz(JzJb)
where q~,is given in eq. (2.8a),
is given in eq. (2.9.2.2e) and
~b
c a(rB9)
ar
(2.9.8b)
2.9.9. Collisional energy interchange Collisions between ions and electrons produce the sources QIE
3CA(
QIE
~ a=1,2,,~,h
2>a [~eq.
(2.9.9a)
flaa/Aa)(Te_ T1)/i~, (2d’)],
Aa
(4-eqs. D.2328—D.2331) and ; (4—eq. B.73) are defined
where Ac (*—eq. in appendix B. B.4),
arccos( C
75) from the normal to the cylindrical surface of C77) with a specified energy (4— C76). This results in a volume source
C~[c7, max(0,F~) + C72F~]ne
C
ii
,-‘
~
‘-70
U
70> 0 c’He
(4—
=
n —t’
and (rHe or
f
—
r)
C73 rwall, otherwise
—
r1
~73r~511,
/
Hedr /(C7SAHe)1,
3He or 4He a=
(2.9.lOa)
where ~ is a normalization constant (.~—eq. B.60a), AH (~— eq. B.60) is the local mean free path for ionization of helium by electron impact, F~ is the parallel helium loss in the scrapeoff in particles/ second,
C.E. Singer et at /
One-dimensional plasma transport code
299
and F~~i1s the outfiux to the wall in particles/ second. The electron energy source due to the ionization of these helium neutrals is 3He or 4He, (29 lOb) or 4He. QHe —(iO~eV)C74S~, a=3He a=
J
2.9.11. Pellet fueling Spherical solid hydrogen pellets are described by their hydrogen species (4— eqs. D.2340—D.2359), velocity (4— eqs. D.2308—D.2327), pellet radius (4— eqs. D.2087—D.2106) and tangency radius (4— eqs. D.2067—D.2086). Pellets can be injected in the tokamak midplane at up to twenty specified times (4— eqs. D.2278—D.2287). The ablation of the pellets is calculated according to an extension of the model of Milora and Foster [44]. The pellet affects the plasma in a direct fashion by changing the particle density and electron energy density (pellet terms do not appear in the diffusion equations): i~n 0(r)= S~~dt(r), a
= 1
or 2,
(2.9.lla)
i~E~(r)=—(3OeV)An0(r),
(2.9.llb)
where the determination of i~n0(r)is described in detail elsewhere [46], and 1~Ee(r)accounts for electron energy loss during ionization of pellet neutrals. Apart from the self-limiting features of the ablation calculation plasma parameters are assumed to be held fixed during pellet ablation. The contribution of fast ions and alphas to pellet ablation is included. An essentially unlimited number of pellets may be injected by using the pellet-cluster model (+— C230). In this model, a cluster of pellets, rather than a single pellet, is injected at up to twenty specified times (— eq. D.2278—2287). A cluster consists of a stream of identical pellets whose firings are separated by a specified time interval (f— C231—C250). Pellet properties can be changed only from cluster to cluster; strictly speaking, when the cluster model is used, one should read “pellets in cluster” for “pellet” in the definitions of these properties (~— eqs. D.2067—2106, D.2087—2106, D.2278—2287, D.2308—2327, D.2340—2359). Within a cluster, pellet firings continue until the next cluster begins or until a specified maximum number of pellets has been fired (~— C252). 2.9.12. Recombination The particle sink due to recombination of hydrogen ions is —nane
5REC
where
3/
\‘r
/
(2.9.12a) (~—
eq. B.57). The corresponding ion energy loss
\
Q —
~2~n1+n2)1ine\ave/r~COm.
(Note that the code combines Q~EC with the charge-exchange power in subroutine NEUGAS.; cf. also section 5.2.5.) Electron energy losses due to recombination are typically small and neglected. Recombination contributes a volume source of neutrals (cf. section 2.9.1). The input switch NLRCOM (4— eq. D.1030) is used to omit all recombination effects. 2.9.13. Neutral impurity influxes Influxes of neutrals of one or two impurity species (*— eqs. D.2254—2255) may be prescribed by input (C2~—C226eqs. D.1024, D.1848—1887, D.2256—2275) for impurity species with atomic numbers 36. The neutral impurity atoms are subject to ionization, and thus contribute an impurity ion source term SMP
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One-dimensional plasma transport code
(a = 1,h) and an electron-energy loss term due to initial ionization QIMP In addition, the coronal radiation produced by each influxing impurity species [cf. eq. (2.9.4)] can be multiplied by an empirical nonequilibrium enhancement factor (C205—C206). The physical model assumes that the neutral impurity density falls off exponentially with increasing penetration depth. This approximates a steady-state condition where most of the neutral-impurity influx involves recycling impurity-ion outfiux off walls, and is correct for any portion of the influx due to controlled gas puffing. Using realistic ionization rates, neutral impurity atoms ionize within a few centimeters of their injection point. Hence, slab-geometry conditions are a good approximation, and the source rate becomes n°(s) n~(r~Ml’) exp[ —A(s)/v], A(s)
= fSfle(X)KGV>iz
(2.9.13a)
dx,
(2.9.13b)
where Koz>1~ is the electron impact ionization rate coefficient [51] which is correct in the case of a directional stream of neutral gas directed normal to the wall. The local source rate of impurity ions due to the neutral influx is then 51MP =
(2.9.13c)
C~MI~F]MISneKov)~nO(s), a = 1,h,
where C~MP is a normalization constant, and FIMP will be defined shortly. This production of impurity ions, due largely to electron-impact ionization of impurity neutrals, involves energy losses which affect the electron energy QIMP =
C203 SLMP e’ 5e
,
aa—~t ,
(2.9.13d)
204
The total neutral influx
FIMP
of species may be determined
by
either of two methods
(~—C
2~).In the first method, the flux is prescribed directly by linear interpolation between input values F10 (~— eqs. D.1024, D.1848—1887, D.2256—2275). As generally accepted models of the influxes are not available for realistic wall materials, the user usually has to make an externally supplied estimate of these influxes (cf., however, section 3.1). In the second method, the flux is determined by a feedback algorithm which attempts (not exactly, or always successfully if this requires excessive influxes) to achieve prescribed total number of impurity ions at specified times MP=max{O.,(4~2R,.walI)_1[Na~No
+L~]}.
(2.9.13e)
1) of neutral impurity a. Na4 is the prescribed total number of Here FOIMI’ the in influx rate (cm 2 scenter of the present time step. N~° impurity a ions the plasma at the is the calculated total number of impurity a ions at the beginning of the time step Na =
4~2Rf~ W,,l (r)r dr.
(2.9.13f)
L~is the estimated total loss rate (s1) for impurity a. ~t” is the present time step (cf. section 6). Na* is computed by linear interpolation from input arrays (f— eqs. D.1848—1887, D.2256—2275). This method is appropriate when an experimental measure of the average impurity content is available. Besides the initial ionization losses, it is possible to include impurity-radiation enhancement due to nonequilibrium excitation and deexcitation. The present treatment of radiation enhancement is quite
G.E. Singer ciat
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One-dimensional plasma transport code
primitive. For each impurity influx a, there are certain prescribed energies
301
(~—
C205, C2~)which are used
ad hoc to enhance the coronal-equilibrium radiation as follows min[C225,1
C2osf~~S~MI’rdr/f~nenaR~0~~a1].a =
+
0
min[C226~1 +
IMP
0
1,
(2.9.13g)
IMP
C2o~j”~S~MI’rdr/f’~ nenaR~01~j,
a
= h,
where R~ot~~ is discussed in section 2.9.4. Note that this enhancement affects the coronal radiation only, and does not affect QIMP or noncoronal contributions to Q~D (~— eq. (2.9.4a)). 2.10. Minimum densities and temperatures Prescribed minimum values of density ne,,.~n= C129 and temperatures 1~ = C130 and T1T~= C131 can be set in the scrape-off region to avoid numerical overflows if the scrape-off region chosen is too large. The effect on the diffusion equations is equivalent to adding local source terms of the appropriate value to maintain these minimum values. A diagnostic for the activation of one of these equivalent source terms is the energy conservation check listed on the printout. Since these equivalent source terms are not included in the conservation checks, it will appear as if conservation is violated when they are activated. 2.11. Compression The purpose of the compression model is to calculate the effects of adiabatic compression, and the effects of changing the limiter radius with or without compression. Adiabatic compression effects are computed based on the major radius R specified as a function of time. The degree of compression from a time t1 to a later time t2 is described by the compression factor c c=R(t1)/R(t2). (2.lla) When the major radius is compressed, the flux surfaces 1”2r. are compressed in minorare radius, so that a flux (The flux surfaces assumed to remain surface which was at minor radius is remappedIf to concentric circular-cross section tori rthroughout.) thec mirror radius of the limiter varies so that 2rwaii(ti),
Twaii(t 2)
(2.llb)
=c~”
then the flux surface next to the limiter at time
t 1
remains next to the limiter at time
t2.
moves with respect to the flux surfaces. We define the “limiter compression” factor as 2rwaii(ti)/rwa cL = c~” 11(t2).
If not, the limiter (2.llc)
If CL < 1, the limiter is moving outward relative to the flux surfaces, and cold low-density plasma will fill the space between the limiter and the flux surface which used to be next to the limiter. If cL> 1, the limiter is moving inward relative to the flux surfaces, and some plasma will be “scraped off”. R (t) and rwaii(t) are linearly interpolated from input values (- eqs. D.2027—2066 and D.2131—2150). Under cL < 1 compression, the number of particles is conserved, while the energy of electrons and ions is increased to account for the work done in decreasing the volume. The toroidal and poloidal magnetic fields increase, with poloidal flux being conserved. These conditions lead to the following transformations under CL = 1 compression of plasma quantities 2, a=1,2,”,h,b, B 3”2, n0cxc 9crc E.ccc10”3, j=e,i, Jccc2, 4 3 . . (2.lld) j=e,1, B~ccc,
qc(c°.
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Thermalization of the thermal compression energy is assumed fast enough that the usual transport coefficients determine the subsequent effects on confinement. Compression is discussed further in section 5.3. Effects of compression on fast ions due to neutral beams and on fast alpha particles are discussed in sections 2.9.2 and 2.9.3, respectively.
3. Boundary conditions and density control 3.1.
Boundary conditions
The inner boundary conditions on the diffusion eq. (1) are Bn
~E.
ar
Br
—~=--—-~=B
9=0 at
r=0,
a=1,2,e’,h; j=i,e.
(3a)
At the outer boundary, the value or radial gradient of each variable is specified. The following combinations are available for na and E~. Pedestal boundary conditions: NBOUND=O:
~lania
and
7~=T11, a=1,2,1,h; j=i,e.
Pedestal unless there are abnormal gradients at the wall: NBOUND=1:
na=nla
BE1
an0 —
Br
= 0
—
Br
=
and
T~=T1~, a=1,2,i,h;
0 for each a or
.
j
Bfla
where
—
Br
r,~,11
j=i,e. 0, or
ai~ Br ~
0.
Specified impurity influxes (with pedestals for hydrogen isotopes and energy): NBOUND
= 2:
na = nla
and
F0 = F10(t) +
As with NBOUND
=
NBOUND=3:
~
T’sputter(t),
a
= 1,2;
j= i,e,
= I,h.
2, unless there are abnormal gradients at the wall: n0=n10 Fa
except
T~=T1~, a
=
and
F10(t)
~
a=1,2; j=i,e,
+ T’~~~4101(t), a
=0,~.L=0foreach a orj where
=
~!H>o,~ or
(3b)
“Pedestal” boundary conditions are replaced by “zero flux” boundary conditions when the gradients at the wall are in the “wrong” direction. n10 (4—eqs. D.1888—1891), T1~(4—eqs. D.2208, D.2250) and Fia(t) eqs. D. 1808—D.1887) are prescribed boundary densities, temperatures and fluxes. Tputter(t) can be set to zero or to the flux of sputtered iron calculated from the Monte Carlo generated neutral spectrum incident on a stainless steel vessel at r = r~~1 using the sputtering coefficients of ref. [50]. The options NBOUND = 1 and NBOUND = 3 can be used to avoid unphysical conductive or convective energy influxes. (4—
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303
The following combinations are available to determine the outer boundary condition on the poloidal field eq. (2c). The evolution of the total toroidal current can be fixed i~=i~(t) at
(3c)
r=rwall,
or the poloidal field can be given B9=B9(t)
at
(3d)
r=rwall,
as described in appendix D (4—eqs. D.1441—D.1460 and D.1462—D.1481). Alternatively, the surface voltage can be specified C396). (4—
3.2. Density control The flux of neutral hydrogen atoms entering across the plasma boundary can be altered during the computation. First, the fraction ~ (~eq. D. 997) of outgoing neutrals returned to the plasma with an energy ~ (~— eq. D.2129) can be changed to a new prescribed value C102 at a prescribed time ~ = C89. Second, influx of gas puffing neutrals entering the plasma with energy TCOLDP eq. D.2130) can be prescribed as a function 1a .La~tj ~ at r—rwall, a—x,.~., rO_rO/ H\ — (4—
....1
‘,
/~i
where r~(tr) are input arrays eqs. D.18O8—1847, D.2224—2248). Finally, an additional neutral influx can be included for feedback control (“density monitoring”) of the volume-average electron density
4. Initial conditions The initial particle densities and temperatures can be prescribed in three ways. First, any profile can be given as an input numerical array. Second, profiles of the form n 0
=
nla +
(flOa
—
n10)[1
—
(r/rwaii)~’]“,
a
=
1,2,1,h,
(4a)
j=e,i
7.=Tij+(Toj_Tij)[1_(r/rwaii)xTuJ)~TJ,
(4b)
can be specified. Third, some or all of the starting densities can be specified as a given fraction P~,of the initial total ion density =
Fa(nij + (nt,1
—
n11)[1
—
~),
(r/rwaii)~]
a
(4c)
= 1,2,e’,h, 1~i =
fa/>fa,
where
f~,are input values
or n0eqs. = D.1888—1891). n where n1 is set by tabular input. In these expressions, The initial poloidal magnetic field profile can be specified either from an initial toroidal current profile (4-.-
Jcc [1— (r/rwaii)~]~’,
(4d)
jçg7~3/2,
(4e)
or as
with the normalization of B 9 determined by the boundary condition discussed in section 3.1. These profiles will be modified in the scrape-off region. The method of specifying these options is detailed in appendix D, starting at (eq. D.1482) for densities, (eq. D.2151) for temperatures and (eq. D.1441) for B9.
304 5.
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Solutions of the diffusion equations
Here, we give a summary of the numerical methods and then write the difference equations used. The differential equations are first transformed to conservative difference equations of the Crank—Nicholson type with a variable degree of implicitness. In a given time step, the B9 equation is first advanced using the previous values of na and E1, and then the n0 and E~equations are advanced simultaneously using the old values of B9. The whole procedure can be done in a simple predictor—corrector sequence or in an extrapolation scheme. As the extrapolation scheme has not been found particularly useful, it is not discussed in detail here. The difference equations solved are as described in sections 5.1 and 5.2. 5.1. Space and time mesh 5.1.1. Space grid The code is one-dimensional in the minor radius r. The plasma is divided into N concentric cylindrical zones about the magnetic axis (r = 0). The plasma zones are numbered, in order of increasing r, from 2 to N + 1. In addition to these so-called real zones, there are two dummy zones, one inside the origin (r = 0) and one outside of r = rwal,, making N + 2 zones in all. The values of variables in the dummy zones are used to impose the boundary conditions on the var~.ablesin the real zones (cf. section 3). Note that rb refers to the inner boundary of zone j. The space grid for a particular run is set by input variables described in appendix D (~— eqs. D.1972—2026, D.2361 and D. 2366). r
0
Inner dummy zone BALDUR zone:
4
1
Main plasma region 3
LCENTR
Physical zone:
1
rb 1
r~
z
=
r r
z
—r
r
=
b
Z
b
r
,
ISEP
LEDGE—i
LEDGE
M
N—i
N
[1,,
rb 4
=
r
wall
rM+i
rb M
rb M+i
(r
+
wall
—r
,
b r 1
z
(r, J
+
= r
b —r 3
z
)/2
,
,
j—l
Number of physical zones Physical
b r N+2
=
b r N+3
=
=
N
zone number of first
=
Iz
z
rM
rb N
j
r
S
Outer dummy zone
wall
MZONES
z
r N+1 rb N+i
RADIUS(J)
=
Iz
rN
rb M+2
=
RNINS
rN+2 rb N+2
*
z N+l
j
radius of inner boundary of BALDUR zone
0
=
z
N+2
2 r, J
Scrapeoff
Iz
3
rb 3
2
b
4
—*
radius of zone center of BALDUR zone
1 r
r
r scr
M—i
I5
rZ
r
ISEPM1
2
I
i
rZ
=
r
=
r wall r
+
wall
=
RMINS
*
XBOUNI(J)
P.MINS
(r
—
wall
r N+i
NZONES
scrapeoff
zone
M
Fig. 1. BALDUR spatial grid.
CFUTZ(120)—1
rb N+3
XZONI(J)
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305
code
Physical quantities are described at zones and boundaries in a typical region of the spatial grid (cf. fig. 1). Densities n0, temperatures T1 and 7~,and energy densities E, and Ee are defined at zone centers. They are then averaged to get values at boundary points. For example,
T0b,= ~
~ 7~+ 1—
(5.1.la)
~ ‘:~ ‘:,-~
‘:~-~
‘)
r.Z_ri~
where j is the zone index and the superscript b or z indicates quantities defined at zones or boundaries. Conversely, B9 is defined at zone boundaries and interpolated to get values at the zone centers, assuming that the toroidal current density is constant across a zone. Thus, B in zone j is 2 b I b \2 — ~ ~2 b ~ ~~,÷1i ~ /2 ‘:~ ( z\2 — ~‘ ( b\-‘ B — ‘‘.. 2 rZ B r~B 91 — b \2 / b\ 91~1+ / b \2 / b\ 91. (5.1.lb) .‘
—
~‘i
)
~
—
~
)
~
Boundary-centered quantities (derived from the ofzone variables n0, in1~,T~,Ec, 2 ~a) are computed using averages the centered values of fundamental the fundamental variables the center E1, on (Z ),~) and
3r~
n~~— flZ;.i_1
(5.1.lc)
.
ijrj_i
5.1.2. Time grid The time grid is determined by a variable time step, chosen before a cycle: (5.1.2a) Quantities defined as “time-centered’ are appropriately interpolated between the values of depending on the solution method described below (cf. also section 6).
t,,
and
t” +
5.1.3. Variables used in code The variables used to describe the space—time mesh in the actual coding differ in trivial ways from those described in sections 5.1.1 and 5.1.2, as summarized elsewhere [17]. 5.2.
Differencing
We now describe the differencing scheme used for the diffusion equations. Crank—Nicholson parameter 0CN We solve a set of diffusion equations of the form
5.2.1.
BX =
131 ax —~--r~A -~j---+B
x) +Cx+D+~9(f),
where x is a vector of independent variables and A’, B’, C are matrices of nonlinear coefficients, and D is a source vector. When solving such equations numerically, two problems generally emerge, namely, stability and accuracy.
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A finite difference scheme can be made stable by making the difference equations implicit. The difference equations then have the form —OCN~(X)
+
(1 —OcN)~(x~),
(5.2.lb)
where f is the value of the variables at the new time n +1 x” is the value of the variables at the old ~ is the differenced form of e’(x) for time t~, ~(~‘) is the differenced form of t9(~)for t = t’2, = ~ ‘~+1, and the Crank—Nicholson parameter 0CN is a factor between 0 and 1 which controls the degree of implicitness. 0CN (4—eq. D.1020) is specified by input to the code. Setting 1/2 < <1 usually guarantees stability. A discussion of the second problem, accuracy, will be found in section 5.2.4, along with a further discussion of stability. +
Poloidal field equation The poloidal field equation used is
5.2.2.
3B
2 B 9
[4—
eq. (2c)]
~ 3(rB
c
9) ~
B
Br
(5.2.2a)
This is differenced conservatively to obey Ampere’s law JJ. dA = (c/4ii)fB9. dl exactly. This is done by defining B9 on the boundary and the toroidal component of J in the zone center. The integral from of Ampere’s law is
JJdA
=
~—JB.dl.
(5.2.2b)
In difference form, the toroidal contribution to this equation is —
rb2) =
~—(2iii~1B91+1
—
2ru~B91),
(5.2.2c)
or ~
—
(5.2 .2d)
(~1~~9,j+1 — ,,bB9~) 1’2 _rb2 j+1
I
r where we denote the toroidal component of J The central boundary condition is that B
by
J. The differencing of the other terms is straightforward.
9(O) = 0, see section 3.1 for the outer boundary conditions. The equation is set up as a variable implicitness Crank—Nicholson diffusion equation and solved by a tridiagonal solution technique. Ohmic heating is computed with a conservative technique developed by Roberts. The basic constraint is that
2~ f IJ.E+——IdV= 1 BB 81T
Btj
—f--~--ExB.dA.
(5.2.2e)
A4~
Taking the toroidal component of Ohm’s law to be E =
~1h [~
—
~beam]
and differencing in time, we write
(B~~±1)2 ~B~2 ~
2~t
~
n+1
=_c{V.~h[OcNJ”~’+(1_OcN)J~_J~eam]( ~
2
B~ o)
+4~11h[OCNJfl+1+(l_OCN)Jfl_J~eam](J 2+J)},
(5.2.2f)
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307
for the change in magnetic field energy. The first term on the r.h.s. is the Poynting flux and the second is the ohmic heating, which we can write as Q~3H=71h[OCNJfl+1+(1_OCN)Jn_J~am](J
(5.2.2g)
2+J).
When J is expressed in the conservative difference form in space, the above term is conservative in space and time. A conservation check can be computed by summing eq. (5.2.2g) over the plasma.1 There a slight + (1 — is0CN)~ difficulty with the Poynting flux in eq. (5.2.21) since (E x B)toroidal = h1[0cN1’~ ~beam]( B + + B )/2 and, while B 9 is naturally defined on the outside boundary, J and ~ are naturally defined at the zone center. While a scheme which conserves poloidal field energy to machine accuracy was not found, interpolation schemes are used to adequately minimize the errors in computing the energy conservation check. —
5.2.3. Particle and energy diffusion equations The other dependent variables besides B9 are the particle and energy densities. The equations for their time development [eqs. (2a,b)] form a set of nonlinear parabolic initial value problems. The equations are conservatively differenced in space, and set in a variable implicitness Crank—Nicholson form. The dependent variables are the hydrogen isotope densities (2 allowed), impurity densities (4 allowed), and the electron and ion energy density (cf. section 2). We write these variables in a vector x = transpose(nl,n2,n(,nh,ne,Ej). The particle and energy transport equations are
_v~.TP~ocN+C1x~ecN+D~.,
~
(5.2.3a)
is the divergence of the flux through zone j, C, is a matrix containing the coefficients of where terms T” which are to be differenced implicitly in ~ and source is a vector of explicit source terms. C and are defined in appendix A. The n + 9CN refers to the variable degree of implicitness: i.e., 9CN
+
X”~°~ (1
—
(5.2.3b)
OcN)x~+ ~CNX~’~
The fluxes F are defined as
~
(1 —OcN)(Bfx~+AJx~...l) +OCN(BfXJ
+A
1X~t~),
(5.2.3b’)
where the matrices A1 and B~are defined in appendix A. This approach was chosen so that we could difference implicitly n~Z~ — n~_ ~ as the difference of the gradient term (B
2 BE~. 3nj Br n7~Bn1 1 Br
—=—————,
j=i,e,
(5.2.3c)
from the chain rule (see also appendix A). The coefficients, 2/(3n~),7~/n1,are nonlinear, of course, and have to be time-centered. The gradients are differenced in the usual way eq. (5.1.lc)] [f—
~
n~—n~_~ r
(5.2.3d)
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/ One-dimensional plasma transport code
The divergence is handled in a conservative scheme: ~!~rF)+CX+D,
(5.2.3e)
or in difference form =
a~+1F1+1—a~F1 +C1~1+D1,
(5.2.3f)
2 2]2.,TR] which will where a~ is the area of boundary j (21ri~2TrR) a nd J’~is the volume of zone j [(rb+ — ,.b 1 automatically conserve particles and energy [4]. The matrices A, B, C and the vector D are set up in subroutine CONVRT. They are then recast in subroutine REDUCE into the form
P
1x~+~ +
+
+
S~,=
0.
The boundary conditions are expressed in the form at r
3x = y,
(5.2.3g)
= 0
and r
=
a (5.2.3h)
aB~/Br+ 1 which translates into
(A
1
+
C1)~1+ B1~2=D1,
(5.2.3i)
at the center and
AN+2xN+1 +(BN+2+CN+2)xN+2=DN+2,
(5.2.3j)
at the edge. The boundary conditions described in section 3 are translated into A, B, C and D ‘s in subroutine BOUNDS, and cast into the P, 0, R, S framework in REDUCE. This gives us the usual tridiagonal matrix equations which are solved in the usual way [17]. The only complication is that P, 0, R, S are matrices, and multiplication and division in the usual method are replaced by matrix multiplication and matrix inversion. One ends up with solving a 3 x 3 up to 8 x 8 linear system in each zone by Gaussian elimination with partial pivoting. 5.2.4. Predictor—corrector and extrapolation For (1/2) ~ O~ 1, a linear set of equations, such as eq. (5.2.lb), would be unconditionally 0CN = 1/2 stable. is not However, eq. (5.2.3a) has nonlinear coefficients which will change during a time step, and necessarily the best choice for stability and accuracy. Stability often then requires 0CN to be close to one. Second order accuracy in time should then be obtained by time-centering the nonlinear coefficients. This is done in two different ways in the code. The most commonly used method is a two-step predictor—corrector scheme. First (predictor), a trial solution x~1is computed, using A, B, C, D evaluated at the old time t0. Next (corrector), A, B, most of C and pieces of D (cf. section 5.2.5) and the linear combination = ~“(l — 9~) + x~’~ 1, are computed, where 0 is a 1 time-centering parameter between 0 and 1A,setB,byC,input to the code D.1021). is computed using the Or-time-centered D and using x (—*~ eq. rather than The then final solution x”~ for use in the next time step. Typical problems are run with O~,and 0CN = 1 to x0 is stored make the problem as stable (implicit) as possible. It has been our experience with real problems that stability has been a much more severe problem than accuracy. Note that the computation of B 9 is done using the same time-centering scheme used in the computation of x. ~
+
~“~‘
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309
Numerical form for sources As various source terms present different stability and computational overhead, different numerical strategies have been adopted for different sources. Source terms are associated with the matrix C, the vector D, or both, and their time-centering depends in part on this association. The C-source is C~5 and thus is time-centered to the extent that ~ ~9CN is. D on the other hand, does not multiply x~On the corrector portion of a predictor—corrector time step, C is recomputed based on x~1 except for the WICHXS term discussed below. The only terms recomputed in D are S,~ [4— eq. (2.9.lOa)], Q~~E[~— eq. (2.9.lOb)], QIMP [. eq. (2.9.13j)], QRAD [- eq. (2.9.4a)], Q~H [4— eq. (2.9.8a)], and the WEIRS term discussed below (cf. appendix A). Note that 5,~EC[4— eq. (2.9.12a)], the particle sink due to recombination, is associated with C and is time-centered, while the corresponding neutral source rate is in D and is not time-centered. Also note that the corresponding ion energy sink Q~EC [~— eq. (2.9.12b)] is treated differently, as described below. Sources due to pellets (cf. section 2.9.11) are computed “instantaneously” as direct additions to or subtractions from x. They do not appear in C or D. They are computed only after all iterations due to time step repetition and predictor—corrector and extrapolation have been completed, and they bypass the conservation checks. In the pellet cluster model (~— C230), if the prescribed interval between successive pellet firings in a cluster is less than the current itt, the firing interval used by the code is set equal to ~t, and to compensate the ion deposition due to pellet ablation is multiplied by an enhancement factor equal to the minimum of i~t/(prescribedinterval) and C252. The charge-exchange loss term can be written as n0 n H (~ — T1 )( at’ )~. (n0 is the neutral density, n H is the hydrogen ion density, 7, is the neutral temperature, T1 is the ion temperature, and (av)~5the charge-exchange collision rate.) If n0 n H T~( av )~ is computed explicitly and 7~ T~as is normally the case, small changes in T1 can make large changes in this term, which in turn limits the time step. This can be especially “painful” since the neutral gas calculation is expensive and called only once every NGPROF (~— eq. D.lOlOa) time steps. Physically, 1~,/T1 changes much more slowly, so rewriting the equation as 5.2.5.
+ 9CN,
~
—
1)T1,
(5.2.5a)
allows one to make the equation implicit in T1, to compute the slowly varying (at’ )~[(7~/T — 1] term from the neutrals package, and to compute nOnH from the previous time step. QREC and the ion energy sink due to charge exchange of thermal hydrogen ions with fast beam neutrals are also formulated linearly in T1. The sum of these two terms plus the charge-exchange loss term is 0+ called WICHXS in the code. WICHXS is time-centered by linear interpolation in T1 as (WICHXS) [(B/BT 1)(WICHXS)]0[(E,’~°cN/n~ 7”)]. Note that WICHXS is associated with both C and D [8]. Under predictor—corrector or extrapolation, only the factor [E°~/n~ — 7”)] is recomputed: e.g., for the corrector portion of a predictor—corrector time step, we have —
9cN/fln+Op —
(WICHXS)~+ [a(WICHxS)/aT1] ~[(E
where
o
—0 ‘~E~ + 0 E~hl + 0 E~~1 p1 i p i J CN i The impurity radiation loss rate depends strongly on T~and can limit the time step, especially in problems with large radiation losses at the edge. The impurity radiation rate evaluated from fits to the coronal radiation rates (cf. section 2.9.4) is summed with Q~IE[~— eq. (2.9.lOb)] and Q~MP[4- eq. (2.9.13j)] into the code variable WEIRS. WEIRS is time-centered by linear interpolation of 7~as (WEIRS)5 + [(a/a7~)(wEIRs)1”[(E:~/n: — T~”)].Like WICHXS, WEIRS is associated with both C and D. Under predictor—corrector or extrapolation, all factors are recomputed; e.g., for the corrector portion of a predictor—correction time step, we have =
—
~.
~
CNJ
p~i4.O~ +
i
o
Op+
‘~“~
CN
i
=
(1 —0 ‘~
CNI[\
(WEIRS)~~°P + [((a/ai~)(WEIRs))”~°P[(E:+9P+6cN/n:+9P —
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In addition to time centering, WEIRS may be modified by the smoothing of sharp peaks in its spatial variation. This smoothing is activated by setting C199 = 1 or 2. 5.2.6. Convection Another standard difficulty is that, near the plasma edge, the character of the heat conduction equations often changes. The equations are of the form (5.2.6a) is the conduction (or diffusion) and B is a flow velocity. e represents Ee or E, (cf. section 2). When A(Be/Br) >> Be, the equation is clearly second order, and second order boundary conditions apply. When & >> A (Be/Br) convection dominates conduction, the equation is close to first order, and applying second order boundary conditions can give difficulties. Numerically, the boundary-centered flux is differenced as A
(e~+,—
— —
—AJ~1/2
L~r
—
BJ~1/2J±1/2.
(5.
.
)
The problem is that ~j+1/2 is not well defined in terms of the zone-centered quantities e~and Since B1 + 1/2 has the dimension of velocity, we can think of it as a “convection” velocity. If we average + 1/2 = ~(e~+~ + ~), the self-compensating nature of the diffusion equation is lost. For example, if t’j+I/2 = <0 and e 0, the “convective” flux does not go to zero, but to eJ+l/2v1+1/2, and the density in zone j can go negative. Several methods have been used to treat this problem [4]. The one we use is to construct the centering variable j+1/2 = ‘~~+1/2~J+ (1 — ~+1/2)eJ+1, where 61+1/2 is always bounded between 0 and 1 and 6, = 1/2 is the default condition. The actual 6 used is ~
j
=
—
signEt’] min{~, v~L~t/aL~r},
(5.2.6c)
,,
with i.~r = + — i~, where v is an appropriate velocity defined in subroutine CONVRT for each element of X and a can be fixed by input as a = C81. The usual value of a varies from 2 to 4. If a is 0, 6 is fixed at 1/2. The most important implementation of this scheme is for ion heat transport, where “convection” at the edge dominates conduction. The algorithm is also applied to electron “convection” and particle “convection” (primarily, due to the neoclassical “Ware” pinching). 5.3. Compression
Compression is modeled as a series of “instantaneous” compressions. The transport equations are used to compute the plasma state at t”~ from the state at 1” ignoring compression. Then the effects of compression in major radius are computed as if the compression which is supposed to have happened over the center time step occurred instantaneously at t~~ Then any effects due to limiter motion relative to the plasma are computed, as if the “limiter compression” had also occurred instantaneously at ~‘i ~ Note that the compression computations are done only after all iterations due to time-step repetition, predictor—corrector, and extrapolation have been completed. If cL < 1 [4— eq. (2.llc)], then cold, low-density plasma must be added. BALDUR does this by adding more zones all with cold plasma. Zones are added by the same method as was originally used to compute the mesh (~— eqs. D.1972—2026, D.2361). Zones are added until a zone is reached with an outer boundary radius r~greater than or equal to rwall. If r,~> rwall, we set r,~= rwall. Then we check to see that this “chopping off” has not resulted in too small a zone width. If the ratio of the width of the last zones to the width of the zone immediately inside is less than 0.2, this can mean that the time step must be extremely small in order to keep the finite-time-step errors reasonable. In this case, the small zone is discarded, and
G.E. Singer ci at
/ One-dimensional plasma transport code
3~I
rwall is set equal to the outer boundary radius of the next zone ion. In the newly added zones, x is set to its edge value [~— eq. (3b)]. B9 is extended by assuming the current density is zero in the new plasma bold
~ ~r1— —
.!.±i~01dI r ~ ~rN+2), bold\
where the superscript “old” means after effects dependent on major radius compression have been computed, but before effects of limiter compression have been computed.
6. Time-step control
The initial value of i~tis prescribed by input (~ eq. D.1787). At each new time step, i~tis adjusted as follows. If, at the end of a time step, it is found that z~ t was too large, the plasma parameters are restored to their old values, and the time step is repeated with a reduced i~t.We describe here the checks used for time-step repetition and the method used to compute ~t. A time step is repeated whenever ~a’
E1’
or
6err/6max
>
B9<0, a1,2,e,h; x~,where
(6a)
j=i,e,
or when
n~1_n~I Efl+1_EP?~ B~1—B~ 6e~=max
n~
‘
E~
B
‘
(6b)
is the maximum change in any zone not bypassed by C 8max 126, isC161.-164 andtime x4 (4— and (e— eq. D.1018) are input constants. When extrapolation used, the stepeq.is D.1036) also repeated if Ceii.> Em~Xl where x 3 (4—eq. D.1035) and m~ (4—eq. D.1019) are prescribed constants and n?~+1_n~~ LEJ~+1_ (err
max
0
~
__________
0
E~ 1=2n~~—n~,a=1,2,e’,h,
n
E7~’=2E)—E~,
~ a=1,2t~’,h;j=e,i,
B90
(6c) (6d)
j=e,i,
(6e)
B’=2B~—B.
(6f)
Here the values labeled “1” are evaluated by calculating for two successive minor time steps at i~t/2and “0” labeled values computed with minor time step i~t. ~ is an estimate of the i~t error. Since extrapolation applies a correction to i~f and to f 0n+1 computed with minor time step i~t (i.e., ~ n +1 = f0n +1 + ~f = 2ff’ + 1 — ~0n + I and ~f = 2[ f~’ + 1 — f0n + 1]), the relative correction is ~f/f~. By requiring this correction to be small (e.g. 5%), the time integration error should be, e.g., (0.05)2. The E~tto be used for a new or a repeated time step is computed as follows. If the time step is to be repeated due to negative n5, E~or B9, the E~~t used is the old t~tmultiplied by x5. If the time step is repeated due to an excessive extrapolation error, the 8t used is the old ~t multiplied by x0 if x6> 0. For all other cases, the ~t used is the old i~t multiplied by a factor C— f1/cei.~ if ceu=max(8ej.~/m~,ee,.r/cmj >1, \min[x2,~tmax/2~t,xi+(1_xi)/cerr]
if
(6 ) g
Ce~
Here x5 and x6 are prescribed constants (4— eqs. D.1037—1038),
4~tm~ (4—
eq. D.1788) is a maximum time
312
G.E. Singer
ci at.
/
One-dimensional plasma transport code
step, x2 (f— eq. D.1034) is a specified maximum rate of the time-step restriction and x1 (*— eq. D.1033) is a constant which damps the error controlled relaxation of the time-step restriction. (We note that l/Emax = 0 when extrapolation is not used). In order to avoid running problems where the time step has dropped to the point where the computation is not preceding at a reasonable rate, the run is terminated if z~tdrops below a prescribed minimum value ~tmjfl (~ eq. D.1789).
7. Structure of the code and test run The structure of BALDUR is largely based on the OLYMPUS system [7]. 7.1. Structure of the FOR TRAN program The structure of the program is outlined in appendix E. The function of the main program BALDUR is primarily to call the OLYMPUS subroutine MASTER. Then MASTER calls BASIC and CONTROL, BASIC calls MODIFY, etc., as outlined in appendix E. The bulk of the computation is controlled by the subroutine STEPON which advances time steps. The structure of subroutines called by STEPON is shown also in appendix E. 7.2. COMMON variables An alphabetic list of COMMON blocks and a description of their use is given in appendix C. 7.3. Atomic data file A file (called F0R22 in the present version of BALDUR) contains polynomial fits to data used to compute the mean charge, mean-square charge and radiative loss of impurities, assuming coronal equilibrium. The data for He, C, N, 0, Ne, Mg, Al, Si, 5, Ca and Fe are taken from Tarter [50]. Data for other impurities were produced from the program XSNQ as described in Post and Jensen [51]. Detailed information on the data in file FOR22 is given elsewhere [17]. 7.4. Ripple amplitude input array An input file (called RIPPLE in the present version of BALDUR) is required if diffusivity from toroidal field ripple effects is included [17]. 7.5. Plotting
BALDUR writes output files which can be used by postprocessor programs on CRAY and VAX computers. 7.6. NAMELIST input description and test case sample run A sample NAMELIST input is given in appendix F. An example of some of the output from this run is given in appendix F. A complete list of the NAMELIST variables and their use is given in appendix D.
G.E. Singer et at
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One-dimensional plasma transport code
313
8. Code version nomenclature A system for naming new versions of the BALDUR code will now be described. All codes which use a substantial fraction of the BALDUR coding should be named BALDXNNY
-
X’N’N’Y’- X”N”N”Y”
where X indicates the institution where the version was made or modified, Y is an arbitrary code letter for the person responsible for providing information about the version and NN is a two digit number assigned sequentially to versions with a given X and Y. Whenever X or Y is changed in creating a new version, then a suffix containing the previous value of XNNY should be appended. For example, the version documented in this paper is BALDP47M
- PO9A
prepared at Princeton by Martha Redi, based on an earlier version prepared at Princeton by Alex Silverman. An earlier version in the BALDPXYA series, BALDPO1A, was supplied to R. Wunderlich at the Institute für Plasmaphysik. The standard version of the code made at the Institut für Plasmaphysik has been called BALDIO9R
-
PO1A.
(The code letter R denoted the person responsible for providing information about said modification.) Each institution should contact one of the first four authors of this paper to ensure they choose a unique institution code and to clarify any other questions concerning use of BALDUR. Acknowledgements Comments from G. Bateman and D. Stotler are gratefully acknowledged. This work was supported by the US Department of Energy Contract No. DE-ACO2-76-CHO-3073. Appendix A. Definition of A~,B~,C1, D1 The basic plasma transport eqs. (2a—b) and (5.2.3a—j) are written in terms of the dependent variables na and E’.. These equations are in the form actually solved by BALDUR. In contrast, eqs. (2h—i) are written in terms of (Bn0/Br). To go from one representation to the other, it is necessary to define A1, B~,C~,D1 in terms of the transport coefficients of section 2. 1. A1 and B1
The matrices A1 and B1 together act as a finite-difference operator on the dependent variables vector, x eq. (5.2.3b’)]. Let a be the analogous operator. a candefine be written as the sumB~.Then of zeroth-order 1: ax [a°differential + a1(B/Br)]x. Similarly, A°,, B~, A1, and and first-order pieces a°and a
[4—
A
1x~1+ B1X1 (1
—
is analogous to ax,
~f+l/2)AfXf_l +
+
&B~x~ is analogous to a°x,
B)x~_1)
is analogous to ~
(A1.1) (A.1.2) (A.L3)
[4—eq. (5.2.6d)] is called BOUND in the code. Given the elements of a, A and B may be derived from eqs. (A.1.1—3).
61+1/2
G.E. Singer et a!.
314
/ One-dimensionalplasma transport code
In the definitions below, the subscripts of aK/ refer to the different matrix elements and must not be confused with the radial zone index j. aaa= [t’a_
b=1,2~
b=~,hBr
—
~
D~n,,+ ~
b=1,2
[
a~~=
D~~anb(Z2)bj_~a=1,2,
+
—
+ [Daa +
D~”n~ +
~
(A.1.4)
b=e.h
b=1,2
b=1,2
a=
+
~
DL~nb(Z2)bJ~-,
a=
~‘,h,
(A.1.5)
b-e,h
1,2,t’,h; /3=
1,2,é~,h,e,i; a #/3,
(A.1.6)
aea= [C365t’~are7~]+ [—x~7~]~--, a= 1,2,
(A.1.7)
aea
(A.1.8)
a=
= [Xe7(Z)a]~,
te,h,
aee= [(F~1)Xe]~,
(A.1.9)
(A.1.10)
aej=0,
a1~= =
0,
/3 =
{
~_
[Daa
~
—
(F
(A.1.11)
i)x~+ ~
a=1,2
~
+
1,2,t’,h,e, (D~a(Za)b_
D~0(Z)b)nbI B’~a
b~,h
[Daa_F—1X1+
~
D~n~+
~
DKz2)bnhj.~_~.
b~-1,2
~ [—
+
a=e~,h +
~
D~n,,— ~
b=1,2
D~(Z)bnbI-I_[na(Z)Q]
b=e,h
~D~a((Z2)b/2+(Z)b)nbj~~i
a=1,2[
2)b + a ~th +
b
12
D~((Z)~/2 + 1)nb +
-i--,
[(F— 1)~1]
b
~
(Z)Q(Z)b)nbl n
-~
BT
-~
(h D:~((z
(A.1.12)
modified if C 110> 0. The following symbols are defined in section 2. D00 [4— eq. (2.6a)], v0 [4— eq. (2.7e)], D~ [4— eq. (2.4.ld’)], D~0 [e— eq. (2.4.lc)], D~j’[4— eq. (2.4.lc)], D~ [~eq. (2.4.ld)], D01 [4— eq. (2.4.11)], t’~jare [f— eq. (2.71)], x1 [4— eq. (2.6d’)], Xe [4eq. (2.6d)] and x1 [4—eq. (2.6c)], F, the ratio of specific heats, is set to 5/3. Note that additional “impurity drag” terms not documented here are included in subroutine CONVRT when C217 > 0.]
G.E. Singer ci at
/ One-dimensional plasma transport code
315
2.C~ The radial zone index SREC +
Caa
=
a
j is omitted here for clarity.
ss
na
a
° ,
=
1,2,
(A.2.1)
5S
C00=—.~, a=t”,h, C~~=
+
Q!E
n~(1~— T1)
(A.2.2)
—
E~
(F— i)[
~
a~’,h
B(WEIRS)0]
(A.2.3)
Here, B(WEIRS)0/B7~is computed from analytic derivatives of the radiation tables from subroutine OFIT. ç~IE
C~1= ~‘ n~(1~T1) ~
Cie+
(A.2.4)
,
IE
~
(A.2.5)
,
fle(7~Ti)
C
~
—
QIE
n1(1~—T1)
+
E~
+
F—i B(WICHXS) n1 BT1
(A 26)
All other elements of C1 are zero. The particle and energy sources are defined in section 2.9 and 5.2.5: 5REC [4— eq. (2.9.12a)], Q~EC [4— eq. (2.9.12b)], ~2’ [4—eq. (2.9.3a)], Q~[~- eq. (2.9.5b); j = e,i], QIE [4—eq. (2.9.9a); j = e,i] and WEIRS and WICHXS (cf. section 5.2.5). 3.D~ Again, the radial zone index Da=SaN+S~+S:, Da=S:+SaHE+S~MP,
De=
j is omitted here for clarity.
a=1,2,
(A.3.1)
a=e,h,
Q~+ Q~+ Q~+ QRAD~
(A.3.2)
Q+ QLH~ QEC+
TMP QOH+
~
+ QHE÷ QI
(A.3 .3) D 1
=
+ Q~+ Q~+ QLH + QEC + QREC
—
rB(WICHxS)
(A.3.4)
The sources are defined in sections 2.9 and 5.2.5: S~ [~ eq. (2.9.1—2)], S~ [4—eq. (2.9.2c)], Set’ [4—eq. (2.9.3a—b)], S~IE [4— eq. (2.9.lOa)], 5IMP [4— eq. (2.9.13c)], Q~ [4— eq. (2.9.lc)J, Q~ [4— eq. (2.9.2d)], Q~ [4— eq. (2.9.3c)], QRAD [4— eq. (2.9.4a)], QLH [~— eq. (2.9.6a)], QEC [4— eq. (2.9.7a); j = e,i], QOH [4— eq. (2.9.8a)], Q~IE [4— eq. (2.9.lOb)], Q~MP [4-.. eq. (2.9.13c)], Q~” [4— eq. (2.9.ld)}, Q~ [4— eq. (2.9.2e)], Q~ [4— eq. (2.9.3d)], QLH [4— eq. (2.9.6b)], Q~EC [4— eq. (2.9.12b)], and WEIRS and WICHXS (cf. section 5.2.5).
316
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Singer eta!.
/
One-dimensional plasma transport code
Appendix B. Formulae Notes: 1. In eqs. (B.1—B.5), (B.8—B.22) and (B.24—B.30), the first code variable listed is the coefficient and the second is the exponential in an expression of the form (coefficient) * (10. * * exponent). For example, eq. (B.1) is evaluated as CFEV*10.* *CXEV = 1.60210 x 10120. 2. The Cross-references colunm tells where to find definitions of symbols used in formulae. For example, the notation A~(*— eq. B.7) at eq. (B.42) means that the symbol A~,which is used in the formula for eq. (B.42), is defined in eq. (B.7).
G.E. Singer ci aL / One-dimensionalplasma transport code No.
Code
317
Formula
Units
Subroutine BLOCKDTA B.1 CFEV,CXEV 8.2 CFEVDK,CXEVDK
12° keV=1.60210X10 keV/deg = 1.16049 X 1040
(erg/eV)
B.3
FCAO,FXAO
a
B.4 B.5
FCAE,FXAE FCALFA,FXALFA
B.6
FCAN
B.7
FCAP
A~=1.00727663
(amu)
B.8
FCC,FXC
c = 2.997925 X 1010.0
(cm/s)
B.9
FCC1,FXC1
Gradi = 3.7405X105°
(erg cm2/s)
not used
B.10
FCC2,FXC2
Grad
8.11
FCE,FXE
e =1.60210x 10—20.0
B.12
FCES,FXES
e
B.13 B.14
FCF,FXF FCG,FXG
B.15 B.16
5.29167x iO.~0 4°
0
Comments
(K/eV)
not used
(cm) (amu)
not used
5.48597x10 a= 7.29720x 10-30 =
not used
1.0086654
(amu)
2 = 1.43879 X 10’~°
not used
(cm K)
not used
(emu)
not used
(statcoul) 2g1/2/mol) (cm” (dyne cm2/g2)
not used
Faraday G = 6.670constant x 10 8.0= 9.64870 x 1030
FCH,FXH
h = 6.6256X1027°
(erg s)
FCK,FXK
kB
(erg/K)
not used
B.17
FCLNAE,FXLNAE
ACompton = 2.42621 )< 10— 10.0
(cm)
not used
B.18
FLCNAP,FXLNAP
A
(cm)
not used
B.19
FCME,FXME
1, =1.32140X10 ma = 9.1091 x1028°
(g)
B.20
FCMN,FXMN
m~ 1.67482 x 10_240
(g)
B.21
FCMP,FXNUCL
m~=1.67252X1024°
(g)
B.22
FCNA,FXNA
NA
B.23
FCPI
‘IT
B.24
FCR,FXR
R~= 8.3143 X 1O~°
B.25
FCRE,FXRE
B.26
FCRINF,FXRINF
R,,~=1.0973731 X105°
B.27
FSCB,FXSB
a
11.28
FCSIGT,FXSIGT
~
B.29
FCVO,FXVO
normal value of perfect gas = 2.24136x104°
B.30
FCWIEN,FXWIEN
Wien displacement constant = 2.8978x101°
Subroutine UNITS B.30a FACU,FXAU
4.80298 x 10—10.0
1.38054x 10160
13°
= 6.02252 X 10~~°
ZADEF2(1) ZADEF2(2) ZADEF2(3) ZADEF2(4) ZADEF2(5)
not used
(mol ~‘)
not used
(erg/K mol)
not used
(cm)
not used
(cm1)
not used
(erg/cm2 s K4)
not used
(cm2)
not used
(cm3/mol)
not used
(cm K)
not used
3.1415926536 2.81777X1013°
=
5.6697
X
iO~°
= 6.6516 X 10250
m~/A~ = number of
grams
in 1 amu
Subroutine AUXVAL
B.31 B.32 B.33 B.34 B.35
not used
Aa =1.007825, a H A,, = 2.0140, a = D A,, = 3.01605, a = T A,, = 4.00260, a =4He A,, = 3.0, a =3He
See also WTGAS, WTIMP ( *— eqs. D.2328-2331)
318
No.
G.E. Singer ci at
Code
/
One-dimensionalplasma transport code
Formula
Units
Comments
Subroutine IMPRAD(1)
B.36
B.37
ZATCOF(6,4)
ZAHEO(9)
A~’J~”° =
n = 0,3 — 8.875780 13.06722 — 22.74558 4.332644 —23. 22560 0.07795893 — 23.22962 0.2169734
n =1,4 33.68544 2.780003 2.823806 1.994706 0.4390245 0.1040682 0.4768998 —0.1969521
A~4-~~e = —44.50917
24.42988
2.470931
Subroutine AUXVAL B.38 AIRED(4,4)
~
B.39
A’=
ZIIONM
0.5107924
—
6)/(A~+ A,,)
~
n,,/
a’-1,2,t,h
~
n,,A,
B.40
ZVS
v11 =[(T,
j=2 j=3 j=4
i = 0,7 i = 8,9
(amu)
reduced mass
(amu~’)
inverse
(cm/s)
mass parallel flow
mean ion
/a=1,2.~’,h
2
j =1
0.02505100
(7); 13.74394;
= (A,,A
not used
—10.25714
—0.3426362
0.007438675 —
n = 2,5 30.04414 0.2216913; 4.693808 0.3405031; 0.03994050 0.1536833; — 0.09974274 0.06531130;
+ 7~)/(A’)I”
velocity, cf.
A1 (4—eq. B.39)
No.
Code
Formula
B.41
ZJS
~
I,=~.= 0.25e(
n~)vii
~
Units
Comments
Cross-refs.
(statcoul/cm2)
Twice the
v 11
1”2e4
B.42
CTIONS
G.,
B.43
CTELES
G.E =
1”2G~ 1
1/2 B.44 B.45
CNUHYD CNUEL
GrH = (m1,/A1,)
Gr~= GrHA~/2
eq. B.40)
(i_
A 1,
0.75[m1,/(A1,’TT)]
C... (A~/2)
imtial current density in scrapeoff
(‘—
(4—
eq. B.42),
A~ B.4) A (~eq. 4-eq. B.7) 1,( ‘—eq. B.44),
CrH(
— B.46
CETA
C,, = m,,e2
B.47 B.48
CDNHS CDNHIS
G~= 8(2i~me)1”2(ec)2/3 CHI =
GHA,~1”2
eq. B.7)
GH(
eq. B.4)
‘—eq. B.47), eq. B.4)
‘—
B.49
CDNIIS
C 1 = C~ 112
GHI( GHI(
4‘—
eq. B.48) eq. B.48)
B.50 B.51
CDITIS CDETES
GHI/(2) CE = CH
CH( 4—eq. B.47)
B.52
CCNU
Cr = 3A~
A,( 4—eq. B.4)
11.53
CDBOHM
GB = 0.0625 c/c
G.E. Singer et at / One-di mensional plasma transport code No.
Code
319
Formula
Cross-refs.
Subroutine IMPRAD 8A~ ~ 1o~(A~1] (1,kav/0.1)”~1Z,2(01), ~kaV<0.1
B.54
(none)
L~,(1) = antilog
B.55
ZPLOSS
R”I’(7,)
A~’]~”~( ~—eq.11.36)
L
0.1< 0.2 < 2. < 20. <
22(7), Lz3(1~), L 54(1~),
Lz4(2O.), B.56
ZBETA
B.57
ZRECOM
B.58
ZVO
L~,~(l)( i—eq. B.54)
20.
~,keV < ~kcv
n~(~— eq. J.3.6e),
= [n~T~ + n,T1 + ~(n~+ nE,,)J/(B~/8’TT) (Note: nE,, term is not currently included) 13n,,([0.073529max(0.001,7k,V)]1”2
nE,,( 4—eqs.
=
[(2 x 103k,,vC
ZSIGV
2,
antilog
=
a = 3He,4He
76)/(A,,mp)]”
~,cV<2
~A~~~(log
antilog
A;+HC +
~,v)”
A~Mc (log ~
2.
<
~,eV
<10~
io~< ~—
B.60
1./ZIONIZ
AMa =
1./ZTOTAL
eq. B.37)
4-eq. B.58),
VHeo/(nc(tJVe)Hco)
VHeO(
Subroutine IMPRAD B.60a
B.233—4)
nc~mb= 1.27 X 10— XE10+0.59(0.0735297~keV)]}~1for ~ 400 = 5.2179X1012n,,1~,,3S8 for Ta,ev> 400 0,
B.59
Tekev <2. <0.2 7~kaV
-1
C~’=
Ha dr ne
f
Hr
dr/(G 7SAHe)I dv)
r1 = maxEo,r —(1— G73)r~,,,1]
rHe(
‘—C77)
(OVC)HCO( 4—eq.
B.59),
AHe( 4—eq. B.60) I
IMP
B.60b
1./ZTOTAL
C~”~ = [4~2Rfr.
SMPT
drl
5IMP( ‘—eq. 2.9.13c),
‘—CC)
r,~Ml’(
Subroutine INITIAL
B.61
RHOINS(2,55)
Subroutine GETCHI B.62 Q(55)
n1=
a
q=(rB~)/(RB9)
B.63
AJZS(2,55)
J = c(arB9/ar)/(4lTr)
B.64
XZEFF(2,55)
Zeif~{(ni+n2)I~”~a + ~
/( Subroutine TRCOEF B.65 GSPITZ(2,55) CALPH(55)
B.67
CLOGIS(2,55)
8.68
CLOGES(2,55)
D.994), t(4-eq.D.995)
a — 1,2/,h ~ n~~)+(Z7-1.0)
A~’= [Z~(2.67+
Zc
3.4O(l.l3+ 8r)j/[
B.66
~
n~~}
= [71cZeci/(thi
Z~
Zc Zct8( ‘—eq. B.64)
8~)]
Z~1,(‘—eq. ~ ‘—eq.
+ n2)] —1.
A~ = max[(1.0, ~
B.64) B.64)
5°~
A = (max(1.,ln[1.54x10b0(T3v/ne)1~~2/Zeff]}, T v< ~max(1.,ln[(1.O9x1O1h7~)(nh/2Zcr 8)] },
Zat4(
‘—
eq.
B.64)
320
G.E. Singer ci a!.
No.
Code
Formula
B.69
AHME.AN(2.55)
A~= ~
AIMASS(2,55)
A
1=
One-dimensional plasma transpori code Cross-refs.
A,,na/ ~
a —1,2
B.70
/
Ia =1,2
Aana/nj
~ a — 1,2,1,h
B.71
Z2
=
~
3”2AH)/(Z
a
B.72
TIONS(2,55)
= (CrjTj
2Aj)
G,~(‘—eq. B.42), AH( ‘—eq. B.69), Z2( ‘— eq. B.71), eq. B.67) C.,c(—‘—eq. B.43),
3~2)/(flcAc)
X~(‘—eq. B.68)
B.73
TELECS(2,55)
; = (C~7
B.74
XNUHYD(2,55)
v~= [(GrHBT)/(Tj
IB
3AH)/(rT~)]1~2
cH( ‘—eq. B.44), AH( ‘—eq. B.69)
0 I)]E(R B.75
XNUEL(2,55)
IB
~ = L(CreBT)/(;
3/(r7)]1’~2
Cr~(‘—eq. B.45), ;( ‘—eq. B.73)
9 I)][R B.76
ZD,ASPINV(55)
B.77
ZFT(55)
B.78
ZXI
B.79
ZC
B.80
ZO
= r/R
= 0.58 +O.2Z~,,
C~=[.56(3.—
Zcit( ‘—eq. B.64)
Z~,,(—
Z~,,)J
Zaii)]/[Zcti(3.+
fTi = fT/(1 + EI~c*)
eq. B.64)
fT( ‘—eq. B.77),
eq. B.78), — eq. B.75)
‘—
8.81
FTRAP(2,55)
B.82
ZYO
CR( ‘—eq. B.79)
fT~( ‘—eq.
= (1—[min(C
B.80)
1/~’)1/2 57,C58)/C58]
Subroutine XSCALE B.83 SLTES B.84 SLNES B.85 SLBPS
rr,= 11/(81/8r)l i, = n,/(~ln,,/8r)I r l(8/arXB
These gradient scale lengths are numerically
9/r) B, rTI—IT~/(aT,/ar)I
smoothed except for r3-c
B.86
SLTIS
Subroutine TRCOEF B.87 ZGYROI B.88 ZESV
~ = c(2AHm1,)” v~,=n1J/e
B.89
ZVTHE
Va = (1~/m~)’/2
B.90
ZVSDI
V,
B.91
ThE 1 ‘ EXBETA
2/e
—
2 1 = [1/A~m1,f/ (t) = rIp
(P. C
145 B.92
ZRIPL1(55)
AH( — ‘—eq. B.69)
~(r, 0,
AH(
variable ripple -
ZMSO,ZMSOO
i) = ~0(i)+[&,or(i)~
<~2)o~
$~~(t)(‘—eq. D.2437—D.2O46)
~o(i)j(r/r,crf’47
~o(i)(
lo, for N,~,,11,4= ‘ff(2$)’/2 .IT/2[1 2. —O.278443269.IT(2$) (0.8862269255/I IT(2$)1”2]}
for B.94
XX
eq. B.90)
fixed ripple
1121 8.93
‘—
1T(2~)’/2> 2.
‘—eqs. D.1451—1460),
‘~,~,(‘X where ‘—eqs.
D.1472—1481)
— eq. 2.3h), 8v,,ñ,,blc( ‘—eq. 2.3i), $( ‘—eq. B.91) 8=
8fixcd(
a = Be/[N,,,, 11,Bz6(r, 0, z)]
6(r, 0, t)( ‘—eq. B.92)
G.E. Singer ci at / One-dimensionalplasma transport No.
Code
Formula
a*
8.96
G = [(3X211’2)/(161T)Jf dO{sin
(c sin ~
=
here, G(a*) [10] B.97
ZG
=
for 0 8 ir 28(1 + t cos 8)[8( r, O)/8(r, 0)]3”2C( a*) [(32X21”2)/(3’IT)](cos 1a~))(1 — (a* )2)1/2 — a*cos la* ]3/2) a*( ‘—eq. B.95)
G = antilog 2 — $1~~2(1. —0.243a — 0.13a2)] x antilog 10[11.641a +0.234a 10[ — (2.346 + o.462pu/2 )][(1 + 0.64379( a ~2)10.317 for variable ripple or for fixed ripple when C149 0.
B.98
ZO
Th, = m~A,/A~ 1”2
8.99 B.100
Z1 Z2
LY.
ZO
i~i,(‘-eq. B.98)
= (T/Th.) = min[r,cr,(c/e)8iji~j/Be]
DR = 3(ir/2)
ñ1 1( 4-eq. B.98), ii,( ‘-eq. B.99) i~(‘— eq. B.99), p~~(‘-eq. B.100)
1~~/R
2mc)/(41Te2) B.102 B.103
ZKADO ZYRFE
Wee =
B.104
ZGYROE
c~[c(2me)112]/e
B.105
ZGYRFI
W~
GK=(c (Bze)/(mcc) =
A~( ‘—eq. B.69) Cpe( ‘—eq. B.104)
1= (BZe)/(AHmPc)
2/Bz
B.106 B.107
ZRHOET ZRHOIT
PZc = PZI =
cc~” G~
2/B~
G~
1T~” B.108 B.109
ZK1 ZK.2
a( ‘—eq. B.94), $( ‘—eq. B.91) A1( ‘-eq. B.70)
1”2~ B.101
321 Cross-refs.
Subroutine AVERGE B.95 ASTAR1 GALPHA(55)
code
32Zcii)]
K1 = (0.53+ Zcff)/[Zeff(l + l. K 2 = 0.82(3— Zcif)/[Zcff(2.57+
$~=
B.110
Z1BETA
B.111
Z1ZETA
8.112
Z2BETA
P2
B.113
Z2ZETA
~2 = 0.25 +O.l 1T/[’. +(/3jVc*)I/2 + ~
B.114
ZK(11), ZK(12)
K1~
B.115
ZL13
L
=
Zcff)]
O.4+O.llZ~
1(‘4-eq. -eq. B.87) Zcir( B.64) Zcii(
*—eq. B.64)
Zcit(
4-eq. B.64)
0.55 +0.255Zcff
Z~,,(‘—eq. B.64)
0.05 +0.O345Z~ 43Zcff
Z~,,( Z,, ‘- eq. B.64) eq.75), B.64) 1,e*( 1(~ ‘-eq.
=
j = 1,2
fT(4- eq. B.77), $~(4-eqs. B.110, B.112), ~ ‘-eqs. B.111, B.113)
13 = (1 + K1)K11[1 B.116
ZL23
L23 = (2.5
B.117
ZNV
~‘cZ =
—
+ K2)K12[1
(K1K11)/(1 + K1)] —
(K2K12)/(2.5 + K2)]
(1 + Zcrr)/; 12GNc~,7,cvT,ev(AH)1”3
B.118
ZNCRIT
n~1=
2.2X10
B.119
ZCRITO
CNCfl,
=
B.120
ZNCRIT
~
6.Ox 10’3CNcnt1~v
=
E3/2[qRA~(1+ Zeff)]~’
K1( ‘-eq. 0.108), K11( ‘-eq. B.114) K2( 4- eq. B.109), K12( ‘- eq. B.114) Zcff(
‘-eq. B.64),
B.73)B.119), CNC~I(‘—eq. AH( ‘-eq. B.64) i~(‘-eq.
Ac( 4- eq. B.68), Zdf( — eq. B.64) GN,,,,I( ‘—eq. B.119)
322 No.
G.E. Singer cia!.
/ One-dimensional plasma transport code
Code
Formula
Cross-refs.
Subroutine HRCAL B.121a HRB(6,11,26)
H,,(~i,r)
neutral beam deposition function for nth energy component
B.121b TAUMAJ(6) B.122
TAUMIN(6)
see last entries in appendix F.3.8 r
6”
see last entries in appendix F.3.8
Subroutine HRSET B.123
HENER(30)
energy of nth energy component of the neutral beams (see F.3.8a)
E,,NB
Subroutine DPOSIT B.124 HNSRCS(10)
j~NB=
B.125
S,,NBI
HDEPS(10,10)
j for which ~ = W,,/J. Number of particles per second which are injected in the n th energy component of the neutral beam (see F.3.8a)
1,,°f~,,
summed over those b and
local rate at which neutral beams deposit particles as a function of ~t = V1/V for each energy component [see eq. (2.9.2.2a)]. This array is reset in each BALDUR zone by a call to DPOSIT which interpolates between elements of H,(~s,r)3 (see B.121a above)
B.126
SHCHXS(2,55)
ñ~,(r)
sink rate of thermal hydrogen due to charge exchange between neutral beam atoms and plasma ion [see eq. (2.9.2.2h)]
B.127
YFRACT(6,3)
~
current fraction in the nth energy component of the bth neutral beam ‘—eqs. D.1234_1243), W~,(‘—eqs. D.1159—1168) injected power in each energy component (see F.3.8a) J,,0(
5
B.128 B.129
ZPOWR(4,10) YR(26)
p~’ ,~
B.130
ZHR(10)
H,,(r)
B.131
YRLOSE(6)
L~~B
(this is R2 in HRCAL, HRSET and IJECT, it is set by DPOSIT) =
f1 H~(it,r) d~s
H(r) for the nth energy component (see B.121a) this is set by DPOSIT from HRLOSE which is calculated
in HRCAL (see F.3.8b)
(
B.132
ZHRT
H( r)
B.133
ZMU
~jNB(r)
B.134
ZDEV
aN11(r)={~ENB!J~f1H(~ r)[~s2
B.135
ZDN
ñ~~B(r)
(see F.3.8c)
B.136
ZCEX
ñNBacx(r)
(see
B.137
ZLOSS
LNB
(see F.3.8d)
B.138
ZMU
<~NB) =(2/a2)f,L~ih(r)rdr
B.139
ZDEV
(aj~> = (2/a2)f o,~B(r)rdr
B.140
ZZDN
B.141
ZZCEX
=
8.142
ZDN
p~NB= (2i~)2Rof
B.143
ZCEX
Nacx = (2i~)2Rof ñ~,(r)r dr
= >.E,~BNJ~JBH,,
=~E~
f’H,,(IL r)jz d~/~E!~’~H,,(r)
ñ~(r)r dr
(2/a2)f ñ~,(r)rdr nNB(r)r dr
~NB(r)2]d~/~EN8P~NBH(r)}
B.126)
G.E. Singer ciat No.
Code
/
323
One-dimensional plasma transport code
Formula
Cross-refs.
Subroutine BEAMS B.144 LHEMAX
Ea,, = HEI(LHEMAX)
B.145
LHEMIN(55)
E~,,= HEI[LHEMIN(J2)]
B.146
HEI(10)
E~the upper boundary of each fast ion energy bin
B.147
HFI(10,10,55)
fB(~
B.148
HNPLAS(55)
EJ’~S
B.149
HEPLAS(55)
]V,P’~’
(see B.213)
B.150
SHBEMS(2,55)
h~,
(see eq. 2.9.2.2fl
B.151
WEBEMS(55)
Q~
(see eq. 2.9.2.2k)
B.152
WIBEMS(55)
Q~
(see eq. 2.9.2.2e)
B.153
ZSIGCX
(see eq. 2.9.2.2a)
E, r)
(see 8.211)
where the charge-exchange cross section is: 2 [1—0.0673 log(E/Ab)] a 15 B.154
ZLOGE
B.155
ZLOGI
8.156
ZBARLG
B.157
ZEFFLG
cm2
[1 + 1.112x 10_15(E/cv)33] 55(E)= 6.937x10 log A~,Coulomb logarithm for ion—electron collisions log A~= log{(2’trm~/nc)”2[k1~,/(he)])—1/2 log A,, Coulomb logarithm for ion—ion collisions if E Ab(ZbZI)2 100 keV: 0/(4’ne log A 1/22A,,Ai/[(A,, + A,)hel) — 1/2 1 =log([2’nEmpk7~/(A,,n~)] ~ n,(Z~/A~)logA, all ions
(n~/n
~
0)Z~logA1 allions 0 total neutral hydrogen density
B.158
ZNO
n
B.159
HSCATS(10,55)
!,~=
ire~Z,,2~(n
(see eq. 2.9.lj)
log2A,
1/n~)Z, 1 2
(see eq. 2.9.2.2a)
(2AbmbE3) / 8.160
ZNUE
4(2gm )“2e~Z,,~nlogA a 3 2 a
~-~e=
3A,,rnp(kTa) B.161
12A
ZGAMMA
\1/2
‘ITe4( —k J
=
(see eq. 2.9.2.2a)
/
Z~~n
B.162
HSLOWS(10,55)
1(Z,2/A,) log A~ \ 1,/ j H53v,~/log(y,,+ v,~E])/(’y,i+ v,0E~.1))
B.163
HCHEXS(10,55)
e,,,
B.164
ZSIGMA
2 1n°a~,[2E/(A,,m~)]” aDT(E) = (4.09 X i05 + 5.02 X 10~/[1+ (1.368 X 105E
(see eq. 2.9.2.2a)
8(r)( ‘- F.3.7c) is calculated ;from HSLOWS (see eq. 2.9.2.2a)
=H
—
1.076)12)
/(E[exp(1453./V~)— 1])x 10— 24 cm2 where E is in eV
B.165
HEINJS(55)
EJ”J
(see B.211)
8.166
HNINJS(55) ZNUDT
N~ 1~ 7 = NDorTa~(E)[2E/(Abmp)1/2
(see B.213)
B.167 B.168
HALFAS(55)
ñ~T(r)
(see eq. 2.9.2.2g)
B.169
SHBLOS(2,55)
n~
8.170
HELOSS(55)
EJ!055
5
(see eq. 2.9.2.2a)
(see eq. 2.9.2.20 (see B.211)
324
G.E. Singer ci at
/ One-dimensional plasma transport code
No.
Code
Formula
Cross-refs.
B.171
HNLOSS(55)
NI~55
(see 8.213)
B.172
ZDDSIG(10)
a
B.173
HDDS
Ni~
2,where °DD( 00(EX2E/A,,m~)~’ E) = (482./([1 + (3.08 x 104E — 1.177)2] x E[exp(47.88/E1/2) —1])) x 10—24 cm2 where E is in keV 0n= (211)2Rof nDDfl(r)r dr
(see eqs. 2.9.2.2n and B.320)
J Nrnn dt
B.174
HDDTOT
N~0~ =
(see F.3.4t)
B.175
RHOBIS(2,55)
n~
B.176
RHOBES(2,55)
Z,,nB
B.177
HEBEMS(55)
EB(r)
(see F.3.7b)
B.178
ZBJ
~~aan,
(see eq. 2.9.2.2c)
(see eq. 2.9.2.2j)
B.179
ZEPSLN
B.180 B.181
ZRTANG ZMUB
Rtang = ~
(R~~ +~
cos
B.182
ZM1
1—rn
1— 2/[(1
+ e)js~]
B.183
ZKM
K’(rn)
(see eq. 2.9.2.2d)
B.184
AJBS(55)
J,,~,,,/CJ
(see eq. 2.9.2.2e)
8.185
HECMPS
EJ°mP
(see B.211)
8.186
HNCMPS
Ncornp
(see B.213)
Subroutine HPRINT B.187 ZZNB
NB
(see F.3.7m)
8.188
ZZEZ
U5
(see
B.189
ZZEP
UI’
(see F.3.7u)
B.190
ZZEZP
(see F.3.7k)
B.191
ZZMUB
(see F.3.7o)
B.192
ZZETP
~
(see F.3.7e)
B.193
ZZET
UB
(see F.3.7n,w)
B.194
ZTAUS
rB(r)
(see F.3.7c)
B.195
ZDEN
nB(r)
(see F.3.7d)
B.196
ZFUSE
h~T(r)
(see F.3.7i)
B.197
ZCHEX
f~~(r)
(see F.3.7h)
B.198
ZSOURC
ñ~,,,(r)
(see eq. 2.9.2.20
B.199
ZIFRAC
f.B(r)
(see F.3.7j)
8.200
ZEPAR
Ei?(r)
(see F.3.7a)
B.201
ZETOT
EB(r)
(see F.3.7b)
83b
=
~
sin (see eq. 2.9.2.2d)
F.3.7v)
B.202
ZEDEN
n~(r)
(see F.3.7e)
B.203
ZMUBAR
~B(r)
(see F.3.7f)
B.204
ZMUDEV
OB(r)
(see F.3.7g)
B.205
ZZMUDV
(see F.3.7p)
8.206
ZZCXFR
(see F.3.7q)
G.E. Singer ci a!.
/
325
One-dimensional plasma transport code
No.
Code
Formula
Cross-refs.
B.207
ZZSORC
N,B
(see F.3.7r)
B.208
ZZFUSE
NDT
(see F.3.7s)
11.209
ZZFRAC
F?
(see F.3.7t)
B.210
ZZCHEX
(see F.3.7x) m~)
Eyias + E~’°
B.211
Zi
B.212
ZZECHK
B.213
ZO
N’~”~ =
B.214
ZZNCHK
F~= 1— N~”/N5
(see F.3.7A)
B.215
ZZLOSS
0.
(not used)
B.216
ZZHEAT
pB
(see F.3.7y)
B.217
ZZALFA
NOT
(see F.3.7s)
~
= =
~ (E”’J = E!oss — all zones j
1— Ek/UB ~
(see F.3.7z)
(Jy~flJ—
lOSS
—
~pIas + NCOrnP)
all zones j
Subroutine ALPHAS B.218 EFUSEI
E,, = 3.5 MeV DT = exp[ — 22.712/T,°275 — 23.836 — 9.393 x 10 2~ +7.994x104T,2 —3.144x106T 3] (T, in keV), see ref. [37] 1DDfl = exp[ — 15.315/T,°406— 35.904+ 0.195/T,1574] (T, in keV), see ref. [37]
B.219
ZSIGV
B.220
ZSIGV
B.221
AFUSES(55)
ñ~T(r)
(see eq. 2.9.3.lb)
B.222
SHFUS(2,55)
5,,”, a = D,T
(see eq. 2.9.3.lb)
B.223
ZDDFUS
n~,,,(r)
(see eq. 2.9.3.lc)
B.224
ADDS
NDDn
(see F.3.4q)
B.225
ADDTOT
NI~’Dfl
(see F.3.4s)
B.226
AOLOSS(55)
L,,(r) local fraction of alphas born on unconfined orbits
8.227
ZCOEFE(75)
Gec
B.228
ZCOEFX(75)
l/Vthc
B.229
ZCOEFI(75)
G,,
ZECRIT
(see eq. 2.9.3.ld)
=1/V21/m~
(see eq. 2.9.3.ld)
4log A,/mp)~n,Z,2/Aj
1 B.230
=16V’A,,Z~e~n~ log Ac/(3me)
=
(4.sA,,Z~e
E~= AaTa[~I1Trn 1,rnc
(see eq. 2.9.3.ld)
2/3
~ ~ ~‘
(see eq. 2.9.3.le)
B.231
ZBIFRC(55)
h1(X,,) when NTYPE = 1
(see eq. 2.9.3.le)
B.232
ASLOWS(55)
1/;,,
(see eq. 2.9.3.1!)
B.233
ALPHAI(55)
n,,(r)
(see eq. 2.9.3.10
B.234
EALFAI(55)
(see eq. 2.9.3.lj)
B.235 B.236
SALFS(55) WEALFS(55)
nE,,/n,,(r) 1’, a =4He S,, Q~’
B.237
WIALFS(55)
Q~’
(see eq. 2.9.3.li)
Qe
(see F.3.9a)
Subroutine APRINT 0.238 ZQE
(see eq. 2.9.3.lh)
B.241
ZTFUSE
nDT(r)
(see F.3.9b)
8.242
ZFALFI
f~,,(r)
(see F.3.9c)
326
G.E. Singer ct at
/ One-dimensional plasma transport code
No.
Code
Formula
Cross-refs.
8.243
ZNA
(see F.3.9d)
B.244
ZEA
(see F.3.9e)
B.245
ZTHERM
<~DT)
(see F.3.9f)
B.246
ZFUSN
<~DT>
(see F.3.9g)
B.247
ZOLOSS
L,,,,
(see F.3.9h)
B.248
ZTFUSE
(see F.3.9i)
8.249
ZTOTFI
F,,,
(see F.3.9j)
B.250
ZNA
N,,
(see F.3.9k)
B.251
ZEA
U,,
(see F.3.9()
B.252
ZTHERM
NOT
(see F.3.9m)
B.253
ZFUSN
NOT
(see F.3.9n)
B.254
ZLOSS
‘~DT,Ioss
(see F.3.9o)
B.258
ZTFUSE
NOT
(see F.3.9p)
8.259
ZPIALF
(see F.3.9q)
Subroutine HE3
B.260
ZENE(6)
8.261
ZAMU(6)
B.262
ZRHO2S
8.263
ZRHO3S
B.264
ZRHO4S
B.265
Z11
13.266
Z12
nOD,, 1D~He
B.267 B.268
Z13 SHFUS(2,55)
‘
8.269
SALFS(55)
4He
(see eq. 2.9.3.2j)
B.270
ALOSS(55)
5F, a a =3He = S,,”,
(see eq. 2.9.3.2k)
B.271
ZLAMDA
h(x,)
(see eqs. 2.9.3.2e,m)
B.272
WEALFS(55)
Q~’
(see eq. 2.9.3.2~’)
E 01 A,
initial energy and mass number
of the 6 charged fusion reaction products in eqs. (2.9.3.2a—d)
3Hc = exp[ — 27.76/1’,°~~ —31.02 + (cm3 s )2.79 X 10~2~-• (oc) —5.53x1O4T,2 D +3.03x106T,3 —2.52X109T4] (cm3 s1) for T, <80 keY (T, in key) = exp[—52.1 +5.84 log T 25(log 7..)2] for T, > 80 keV 1 —0.5 but = 0 for T 1> 240 keV 3T 095] exp(— 19.3/T3)/1~6(cm3 s~)
+ 5.4 x 103T,°92]exp( — 19.8/73)/T 6 for T, <80 keV (T, in keY) 1 = exp[—46.51+2.561og T 6(log T)2l for T, <80 keV 1 —0.1 but = 0 for T 1> 240 keV
(see eq. 2.9.3.2e) (see (see eq. eq. 2.9.3.21) 2.9.3.2g)
S,~’, a = D, P
(see eqs. 2.9.3.2h,i)
B.273
WIALFS(55)
Q,l~
(see eq. 2.9.3.2m)
B.274
ZX
x~T= Eo,/(A,7[3/4(~ITmp/rnc)1/2~(njZj/Aj)]2/3)
(see eq. 2.9.3.2e)
CE. Singer ciat No.
Code
/
327
One-dimensional plasma transport code
Formula
Cross-refs.
Subroutine PDX B.309
ZIIONM
Th~=—~ ~
n,,/ ~
Pa—1,2,~,h
B.310
ZTNEU
~ noE,,°/ ~
Tj~=
a—1,2
B.311
ZMNEU
ZVO
B.313
ZYI
E,,°(4-eq. 2.9.ld),
fa—1.2
ffi~=rn~~
~A,,/AP I
a—1,2
8.312
A~(’-eq. B.7)
n,,A,,
a—1,2,~’,l,
(ñ~(‘— eq. 2.9.le) (n~(’-eq.2.9.le)
~ a=1,2
A~(‘—eq. B.7) ~~5( 4- eq. 11.310), 4-eq. B.311)
2
= (3T
0s/Th~)l~’
i5~= [2APTI/(AHmP)11”2
A~,(4-eq. B.7),
AH( ‘-eq. B.69) B.314
ZE
v~[(13~)2+(i3~)2I
E5i
A~,(4-eq.B.7), ~o(‘—eq. B.312),
2A~
4-eq. B.313) (0.6937x 10~~)(1 —0.155 log B.315
8.316
ZSIG
=
10E 1 . ~) 1+0.1112X10_14(E~iev)3~3
[32] —
E~icv( ‘-eq. B.314)
(
ZFREQ
2
5
~ a—1,2
)~a:j~i~2 +(ii~)]i~~2
ñ~(‘-eq. 2.9.le), i3~(‘-eq. B.312), ii~(‘-eq. B.313), ‘-eq. 11.315)
Subroutine HEAT B.317 (none)
Wpe(4lTflce2/rna)”2
B.318
w
ZWPIHZ
2/mi)1/2 1,,
B.319
ZWRAT
Subroutine MPRINT B.320 HDDS
=
rn
(4ITn~e
1( ‘-eq. D.2252)
ec[41Tm~n~/(B~ + By)]
W~/Wce
nD(r)r dr 1”2 (E)[2E/(Am)]
NDDn = 8,,o(21r)2f
x fEm=o
dEJ1 f,~Qi,E, r) d~s Emax(
‘--
eq. B.144),
E,,~,(‘-eq. B.145), °D[),,( ‘-eq. B.172), A,,( ‘-eqs. D.1069—1078, D.1401—1410), ‘— eq. B.147) all beams
B.321
ZPOWR
=
~
f 15IW,,/j
b—I
fjb( ‘-
eq. B.127)
j—1,3 eqs. D.1234—1243) ‘-eqs. D.1159—1168)
4-
B.322
ZFUSN
(P,/l7.6
MeV) = NDT + NOT
N~T(4-eq. B.208) N~’T(4-eq. B.252)
328
G.E. Singer eta!.
/
One-dimensional plasma iranspori code
Appendix C. COMMON blocks; listing of common blocks by affiliation with descriptions ALPHAS (alpha particle package) COMBIG COMBNC BALDUR (main calculation) COMALF — alpha particle slowing down variables. COMBAS — basic system parameters. COMBEM — standard neutral beam variables. COMCNS — conservation checks. All COMCNS quantities beginning in A- and F- are computed in subroutine SOLVE and are CHI conservation checks, as in TOTAL!. All quantities beginning in E- and W- and B-poloidal energy check variables and are computed in subroutine SOLVEB, except EBIN!. BEGIN! and EBINI are computed in subroutine START. COMCNV — conversion factors (see subroutine UNITS). COMCOE — transport equation and boundary condition coefficients. COMDDP — diagnostic and development parameters. COMDF2 — plasma quantities computed by TRCOEF or used by TRCOEF. IH is a hydrogen species index, !I and 112 are impurity species indices, J is boundary/zone index, IZ selects boundary-centered value if !Z = 1, zone centered if IZ = 2. COMDIF — diffusion coefficients. All diffusion coefficients are in standard units. In this common block, J is a zone index, !H and 1H2 are hydrogen species indices (density of species IH is parameter IH — LHYD1 + 1 in CHI), II and 112 are impurity species indices (density of impurities species II is parameter II — LIMP1 = 1 in CHI). COMDIM — dimension of variables and important indices (see subroutine AUXVAL). COMEXT — extrapolation variables (see subroutine RESOLV). COMFLG — code flags and variables. COMFUN — fundamental constants. Each fundamental constant is stored as two real numbers, which are the coefficient and exponent of the base-lO floating point representation of the constant. They are stored separately because some constants are close to the largest or smallest number that can be represented on some machines. COMIN — inputted variables (see subroutine AUXVAL). All variables in this common block represent values in external units. COMMSC — miscellaneous constants. COMMSH - mesh variables. COMNEU — standard neutral gas variables. COMOUT — output parameters. COMOVF - overflow variables. COMSTA — arrays describing state of plasma. COMTIM — time dependent conditions. COMTOK — tokamak and plasma parameters. CORONA (coronal radiation package) COMCOR — coronal equilibrium fit coefficients. FOKKER (neutral beam injection) COMFOK — Fokker—Planck neutral beam. FREYA (neutral beam deposition package) COMLOC - input variables for FREYA. COMMCP — local Monte Carlo particle parameters.
G.E. Singer ci a!.
/
One-dimensional plasma transport code
329
neutral atom parameters. COMNUP neutral beam injector parameters. HE3 (D—D fusions) COMTST MONTE (Monte Carlo neutral gas package) COMUSR user interface. COMPRT particle variables. COMSPL — splitting variables. COMXS cross-section variables. PELLET (pellet injection package) COMPEL MISCELLANEOUS NEOION quantities having to do with nearly-exact neoclassical transport model. APOLAR — quantities having to do with reading of toroidal ripple input file [17]. TEMPRY — miscellaneous calculations. COMNUB
— —
— —
—
—
Appendix D. Input variables
Notes: 1. * * or * preceding a variable name means that the variable is in NAMELIST/NURUN1/, $$ or $ means it is in NURUN2 and * * or $$ means it is in RESET. 2. A variable is referred to (“USED IN EQS. *“) a particular equation whenever possible. If it is not used in any particular equation, it is referred to the section to which it applies. 3. Appendix numbers in the cross references refer to the extended user’s manual [17]. 4. Index of NAMELIST input variables. Variables are listed in the format D.xxxx?? NAME, where D.xxxx is the equation number associated with the variable, and ?? is one of the symbols * *, *, $$ or $. * * or * indicates that the variable is in NAMELIST/NURUN1/, $$ or $ that it is in NURUN2 and * * or $$ that it is in RESET. D.2527 D.2534 D.1441—1460 D.1461 D.2351 D.494—993 D.1068 D.2529 D.2530 D.1462-1481 D.1018 D.1482-1483 D.1484—1485 D.1486-1595 D.1596—1597 D.1598—1599 D.1600—1709
$ $$ $$ $$ $$ * *
$$ $ $ $$ * *
$ $ $ $ $ $
AFSLOW APRESR BPO!D(IT) BZ BZSTAR CFUTZ(500) CJBEAM CPVELC CPVION CURENT(IT) DELMAX
DENGAO(IH) DENGA1(IH) DENGAS(IH,J) DENIMO(II) DENIM1(II) DENIMP(!I,J)
D.1710-1729 D.1730—1784 D.1785 D.1786 D.2697-2776 D.2777—2856 D.2617—2696 D.2537-2616 D.2881—2904 D.1787 D.1788 D.1789 D.1790 D.1791 D.1792 D.1793—1794 D.1795—1796
$$ $ $ $ * * * * * * * * * *
$ $$ $$ $ $ $ $ $
DENMON(IT) DENS(J) DENSO DENS1 DFUTZE(N,K) DFUTZI(N,K) DFUTZD(N,K) DFUTZV(N,K) DXSEMI(N,IX) DTINIT DTMAX
DTMIN EBFIT EEBFIT EEFIT EEHFIT(!H) EEIFIT(I!)
330
G.E. Singer ci at
/ One-dimensional plasma transport code
D.1797
$
EETEFT
D.1285—1339
$$
HR(J)
D.1798 D.1799 D.1800-1801 D.1802—1803 D.1804 D.2532
$ $ $ $ $$ $$
EETIFT EFIT EHFIT(H) EIF!T(II) EIONIZ ELECDO
D.1340—1349 D.1350—1359 D.1360 D.1361—1370 D.1371—1380 D.1381—1390
$$ $$ $$ $$ $$ $$
HRMAJ(B) HRMIN(IB) HTCHEK HTOFF(IB) HTON(IB) HWIDTH(IB)
D.2533 D.237 D.1019 D.1805 D.1807
$$
ELECTE
*
ELLIPT ERRMAX
D.13-32 D.1023 D.1043
$
ETEFIT
D.2332
$
ETIFIT
D.1024 D.33
D.994 D.995 D.1808—1847 D.1848—1887 D.1888—1889 D.1890—1891 D.1892—1911 D.1912—1951 D.1952—1971 D.996 D.997 D.998 D.2420 D.999—999a D.1000 D.1001 D.1002—1005 D.1006-1009 D.1069—1078 D.1079—1118 D.1119—1128 D.1129—1138 D.1139—1148 D.1149—1158 D.1159—1168 D.1169-1178 D.1170-1188 D.1189-1198 D.1199—1228 D.1229—1233e D.1234—1243 D.1244-1253 D.1254 D.1255—1264 D.1265—1274 D.1275—1284
* *
* * * *
$$ $$ $ $ $$ $$ $$ * * * * * * * * * * * * * * * * * *
$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$
EXTF EXTZEF FLGAS(IH,IT) FLIMP(!!,IT) FRACTH(IH) FRACTI(I!) GFLMAX(!T) GFRACT(IT,!H) GFTIME(IT) GNCHEK GRECYC GTCHEK GWMIN GXDOME(2) GXMAXE GXMINE GXPH!(4) GXTHET(4) HABEAM(IB) HANGLE(4,IB) HAPER(IB) HAPERV(IB) HD!V(!B) HDIVV(IB) HEBEAM(IB) HEIGHT(IB) HFOCL(B) HFOCLV(IB) HFRACT(IE,IB) HFUTZ(1O) HIBEAM((IB) HLENTH(IB) HNCHEK HPBEAM(IB) HPROF(IB) HPROFV(IB)
D.9 D.2251 D.2418 D.2252—2253 D.1O1O DlOlOa D.1O11 D.1012 D.1013 D.1391-1400 D.1401-1410 D.1411 D.1412 D.1413—1422 D.1423a-j D.1424—1433 D.2417 D.1434 D.2254—2255 D.1O D.1435 D.1436 D.1025 D.1437 D.76 D.1028 D.1 D.1029 D.1014 D.1015 D.2528 D.1016 D.1017 D.77—85 D.86—135
* *
NADUMP(20) NATOMC NBDUMP NBFIT NBOUND NCLASS NDIARY NEDIT NFUSN NGAS(IH) NGPART NGPROF NGSPLT NGXENE NGZONE NHAPER(IB) NHBEAM(IB) NHE NHMU NHPRFV(IB) NHPROF(IB) $$NHSHAP(IB) NHSKIP NHSRC NIMP(II) NIN NIPART NIPROF NITMAX NIZONE NLCHED NLDIFF NLEDGE NLEXTR NLGLIM NLGMON NLGPIN NLGREF NLGSPT
* *
NLHEAD(9)
* *
NLOMT1(50)
* * * * * *
$ * * * * * *
$$ *
$ * * * * * * * * * *
$$ $ $ $ $$ $$ $$ $ $ * *
$$ $$ * *
$$ * * * * * * * * * * * * * * * *
G.E. Singer ci at / One-dimensional plasma transpori code
D.136—185 D.186—235 D.1048—1067 D.1030 D.236 D.2 D.1031 D.1032 D.2367—2416 D.2333 D.2334—2335
D.2336—2337 D.3 D.1026 D.4 D.34—53 D.2338 D.2339 D.2340-2359 D.2419 D.1044 D.54 D.5 D.2360 D.11 D.6 D.7 D.8 D.2361 D.1045 D.489 D.490 D.491 D.492 D.493 D.12 D.1046 D.2362 D.55
D.2363 D.1047 D.2364 D.1027 D.2365 D.1440 D.56—75 D.2366 D.2913
* * * * * * * * * * * * * * * *
$$ $ $ $ * * * * * * * * $$ $$ $$
$$ * * * * * *
$$ * * * * * * * *
$ * * * * * * * * * * *
$$ * *
$ * *
$ * *
$$ $ * *
$ * *
NLOMT2(50) NLOMT3(50) NLPOMT(20) NLRCOM NLREPT NLRES NLSORC NLSORD NLXXX(50) NNFIT NNHFIT(IH) NNIFIT(II) NONLIN NOUNIT NOUT NPDUMP(20) NPEL NPEL2 NPELGA(IT) NPELOU NPLOT NPOINT NPRINT NPUFF NPUNCH NREAD NREC NRESUM NRFIT NRGRAF NRLDMP NRLEEX NRLIEX NLRMOD NRLPAR NRUN NSEDIT NSKIP NSUB NTEFIT NTGRAF NTIFIT NTRANS NTTY NTYPE NVDUMP(20) NZONES QMHD1
D.2914 D.2536 D.1972—2026 D.2535 D.1438 D.1439 D.238—257 D.258—277 D.278—297 D.298—317 D.318 D.319 D.320 D.321 D.322 D.323 D.324-378 D.379 D.380 D.381 D.382 D.383-437 D.438 D.439—488 D.2421—2471 D.2027—2046 D.2047-2066 D.2067—2086 D.2087—2106 D.2915 D.2107 D.2108 D.2109—2128 D.2129 D.2130 D.2131 D.2151—2206 D.2207 D.2208 D.2209—2228 D.2229—2248 D.1020 D.1021 D.2472—2526 D.2249 D.2250 D.2256—2275 D.2276
331 * *
$$ $ $$ $ $ * * * * * * * * * * * * * * * * * * *
$$ $$ $$ $$ $ $$ $$ $$ $$ $$ $$ $ $ $ $$ $$ * * * *
$ $ $ $$ $
QMHD2 QSTAR RADIUS(J) RASTAR RCWALS RDWALS REHE(IT) REHI(IT) REOFF(IT) REON(IT) REX1 REX2 REY1 REY2 RELEO RLE1 RLEPRF(J) RLEPWR RLFREQ RLIO RLI1 RLIPRF(J) RLIPWR RLPARA(50) RLPOWR(50) RMAJOR(IT) RMINOR(IT) RPA(IT) RPELA(IT) RRSTAR SEDIT SPLOT TBPOID(IT) TCOLD TCOLDP TCOMP(IT) TE(J) TEO TEl TEDIT(IT) TGAS(IT) THETA THETAP TI(J) TIO TI1 TIMP(IT) TINIT
332
G.E. Singer ci a!.
D.2277 D.2278—2287 D.2288—2307 D.1022 D.2308—2327 D.2857—2880
$$ $$ $$ * *
$$ * *
TMAX TPELA(IT) TPLOT(IT) TSPARE VPELA(IT) VXSEMI(N,IX)
/
One-dimensional plasma transport code
D.2328—2329 D.2330—2331 D.2905—2908 D.1033—1042 D.2909—2912
$ $ * * * * * *
WTGAS(IH) WTIMP(II) XESEMI(N) XFUTZ(1O) XISEMI(N)
C.E. Singer ci al. No.
Symbol
Code (array size)
/
333
One-dimensional plasma transport code Default (logical value or
Used in eqs no.
Description
units)
OLYMPUS variables D.1
a * NLEDGE
0 (false)
Not used in present version of BALDUR (set by formatted READ input as well as NAMELIST)
D.2
*
*NLRE5
0 (false)
Not used in present version of BALDUR (set by formatted READ input as well as NAMELIST)
D.3
* *
NONLIN
5
Channel to write messages to teletype (do not change)
D.4
* *
NOUT
6
Not used for input (current output thannd; cf. ref. [7])
D.5
*
aNPRINT
6
Channel to
write main output (do not
change)
D.6
**
NREAD
1
D.7
* *
NREC
0
Channel to read main input file (do not change) Not used in present version of BALDUR (set by formatted READ input as well as by NAMELIST)
D.8
a aNRESUM
0
Not used in present version of BALDUR (set by formatted READ input as well as
D.9
* *
NDIARY
6
Channel to write short printout (do not change)
D.10
*
1
Not used for input (current input channel; cf. ref. [7])
D.11
NPUNCH
7
D.12
* *
NRUN
10
D.13 —32
* *
NADUMP(20)
0, 0, 100, 0, 1, 1, 100, 1, 0, 0 0
D.33
* *
NCLASS
0
Not
D.34 —53
* *
NPDUMP(20)
0
Not used
D.54
*
a NPOINT
1
Not used for input (most recent point reported, cf. ref. [7])
D.55
* *
NSUB
1
Not used for input (most recent subprogram reported; cf. ref. [7])
D.56 —75
*
0
Not used
D.76
a*NLCHED
D.77 —85
*
by NAMELIST)
a NIN
*NYDUMP(20)
*NLHEAD(9)
Channel for card output (not used) appendix F
Maximum number of time steps This OLYMPUS array has been modified to be used as an internal storage array and should not be used for input used for input (most recent subprogram class reported; cf. ref. [7])
0 0 (false)
Not used
334
G.E. Singer ci a!.
No.
Symbol
D.86 —135
* *
D.136 —185
* *
D.186
Code (array size)
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
NLOMT1(50)
0 (false)
appendix F
True if subroutine is to be skipped (cf. ref. [7]). If NLOMT1(3) = .T., subroutine PRESET is skipped 5 AUXVAL 6 INITAL 8 START 9 ERRCHK 10 UNITS
NLOMT2(50)
0 (false)
appendix F
True if subroutine is to be skipped (cf ref. [7]). If NLOMT2(1) = .T., subroutine(s) STEPON is skipped 2 COEF 3 NEUGAS (Monte package) 4 BEAMS (Fokker package) 5 HEAT,ICRF 6 IMPRAD 7 BOUNDS 8 SOLVEB 9 REDUCE 10 SOLVE 11 RESOLV 12 GETCHI 13 CMPRES 14 TRCOEF 15 CONVRT 16 CNVCOF 17 DPOSIT (Freya package) 18 XSCALE 19 ALPHAS, HE3 20 EMPIRC 21 PDRLVE (pellet package) 22 NCFLUX
0
appendix F
True if subroutine is to be skipped (cf. ref. [7]). If NLOMT3(1) = .T., subroutine(s) OUTPUT is skipped 2 MPRINT 3 SPRINT 4 GPRINT 5 HPRINT 8 IPRINT 9 APRINT,FPRINT
* *
NLOMT3(50)
—235
D.236
/
(false)
*
*NLREPT
0 (false)
Not used
Ellipticity for modifying beam deposition, numerical ripple input table and semiernpirical model D.237 K aELLIPT I 2.3f Ellipticity 3g 2. 2.le section 2.7 section 2.8
G.E. Singer cia!. No.
Symbol
Code (array size)
/
335
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
Total ERCH power to electrons is pFiC between times t~k and t(,ffk
EGRH heating parameters
D.238 —257
P,~
*
REHE(20)
(W)
2.9.7a
D.258 —277
p,~C
*
REHI(20)
(W)
2.9.7a
D.278 —297
t~j~
*
REOFF(20)
(s)
D.238 —277
Times to turn off ECRH
D.298 —317
t~k
*
REON(20)
(s)
D.238 —277
Times to turn on ECRH
D.318
*
REX1
(cm)
section 2.9.7
D.319 D.320 D.321
*
REX2 REY1 * REY2
(cm) (cm) (cm)
Coordinates of vertices of a rectangular window into which ECRH power is unifonnly distributed: the origin (0.,0.) for these coordinates corresponds to the magnetic axis. REY1 and REY2 are height with respect to toroidal midplane and REX1 and REX2 are distance of sides of window in midplane from magnetic axis
p,,EC =
Total ECRH power to ions is p,Ec between times t~k and i(,ffk
*
p~C
Lower hybrid and arbitrarily specified heating D.322 * RLEO
0
section 2.9.6
Relative heating of electrons at magnetic axis [for heating profiles specified by polynomial fit (‘-eq. D.492)]
D.323
*
RLE1
0
section 2.9.6
Relative heating of electron at r = rWall [for heating profiles specified by polynomial fit (‘-eq. D.492)]
D.324 —378
*
RLEPRF(55)
0
section 2.9.6
Relative heating of electrons at each zone center [for tabular input profiles (‘-eq. D.492)]
D.379
*
RLEPWR
106 (W)
section 2.9.6
Total power to electrons [for NRLMOD = 1 or 2 (i— eq. D.492)]
*
RLFREQ
5 X 10~ (Hz)
section 2.9.6
Frequency of lower hybrid heating source
D.381
*
RLIO
0
section 2.9.6
Relative heating of ions at magnetic axis [for heating profiles specified by polynomial fit]
D.382
*
RLI1
0
section
Relative heating of ions at r = ~ [for heating profiles specified by polynomial fit (e— eq. D.492)]
D.380
w
2.9.6 D.383 —437
*
RLIPRF(55)
D.438
*
RLIPWR
*
RLPARA(50)
D.439 —488
n
11
0
section 2.9.6
Relative heating of ions at each zone center (for tabular input profiles (‘-eq. D.492)]
106 (W)
2.9.6a—b
Total power to ions [for NRLMOD = 1 or 2 or to ions plus electrons [for NRLMOD = 3 (‘-eq. (D.492)]
0
2.9.6a—c, 2.9.6e—f
Index of refraction for mode i [1 NRLPAR ( ‘-eq. D.493)]
i
336 No.
G.E. Singer ci aL Symbol
Code (array
size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
D.489
*NRLDMP
3
section 2.9.6
Switch to choose lower hybrid heating model: 1 -. ion heating only — all energy deposited in radial zone containing lower hybrid resonance point, 2 —+ electron Landau damping only, 3 -. other model
D.490
*
NRLEEX
0
section 2.9.6
D.491
*NRLIEX
Exponent in radial variation of polynomials fit to electron and ion heating: P = RLj1 + (RLjO— RLj1) * [1—(r/r~,,ii)s * NRLJEX], j = E,I NRLEEX = electrons; NRLIEX = ions
D.492
*NRLMOD
3
section 2.9.6
Switch to select: 1-. tabular input, 2 —* polynomial heating profiles, 3 —. lower hybrid heating (cf. eqs. D.322—379, D.381—438) (cf. C~—C 63)
D.493
*
NRLPAR
0
section 2.9.6
Number of specified indices of refraction in lower hybrid wave spectrum (maximum is 50)
Plasma modeling parameters D.494 C~ —993
* *
CFUTZ(500)
Specified values multiplying various transport coefficients, and controlling transport, source models, and numerical methods. In what follows, “x” means “multiplies”: “A —* + B” means “B gets added into A”
C1
0
2.lm
x pseudoclassical contribution to DHH; H =1,2
C3
0
2.ln
x pseudoclassical contribution to D,1 I =
x,, x~
0 0
2.lo 2.lp
x pseudoclassical pseudoclassical contribution to
CC4 5
0
2.lm
x Bohm contribution to Dim; H1,2
C6
0
2.ln
x Bohm contribution to D11 I
C7
0
2.lo
x Bohm contribution to
x,,
C8
0
2.lp
x Bohm contribution to
x~
C9
1
2.5.1 a—d
x simplified neoclassical contribution to DHH, diagonal terms only; H = 1,2
C10
1
2.5.1 a—d
x simplified neoclassical contribution to D11, diagonal terms only; I =
C11
1
2.5.1 a—d
x simplified neoclassical contribution to
C12
1
2.5.1 a—d
x simplified neoclassical contribution to
C3
x~
=
G.E. Singer ci at / No.
Symbol
Code (array size)
One-dimensional plasma transport code Default (logical value or
Used in eqs no.
337
Description
units)
C
13
0
Not used (coefficient for nonimplemented model of trapped electron thermal diffusivity)
Drift wave contribution to
x,,
C14
0
2.lo
x~
C15
0
2.lh
D~+n(~5)[1+Cis(r/r~r)c40];
a = (~‘~) (modifiedby “soft Bohns limit”, cf. C51’) C16 C17 C18
0 0 1s~) (cm
2.1k
0
2.lp
x~ by —~ “soft” +(Cj7/n~)[q(r Bohm limit”,=cf. O)]Cis C
(modified
51) x
~T,
ion thermal gradient contribution to
xi 1
C20
0
C21
0 (cm2/s)
2.lc
DHH—s+C21 H=1,2
C22
0 (cm2/s)
2.ld
D,, —+ + C22, I
C23
0 (cm2/s)
2.le
x~—~+ C23
C24
0 (cm~/s)
2.lf
(225
(cm2/s) 0
2.lh
(226
0 2/s) (cm 0
2.lh
C 27
2.7e—f
x v~i~ (factor multiplying Ware pinch of
C19
hydrogen isotopes and electron energy) Not used in TRCOEF
—‘
+
C2~
1[1 + Gis(r/r,~)C40]; H = 1,2 (modified by “soft Bohm limit”, -~+ C25n cf. C 51) IDJ,~~9+GS6n~~[1+GlS(r/r,cr)C10]; = 1,2 (modified by “soft Bohns limit”, cf. C51). Not used in TRCOEF
2.lf
x. —~+
0 (cm~/s)
2.le
X~~+C29(r/r~,)~
(230
0
2.le
(modified by “soft Bohm limit”, cf. C~~)
C31
0
2.lm
x contribution for q ~ 1 to DHH; H
C32 C53
0 0
2.ln 2.lo
x contribution for q ~ 1 to D11 I = x contribution for q 1 to
C34
0
2.lp
x contribution for q ~ I to
(2~~
0 2/s) (cm 0
2.lc
DHH~+C3s(r/r,or)C36; ified by “soft Bohm limit”,H=1,2 cf. C (mod31)
C28 C 29
C~
0 1s (cm
1)
2.lc
C28/n~
= 1,2
x~ x~
338 No.
G.E. Singer ci a!. Symbol
Code (array
size)
C
37
C38 C39
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
0 1s1) (cm
2.lc
DRH + C37/ne; H =C 1,2 (modified by “soft —~ Bohin limit”, cf.
0 2/s) (cm 0
2.ld
D11 -. + Css(r/r~r)’39; by “soft Bohm limit”, cf.I = C !‘,h (modified 51)
0
51)
2.ld 2.lh
Do,, —~ + (~~)[l + a = (~‘~) (modified by “soft Bohm limit”, cf. C55)
C41
0
7~scales as (compression ratio)C~~ (if C41 * 0?)
C42
1s1) 0 (cm~
2.ld
“soft limit”, Icf. D11 -~Bohm + C42/ne; = C !‘,h (modified by 51)
C43
0 (cm2/s)
2.6a
~
0 (cm2/s) 0
2.6c—d
x.,’ maximum
2.1!’
If
C45
(keY/cm)
maximum allowed value; a
(245>
x~ C~ (247
C48 C49 C50 C51
C52
0 (cm2/s) 0 0 0 0 1010
=
1,2,t’,h
i,e
0., C49
DHH
D11
allowed value; j
=
—. +
C50 [C~ C48
+
C45n~17’-C~,
2.1!’ 2.1!’ 2.le 2.lc 2.ld 2.li
0
(modified by “soft Bohm limit”, cf. C351)
= max(C51,1.), DB is the “soft Bohm limit” applied to various transport coefficients (cf. C55, C17, C25, C26, C29, C30, C35_.~,C42, C57, C58) ~j~B
Reserved for use as maximum allowed value overriding C~
C53
0 (cm2/s)
2.5.2a
DHH=C53 for r>
C54
0 (cm~/s)
2.5.2a
D11=C54 for r> r~,if C54>0.; I=!’,h
C55
0 (cm2/s)
2.5.2a
x~=
2.5.2a—c
If C56> 0., the values of these diffusion coefficients are linearly interpolated in radius between Cs6r~rand r~r. Note that C53—C56 have 0.~ forCno effect if C120 = 0. x~—’~ x~ —. + C 1[158=0. ~(r/r~,)2]~”, (r/r,~,)<57n~ y 0 <1., x~—+Css/n~, (r/r~~)y0ory0>l. and is (y0 C the value of r/r,cr where the C5-, 58 terms are equal)
C56
0
C57
0 1s’) (cm
2.1k’
C58
0 1s1) (cm
2.1k’
c~r if
x~
C53>0.; H=1,2
C55 for r> r,,~r if C55> 0
G.E. Singer ci at No.
Symbol
C
59
Code (array size)
/
One-dimensional plasma transport code
339
Default (logical value or units)
Used in eqs no.
Description
0
2.1k’
(modified by “soft Bohm limit”, cf. C51) If 0. C~<1., then the rf lost power fraction, LRF, is G~ (LRF = 0, otherwise). This is used to calculate Q,,
0
on
C61
0 (s)
section 2.9.6
Time auxiliary heating is turned G~)
(cf.
C62
0
section
Time auxiliary heating is turned off (cf.
C63
(s) 0
2.9.6 section 2.9.6
G~) Int(ç3 + 0.1) = time step after which cxtensive printouts of lower hybrid heating [if activated, cf. C60, C61, C62and NRLMOD (‘-eq. D.492)] are made
CM
0
C65
0
CM
0
C67
0
CM
0
C69
0
C70
0
section 2.9.10
~ 0. — turns off He volume source mediated by C7j—C77, and its associated short printout. Note that even if C.,,~> 0., there is 4He willspecified be no He as volume an impurity sourcespecies unless (‘-eq. D.2254—2255)
71
0
2.9.lOa
(If C71 0., we set C71 = 0.9.) Fraction of helium ions crossing out at r = ~ which is recycled as a volume source (cf. C70)
C.,2
0
2.9.lOa
(If C.,2 0., we set C.,2 = 0.9.) Fraction of helium lost by parallel flow in scrapeoff which is recycled as a volume source (cf. C70)
C73
0
2.9.lOa
(If C73 0., we set C.,3 = 0.9.) Helium neutrals penetrating further than C.,3r~~ are distributed in volume source by normalization (cf. C70)
C74
0 (keY)
2.9.lOb
(If C74 0., we set C74 = 0.1). Electron energy lost per He° ionization (cf. C70)
C.,5
0
2.9.lOa
is set the C75 (unique) angle, (If C75 0He 0., we = 1.) C75 = cosfrom where to angle to circular flux surfaces normal at which He° atoms are launched (cf.
Reserved for use by ion cyclotron resonance heating package
Notused Not used
C
C~~) C 76
0 (keV)
section 2.9.10
(If C76 0., we set C76 = 0.04.) (Unique) energy with He°atoms are launched
340 No.
G.E. Singer ci a!. Symbol
(277
Code (array size)
/ One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
0
section
(If C
2.9.10
77 0., we set C77 = 1.) He° atoms are launched inward from the outer boundary of zone number N +2— C77 (cf. C0)
C78
Not used
C79
Not used
C80
Not used
C81
0
C82
0 (s)
C83
0
5.2.2.5d
Flow bias parameter: If C81 * 0., zone centers for some of the differencing formulas are offset by (v~ i)/( C81 ~ r), where v is the convective velocity Time at which BALDUR will re-read the input file with an abbreviated namelist (see subroutine STEPON)
2.5.2d—e
Sets diffusivities in scrapeoff to a multiple of Bohm value. (C83 0., r~,)~ x~
r>
1.5DB; a =1,2,!’,h; j and C120) =
C84
0
Xe
C85
0
Not used
C86
0
Not used
C87
0
Not used
C88
0
Not used
C89
0 (s)
=
i,e(cf. C51
enhancement factor inside a tearing mode island (not used in present version)
section 3.2 2.lp
Recycling coefficient for hydrogen isotopes is set to toC102 time6-regime C89 if C89 > 0 Contribution x, atfrom model is 1.5C~,JD6(cf. C 91)
0 C91
0
2.1
C91
C92
0
2.1B
Arbitrary multiplicative 6 (cf.factor C included in the definition of D” 91)
C93
0
2.IA
Arbitrary multiplicative factor included in the definition of DKP (cf. C91)
C94
0
2.1C
C95
0
2.1w
Arbitrary multiplicative factor included in the definition of ~ (cf. C91) Arbitrary multiplicative factor included in the definition of DTE (cf. C91)
C96
0
C97
0
2.lm
C98
0
2.ln
0. turns off effects of 6-regime model mediated by C~ — C96
Arbitrary multiplicative factor included in the definition of DTI (cf. C91) x Kadomtsev contribution to DHH; H = 1,2 x Kadomtsev contribution to D11 I =
G.E. Singer ci al. No.
341
code
Default (logical value or units)
Used in eqs no.
Description
0
2.lo
x Kadomtsev contribution to
0
2.9.4a
Fraction of local cyclotron emission included as an electron energy loss
0
section 2.1
2C101 — 1 = the number of averagings done in computing scale9,hea, heights(‘~,,eqs. r~7.1 and magnetic shear B.83—88)
102
0
section 3.2
Recycling coefficient for hydrogen isotopes (— eq. D.497) is reset to C102 at time C89
C103
0
Not used
C1~
0
Not used
C105
0
Not used
C1~
0
Not used
C107
0
Not used
C108
0
Not used
C1~
0
Not used
C110
0
2.4.2a
C110 > 0. activities “nearly exact” neoclassical particle diffusivities between each ionic species and all others, and replaces simplified off-diagonal neoclassical diffuSivities
C111
0
2.4.2a
Factor multiplying classical contribution to “nearly exact” neoclassical particle diffusivities
C112
0
2.4.2a
Factor multiplying banana-plateau contribution to “nearly exact” neoclassical particle diffusivities
C113
0
2.4.2a
Factor multiplying Pfirsch—Schluter contribution to “nearly exact” neoclassical particle diffusivities
C114
0
Not used
C115
0
Not used
C116
0
Not used
C117
0
Not used
C118
0
Not used
C119
0
C120
0
C121
0
Symbol
C
99
C101
Code (array size)
/ One-dimensional plasma transport
x~
C
Not used 2.lh,k’; 2.3h—i; 2.5a—e; 2.8e; 2.9.5e 2.9.5e
C120 > 0. activates scrapeoff modeling. r,~ is defined by the inner boundary of physical zone number (C120 —1). If C120 0., r,,~r is set equal to r~~i1 and C53—C56 and C129—C132 have no effect (If C121 = 0., we set C121 = 100.) “Second scrapeoff” with connection length C128 lies between r~~1 and the inner boundary of real spatial zone number (C121 —1)
342 No.
G.E. Singer ci a!. Symbol
C
Code (array size)
/
One-dimensional plasma transpori code Default (logical value or units)
Used in eqs no.
Description
122
0
C123
0
2.9Sf
Charge-exchange parallel frictionis turned on in scrapeoff if C123 = 1
C124
0
2.9.5a
(If C124 = 0., we set C124 = 1.) Hydrogen isotope loss rate due to parallel flow in scrapeoff is multiplied by C124
C125
0
2.9.5a
(If C125 = 0., we set C125 =1.) Impurity loss rates due to parallel flow in scrapeoff are multiplied by C125
C126
0
C127
0 (cm)
2.9.5e
Connection length for parallel flow in inner-most scrape-off layer
C128
0
2.9.5e
C129
0
Connection length for parallel flow in “second scrapeoff”, cf. C121 Minimum n~in scrapeoff (i.e., n~ C129 is forced by the code; if activated, this forcing nullifies conservation checks). Has no effect if C120 0
C1~
0
section 2.9.10
Minimum T~0~ in scrapeoff (i.e., 77,~,,, C130 is forced by the code; if activated this forcing nullifies E,, conservation check). Has no effect if C120 0
section 2.9.10
Minimum ~ in scrapeoff (i.e., T1~~> C131 is activated, this forcing nullifies E, conservation check). Has no effect if C120
(eV)
C131
0 (eV)
Not used
(If C126 = 0., we set C126 = N + 2). Between times specified by C161 and C162, the outermost radial zone subject to normal time-step zone controls is zone number (C126 —1), for C126 > 2
C132_138
0
Not used
C139
0
C139 = —1 sets simplified off-diagonal neoclassical transport coefficients to zero
C~
0
2.3a
Switch to control effects of toroidal field ripple on ion thermal conductivity. C1~ = 0. -. no effects, 1. 2.
3. 4.
+ X~T+X~’+X~, -.
+
-.
+ +
-
x~”~
5.~.+ X~+
C141
0
2.3c,d,h
xi”.
(If C141 = 0., we set C141 = 12.) Toroidal periodicity of fixed component of ripple
G.E. Singer ci at No.
Symbol
Code (array size)
/
One-dimensional plasma transport code
343
Default (logical value or units)
Used in eqs no.
Description
2.3h 2.3h 2.3h 2.3h
&r~d(r,
C145
0 0 0 0
CIM
0
2.3c,d,i
Toroidal periodicity of time varying component of ripple (cf. eqs. D.1451—1460, D.1472—1481 and D.2119—2128). C1~ 0, turns off time-varying ripple component
0
2.3i
(If C147 = 0., we set C14., = 2.) Exponent in radial variation of time-varying ripple; analogous to C1M above (cf. CIM)
0
section 2.3
C~> 0. —e use table for input of fixed ripple contours (cf. section 7.5 and appendix G of ref. [33])
C149
0
2.3b
C149 >1. —# use Goldston—Towner formula (‘-eq. B.96) [21] for coefficient G when computing X~T for fixed ripple (cf.
C150
0 (s)
t~151_160 C161
0 0 (s) 0 (s)
section 6
Not used in present version of BALDUR. (If C162 = 0., we set C162 =10~.) Certain radial zones, as specified by C126, C163 and CIM, are excluded from normal time-step controls at times C161 i C16~
C163 CIM
0
section 6
Between times specified by C161 and C162, radial zones C163 j C~ are excluded from normal timestep controls
Nc,,
If
0s
C142 C143 CIM
~
C147
C145
0., 9,
we set C145 = 0.2.) sin(C ~)[C142 +2)(C143 — C142) (r/?~,cr)C144)[exp(_ C1459 141~)] (ignored if GIM> 0.). [C142and C143 represent the fractional peak-average fixed ripple at r = 0 and r = r,cr, respectively; i.e,. unlike the input file whose reading is triggered by C148 > 0., C142 and C143 are not given in percent] =
Csse)
C162
(If C150 = 0., we 5~tC150 = 5(L~tmax) (4eq. D.1788). Time between successive computations of coefficient G for fixed ripple (‘-eqs. B.96—97; cf. C~49)
C16519.,
0
C198
0 (keY)
section 2.9.1
Not used in present version of BALDUR If C198 > 0., TCOLD (‘-eq. D.2129) is continually made the minimum of (1) the ion temperature at a radial point determined by C198 times the zone index of the outermost non-scrape-off zone, and (2) its originally prescribed value. Note that TCOLDP (‘-eq. D.2130) is not affected by C198
C1~
0
section 2.9.4
If C1~= 1.(2.) peaks in the impurity radiation losses are smoothed by a binomialweighted smoothing performed over 5(7) adjoining radial zones
344 No.
C.E. Singer ci al. Symbol
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
C
0.
2®
Used in eqs no.
Description
section 2.9.13
If C2® 0., prescribed impurity influxes (‘— eqs. D.1848—1887) are considered ionized. Setting C~®>0. activates the neutral impurity influx model, which is mediated by C201216 and C220_226 in conjunction with N~,IMP (‘- eqs. D.2254—2255), t] (‘-eqs. D.2256—2275) and F1~(~— eqs. D.1848—1887). However, only impurity species numbers 1 and 2 are allowed 0 and to NbemP). influxed If C as neutrals, (i.e., N1” 2’ 2® = 1., 1., F1~, prescribes neutral influxes of 2s1). impurity If C species q at times tJ; in (cm 2® = 2., F1,, prescribes the total number (integrated over the plasma volume) of first and second impurity species a ions desired at times 1]; the influx is adjusted (sometimes overstability) to attempt to meet these targets. The third and fourth impurity species are always ionized and treated as for C2® = 0 (cf. C225226). Note effect of C205 and C2®
C201
0 (keY)
section 2.9.13
(If C201 = 0., we set C201 = 0.01.) Prescribed incoming kinetic energy for neutrals of first impurity species species (cf. C2®)
C202
0 (keY)
section 2.9.13
(If C202 = 0., we set C202 = 0.01.) Same as C201, but for second impurity species (cf. C2®). Note: neutral impurity influx can only be prescribed for the first two impurity species
C203
0 (keY)
2.9.13d
(If C203 = 0., we set C203 = 0.0136.) Electron energy loss per initial ionization of a neutral atom of first impurity species (cf. C200)
C2®
0 (keY)
2.9.13d
(If C2® = 0., we set C2® = 0.0136). Same as C203, but for second impurity species (cf. C2®) [If C
C205
0 (keY)
2.9.l3g
0., we used to set C205 = 10 (in before BALDP27M).] Extra radiation loss due to nonequilibrium per atom of first impurity species (cf. C2®) 205
=
versions
C2®
0 (keY)
2.9.13g section
C20.,
0
section 2.9.13
versions as [If C2® before 0., we BALDP27M).] used to set C2®Same = 10 (in C 2®, but for second impurity species (cf. C2®) (If C207 = 0., we set C207 =1.) N +2— C2® = index of outermost radial zone allowing ionization of neutrals of first impurity neutrals of first impurity species (cf. C2®)
C.E. Singer et at / One-dimensional plasma transport code No.
Symbol
Code (array size)
345
Default (logical value or units)
Used in eqs no.
Description
208
0
section 2.9.13
(If C208 = 0., we set C208 = 1.) Same as C207, but for second impurity species (cf. C2®)
C2®
0
section 2.9.13
(If C2® = 0., we set C209 = 0.9.) Prescribed maximum penetration depth for neutrals of first impurity species, expressed as a fraction of r,cr (cf. C120, C2®). Neutral impurity density is set to zero beyond this depth
C210
0
section 2.9.13
(If C210 = 0, we set C210 = 0.9.) Same as C209, but for second impurity species (cf. C2®)
C211
0
section 2.9.13
(If C211 = 0., we set C211 = 1.) Prescribed cosine of the angle between the inwarddirected normal to the outer plasma
C
surface and the influx direction of neu-
trals of first impurity species, Used in a rough approximation of controlled gas feed. Limited so that 0.5 C211 1.0 (cf. C2®) C212
0
section 2.9.13
(If C212 = 0., we set C212 =1.) Same as C211, but for second impurity species (cf. C2®)
C213
0
section 2.9.13
Setting (2213 1. activates an adjustment of the ionization rates determined by an empirical formula so as to approximate within 15% values of Freeman and Jones [40] (cf. C2®)
C214
0 (s)
section 2.9.13
If G2®> 0., impurity ion source rates (before and after normalization with respect to the neutral influx) and nonequilibrium radiation enhancement factors are printed out for three time steps starting at time C214. If C217> 0., average hydrogenic flux velocity and any enhanced drag on impurities are sinularly printed out
C215
0
section 2.9.13
(If C215 = 0., we set C215 = 1.) Multiplies the ionization rates for neutrals of first impurity species (cf. C2®)
C216
0
section 2.9.13
(If C216 = 0., we set C216 = 1.) Same as (.2215, but for second impurity species (cf. C2®)
C217
0
section 2.9.13
Proportionality factor for enhanced drag on impurities attributable to hydrogenic. ion fluxes. The extra drag gives rise to an increment ~ in the flux of impurity species a ions: ~F,, = (C217 — n,,) (flux velocity selected by C214). The enhanced drag affects only the region selected by C219. If C217 0., this enhanced drag is not included
346 No.
G.E. Singer ci al. Symbol
C
218
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units) 0
Used in eqs no.
Description
section 2.9.13
If C218 = 0., enhanced drag on impurity ions is proportional to flux velocity difference between hydrogemc and impurity ions. If C218> 0., enhanced drag is proportional solely to the hydrogenic flux velocity (cf. C217) (If C219 = 0., we set C219 = 0.2) Enhanced drag is active only for radii > (1— C2i9)r,,,~ (cf. C120, C217)
C219
0
section 2.9.13
C220
0
section 2.9.13
(If C220 = 0., we set C220 = 1.) Adjustment factor for influx rate of first impurity species when using model involved by setting (.2209 = 2 (cf. C2®)
C221
0
section 2.9.13
(If C221 = 0., we set C221 =1.) Same as C220, but for second impurity species (cf. C2®)
C222
0 2s1) (cm
2.9.13 section
mum recycled neutral influx of first im(If C222 = 0., we set C222 = 1017.) Maxspurity species (cf. C 2®)
C223
0 (cm —2 s — 1)
section 2.9.13
(If C223 = 0., we set C223 = 1017.) Same as C222, but for second impurity species (cf. C2®)
C224
0
section 2.9.13
Not used (dummy switch available for future used by neutral impurity model; cf. C2®)
C225
0
2.9.13g
(If C225 = 0., we set C225 = 10.) Maximum radiation enhancement factor with respect to coronal equilibrium for first impurity species (cf. C2®)
C226
0
2.9.13g
(If C226 = 0., we set C226 = 10.) Same as C225, but for second impurity species (cf. C2®)
C227
i0~~ (eV)
Prescribed maximum ion temperature (cv) desired at the plasma edge (JSEPX1)
C228
0.90
Factor lowering the prescribed neutralimpurity influx if T,(JSEPX1) is too cold
C229
1.10
Factor raising the prescribed neutralimpurity influx if ~‘1 (JSEPX1) is too hot
C2se
0
section 2.9.11
(If C231 through C250 are all 0., we set C2~= 0.) C230 = 0. —‘ pellets are injected singly. C230 = 1. —. pellets are injected in multiple-pellets clusters. The time interval between the firings of successive pellets within a cluster is the same for all clusters (cf. C231). C2~= 2. —‘ pellets are injected in multiple-pellet clusters. The time interval between the firings of successive pellets within a cluster must be specified individually for each cluster (cf. C231250) (4- eqs. D.2278—2287)
C.E. Singer ci at No.
Symbol
Code (array size)
/
One-dimensional plasma transport code Default (logical value or
347
Used in eqs no.
Description
section 2.9.11
If C230 = 1., C231 is the time interval between the firings of successive pellets within a cluster for all clusters. If C230 = 1., C230.~1is the time interval between the firings of successive pellets within the jth
units)
C
231
0 (s) 0 (s)
cluster for j > 1 C251
0
section 2.9.11
(If C251 = 0., we set C251 = 50.) Maximum deposition enhancement factor employed when intervals between successive pellet firings within a cluster are less than the current time step (cf. (2230)
C252
0
section 2.9.11
(If C252 = 0., we set C252 = 100.) Maximum allowed number of consecutive firings per pellet cluster (cf. C230)
C253
0
section 2.9.11
C253 = 0., —, Send a message to the teletype every time a pellet cluster is injected unless NTTY = 0 (— eq. D.2365). Also do a short printout everyNPELOUth time a cluster is injected (~eq. D.2419). C253 > 0. suppress these outputs
C254
0
section 2.9.11
~‘255
0
section 2.9.11
The fraction of the pellet cloud which is ionized hydrogen. Values greater than 0.1 suppress ablation by fast ion and alphas The fraction of the calculated pellet partide source which is added to the plasma density (the rest vanishes)
C256
0
section 2.9.11
C257_230 C281
The fraction of the pellet cloud which is neutral hydrogen. C254 and C256 do not need to add to 1 Not used
0
Switch for
x~0
—‘
x~”~ 1-. XFH,
2 —e
(~*1 contributions to xF’ and x~’are computed and then added [18]) ~.
C282
0
Fraction of minor radius that magnetic axis is shifted outward for computing xFH (= R~ of ref. [27]). Maximum value = 0.95
C283_289
0
Not used
C2®..292
0
2.lg’
x18=f,= 1, r/aC291, (r/a) 1+(C2® 1) 292
C291 C2®,
<
r/a
—
—
C292,
C292
C291 291
348 No.
CE. Singer ci at Symbol
C
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
293
0
2.lf
X15
IX1~=C293+(r/LTC)
C209
0
2.lf
X19’
X~I=C294+(r/LTC)
C295
0
2.ld
eA~l9 —e eA~l9+ v~./(C295 A = D, i, e, v for all n
C296
0
2.lh +
+
QMHDI q—q~, ~MHD2
~
+ C 296
C297326
0
Not used
C327_329
0
If C327 * 0, ignore coronal equilibrium calculation of impurity radiation. When C327 > 0., the total radiated power becomes C327 times total heating power. When C327> 0., the radiated power density is proportional to =
C328 +(1 — C32s)[r/(r~,5iC329)], r < rwajjC329, 2 0.25 025 [1-(r/rweii) J /[1—C~9] r r~~iC329 C330.
Not used
3’U
C~5
0
2.11’
If C345, C3~and C347> 0.,
2.11’
DHH D1, CX~ 349 — C350
(cm 1~—1)
C3M C347
0 (keY) 0
2.11’
1 C345n [~+ (~kev/C3~)~’I1/C~7
~348
C~8 CM9 C350 C351
0 0 0 1010
2.lc 2.ld 2.le
(modified by “soft Bohm limit”, cf. C351)
2.11”
~B = max(C351,1.) is the “soft Bohm limit” applied to the so-called PLT empirical transport models mediated by C4550 and C345350 Not used
C352_353
0
C354
0
2.4.ld’
Factor multiplying the anomalous hydrogen mixing coefficient which scales like neoclassical coefficients
0
2.11’
D,,,, -. + DPDX = C3SSr~6(BOT/ a =1,2,1,h Not used in present version of BALDUR
2.le
Model: time-averaging parameter for semi-empirical Xe,,, = C3MXe,e,(iJ_l)+(l —
C358363
0
C~
0
C.E. Singer ci at No.
Symbol
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
349
Used in eqs no.
Description
2.7f
C
2.1J 2.1K 2.1K
Not used QSTAR (alternate input form ~— D.2913) QMHD1 (alternate input form e— D.2914) QMHD2 (alternate input form 4- D.2915)
CM5
0
C309_376 C37., C378 C379
0
C380
0
2.9.5i
Effective neutral path length from divertor
C381
0.1
2.9.5i
“Leakage” fraction of divertor reflux which does not contribute to divertor throat blocking (sometimes called ~
2.9.5m
Absolute value of energy loss per ionization in divertor chamber. C382 > 0 energy gain, so default value should not normally be used
C382
40(eV)
365 0 includes additional electron energy pinch [13] and omits the convective energy flow controlled by CPVELC (~— eq. D.2529)
C383
1
2.9.5m
Ion energy-per ion through divertor sheath
C384
1
2.9.5m
Electron energy-per ion through divertor sheath
2.9.5m
Neutral energy in divertor chamber
C385
10(eV)
C386
0
2.9.5n
Heat flux limit factor for parallel electron heat conduction loss
C38.,
0
2.9.5n
C388
0
2.9.5p
Determines electron temperatures used in computing heat flux limit Determines temperature used in computing s~
C3®
0
2.8e
C389,391..309
0
Changes scrape-off resistivity. (C3® = 1 gives problems near r = Not used
section 3.1
If C395of>the tance 0, plasma the effective is C external induc395
0
section 3.1
Switch to determine the type of boundary condition for the poloidal field diffusion equation 0—e total current or B0(a) specified, 4’eo~ specified 21 —e —e surface surface voltage voltagefrom calculated implicitly from specified ~ and ~
0
section 5.2.2
If C397 > 0., then the the B0e,~ diffusion min(C397,1)
C395 ~ C309
02/cm) (s
C 39.,
C398
0
°CN
eq.
Not used in present version
used
to solve (5.2.2a) is
350 No.
C.E. Singer ci at Symbol
C
309
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
1
2.8c
Multiplies resistivity in current diffusion
5.2.2a
Tlb’ equation but not in ohmic heating; s~ = C309s18 the B~,,,[ ‘-eq. (2.8c)]. Note: since si *
1energy conservation may be poor C4®
0 0
Not used 2.8b
If C,,®> 0.,
1~his
C~ 1times Spitzer s~
if C~1 0., ~ is neoclassical s~ C~2— C~®= sawtooth model with magnetic insulation at q = 1 between sawtooth disruptions (C~2, C,,.~3, C~5);enhanced transport during disruptions (C404, C,1®- C,~®)
C,~2_4®
C~~2~13
DflH+C,~2DHH, XcCse3Xe (only on outermost computation zone which contains a q = 1 surface) when d In Pth/d r < C~5,where Pth n,,T~+ n1T1
C4®
Disruption trigger; sawtooth disruption on when din p,h/dr> C4®
C~5
Disruption off when d In Pth/d r < C~5
C4,®
DHH —e C4®DHH for q <1 when d ln p,8/dr> C4® (unless C4® = 0)
C,~7
DHH —e C~7D~11for q > 1 when d ln p,h/dr> C4® (unless C,~7= 0)
C~8 >
C4® >
C410
1
-. ~ for q <1 when d ln p,h/dr C4® (unless C~8= 0) —e ~ for q >1 when d ln p~h/dr C4® (unless G~9= 0)
Multiplier on electron—ion equilibration rate
C415450
Not used in present version eV/cm3))C452 of BALDUR 16
C451455
x~ C4Sl?i~a~~”(plh/(10 X = R~4”I~’, where Pth =
(n,,7,,v +
n ~ Not used C~
0
No sawteeth may occur before
t=
C 4®
(s)
C~1
0
No sawteeth may occur after
t
C~1
(s)
C~2
(s)
0
Sawtooth period for t C,~5_~7
C~ overrides
C~3
0 (s)
Sawtooth period for given by C462 if I C463, by C~ otherwise
C4®
0
Sawtooth period for i> C463 overrides
(s)
C465_
C.E. Singer ci at No.
Symbol
Code (array size)
C
465467
C468
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
.
351
Description
0
If C462 = 0 = C4®, the sawtooth period is given by C46STrXT Y where x = C4®, y = C467
0
Determines what quantities are mixed by the sawtooth model; if C468 0 all the following are mixed. If C468 is exactly divisible by 2 then electron energy is mixed, by 3 then ion energy, by 5 then hydrogen density, by 7 then impurity density, by 11 then fast ion density, by 13 then alpha particle density (if NTYPE = 1).
C469
Not used in present version of BALDUR
C470_474
0
Xe_ + XTFCD [C472+ (r/r,~,) C473] /C~2
C471
>
0,
+ [C471I/ne[C472 +(r/r,cr)C4h3] /C~2,
C471O,
where C4’~2 = C472 +O.5~473, 2] C47i(rm/I~)exp[($t/$~) 2] by 1 47e.0.5) (replace if C exp[(Pt/~m) 470 <0)
XTFCD /.t(.(C=
2)K”2r~~n
$~
/(2 Rq( r~~)), film = C470(1 + K
I~ = 2.5(1 + K2)amBT /[(R /a)Kq(r~~~)], rm= rear/100, am= r~~i/100, BT = Bz/104, K( ‘-eq. D.237) C 475,476,479_5®
Not used in present version of BALDUR
C47.,
0
C478
0
Plasma model, time step control and section 2.8 neutral gas parameters D.994 F,”~,, * *EXTF 1
D.995
z:~
D.996 D.997
B.64
1,,,+ ~ n~(Z>~] Zett = [(ni + n2)I~, a e,h /[(n 1 + n2)F~,,.+ ~b + ~ ne,
EXTZEF
1
~GNCHEK
0
Not used
GRECYC
1
Fraction of ions or neutral leaving outermost zone which are recycled as neutral gas (cf. eq. C®2)
* * *
GRECYC
2.8d
If C477> 0, then sawtooth mixing occurs when q(0) C477. Overrides sawtooth period specification C462, C4®..467 If C478> 0, the resistivity used in eq. (5.2.2a) is determined by eq. (2.8d). Since ~ * fl~,in this case the B~1energy conservation may be poor
* *
+ (Z~—1.)
G.E. Singer ci al.
352
No.
Symbol
/ One-dimensional plasma transport code
Code
Default
Used in
(array
(logical
eqs no.
size)
value or units) 0
Description
D.998
s * GTCHEK
Not used
D.999 —999a D.999 D.999a
* *
D.1000
*
*GXMAXE
io~ (eV)
Maximum energy for neutral outfiux spectrum ( ‘-eq. D.1067)
D.1001
*
*GXMINE
1 (eV)
Maximum energy for neutral outfiux spectrum (‘-eq. D.1067)
D.1002 —1003 D.1002 D.1003
*
*GXPHI(4)
(deg)
Defines azimuthal angle bins used in computation of neutral outfiux spectrum
GXDOME(2)
D.999—D.1009 control neutral outfiux spectrum calculation not implemented in in the present version of BALDUR. Ratio to ion temperature of minimum (‘-eq. D.999) and maximum ( ‘-eq. D.999a) energies used in least squares temperature calculation based on neutral outfiux spectrum ( ‘-eq. D.1067)
4 10
0 15
D.1004
30
D.1005
60
D.1006 —1009 D.1007 D.1008 D.1009
*
*GXTHET(4)
(deg) 0 15 30 60
D.1010
Defines bins for angle between neutral outfiux detector and normal to the wall, used in computation of neutral outfiux spectrum
* *
NGPART
500
section 2.9.1
Number of particles in Monte Carlo cornputation of each source of neutral gas
D.lOlOa
* *
NGPROF
10
section 2.9.1
Neutral profile every NGPROF time steps
D.1011
* *
NGSPLT
5
section
Number of splitting surfaces for neutral gas (cf. section 2.9.1); must be 10
2.9.1
D.1012
*
*NGXENE
31
D.1013
*
*NGZONE
20
Number of energy bins for neutral outflux spectrum printout (~eq. D.1067); must be 100. Not usable in present version of BALDUR section 2.9.1
A minimum for the number of spatial zones in neutral gas computation; NGZONE>
24
-~
we set NGZONE
=
24; [NOTE: NGZONE> NZONES -. NGZONE < min(NZONES,24) (‘-eq. D.2366)] D.1014
*
*NLGLIM
0 (False)
section 2.9.1
.TRUE. —e Include reflection off limiter; ignore TCOLD (‘- eq. D.2129) and NLGMON (‘-eq. D.1015) for recycling neutrals, and launch these neutrals with a Ma.xwellian energy distribution at the ion temperature of the outer dummy zone. Note that gas puffing neutrals are not affected by NLGLIM
G.E. Singer ci at No.
Symbol
Code (array
size)
D.1015
* *
NLGMON
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
—1 (True)
section 2.9.1
TRUE. -e Monoenergetic, (.FALSE. — Maxwellian) recycled neutrals with energy (temperature) TCOLD and gas puff neutrals with energy (temperature) TCOLDP (‘- eqs. D.2129 and D.2130); NLGLIM (‘-eq. D.1014)
D.1016
*
D.1017
*
*NLGREF
*NLG5PT
353
cf.
0 (False)
section
0 (False)
section 2.9.1
TRUE. —e Sputtering of iron by neutrals crossing out over r = r~~ 1 is computed. [Sputtered iron is added to impurities as an influx analogous to FLIMP (‘-eqs. D.1848—1887) if NIMP(1) = 26 or NIMP(2) = 26 (‘-eqs. D.2254—2255)]
2.9.1
.TRUE. —, Neutrals lose energy due to wall reflection; assumes a stainless steel wall
D.1018
8,,,,,
* *
DELMAX
0.1
6g
Maximum relative change allowed in any zone over one time step without reducing the time step (cf. eq. (6b)]
D.1019
~m~e,
* *
ERRMAX
0.01
6g
9CN
* * THETA
1
section 5.2.1
Maximum allowed extrapolated error in any zone [whenextrapolation is used; cf. eq. 6c and NLEXTR ( ‘- eq. D.1029)] Degree of implicitness in Crank—Nicholson differencing schemes. Fully implicit —e °CN 1
O~,
*
0
section 5.2.4
D.1020 D.1021
D.1022
*THETAP
* *
TSPARE
60
Degree of implicitness for iteration in predictor—corrector scheme. Fully corrected —. = 1 Not used in present version of BALDUR
(s)
D.1023
* *
D.1024
*
D.1025
NATOMC
2
2d’ 2.9.4a
*NBOUND
1
section 3
* *
NITMAX
25
section 6
D.1026
* *
NOUNIT
22
D.1027
*
*NTRANS
1
D.1028
* *
NLDIFF
—1 (True)
2 -~eCoronal equilibrium radiation [26,27], 1—e bremsstrahlung only, impurities assumed fully ionized, 0 —e neither bremsstrahlung nor line radiation. Type of boundary condition. Note that impurity influxes (~— C 2®, eqs. D.1848— 1887, D.2256—2275) require NBOUND = 2 or 3 Maximum number of iterations in one time step. If this maximum is exceeded, run is aborted Unit to read impurity radiation ionization (do not change)
section 2.1
If NTRANS <1 —e all diffusion coefficients are set to zero
appendix A
.FALSE. sets A = B = 0
354 No.
G.E. Singer ci at Symbol
D.1029
Code (array size)
*
D.1030
*
*NLEXTR
*NLRCOM
/ One-dimensionalplasma transport code Default (logical value or units)
Used in eqs no.
Description
TRUE. if extrapolation is to be done
—1
section
(True)
5.2.4 of ref. [33]
—1 (True)
sections 2.9.1—
TRUE. if recombination is to be ineluded
2.9.12
D.1031
* *
D.1032
NLSORC
* *
NLSORD
* *
XFUTZ(10)
—1
appendix
(True)
A
—1 (True)
appendix A
FALSE. sets C = 0 FALSE. sets D
0
D.1033 —1042
x~
D.1033
x
1
0.5
6g
Damper on rate of increase of time step; 0. —e no damper, 1. -+ no increase
D.1034
x2
1.5
6g
Maximum rate of increase of time step
D.1035
x3
2
section 6
Maximum on value of iteration or extrapolation error, which can be used to control time step ( ‘-eq. D.1018)
D.1036
.r4
2
section 6
Maximum on value of relative extrapolation error, which can be used to control time step. [~ eqs. (6.6), D.1018]
D.1037
x5
0.333
section 6
D.1038
x6
0.333
section 6
Time step reduction when negative values of dependent variables have been computed * 0. —. override relative extrapolation error method of decreasing time step, and decrease time step when necessary by x6
D.1039 —1042
x7 — x10
0
Not used
*NBDUMp
0
Not used
NPLOT
10
Every NPLOT time steps generate profile
3
plots. Units to write out information for
Output controls D.1043 D.1044 D.1045
*
* *
* * NRGRAF
Extrapolation and time step control factors
plots vs. radius and 3-D plots (do not change) D.1046
* *
NSEDIT
1
D.1047
* *
NTGRAF
4
D.1048 —1067
NLP
*
*NLPOMT(20)
Every NSEDIT time steps generate small edit appendix F
Unit to write out information for plots vs. time (do not change)
appendix F
Controls which pages are printed in long edits. Note that the neutral gas neutral beam, and alpha printouts are automatically omitted when these packages are not active, regardless of how the .N~LP are set
G.E. Singer et at No.
Symbol
D.1048
NILP
D.1049
N
D.1050
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
355
Description
0 (False) 0 (False)
.TRUE. —e omit page 1 of long edits [1(r), T,(r) profile output] TRUE. —~ omit page 2 of long edits [n,,(r), a =1,2,1,h, output]
Nt”
0 (False)
.TRUE. —e omit page 3 of long edits (energy balance output)
D.1051
N 1-” 4
0 (False)
.TRUE. —e omit page 4 of long edits (output of confinement times, profiles averages, etc.)
D.1052
N —1 (True)
.TRUE. —e omit printout of diffusion coefficients vs. radius
1-” 2
5LI’ D.1053
~
—1 (True)
.TRUE. —e omit printout of zone-centering parameters
D.1054
~
—1 (True)
TRUE. —e omit printout of ripple diffusivities
D.1055
N~-~
—1 (True)
1-”
D.1056
N9
0 (False)
.TRUE. —e omit printout of neutral beam distribution in energy and pitch angle in each zone, and fits to the distribution Not used in present version of BALDUR
D.1057
Nh”
—1 (True)
TRUE. —e omit printout of sawtooth quantities
D.1058
Nh”
0 (False)
.TRUE. —e omit first half of neutrals printout (temperatures, densities, chargeexchange losses)
D.1059
N
0 (False)
.TRUE. —e omit second half of neutrals printout (neutral particle source rates)
1’~”
D.1060
~
—1 (True)
.TRUE. — omit printout of neutral beam energy levels and densities at each energy level
D.1061
Nh”
0 (False)
.TRUE. —e omit profiles (average energy per particles, particle source rate, etc.) of main neutral beam printout
D.1062
Nh”
0 (False)
TRUE. —e omit summary at bottom main neutral beam printout
D.1063
~
—1 (True)
TRUE. —e omit printout
D.1064
N1~.,”
0 (False)
.TRUE. -. omit printout of H(r) for 1/1, 1/2, 1/3 energy components of neutral beams
D.1065
~
0 (False)
.TRUE. —e omit printout of total H(r), charge-exchange rates, etc. (neutral beams)
D.1066
Nh”
0 (False)
.TRUE. —e omit alpha particle or D-D fusion printouts
D.1067
Nh”
-1 (True)
.TRUE. -. omit neutral outfiux spectrum printout. Not usable in present version of BALDUR
of
beam driven current
No.
Symbol
Neutral beam parameters D.1068
Code (array size)
Default (logical value or units)
$$CJBEAM
0
Used in eqs no.
2.9.2.2e
Description
Factor multiplying beam-driven current in poloidal magnetic field equation
D.1069 —1078
Ab
D.1079 —1118
6~,,,
$HABEAM(10)
0
2.9.2.2c
(am)
81b
$$HANGLE(4,10)
0 (deg)
section 2.9.2.1
0
92b
(deg) 0
83b
= 0, we set Ab according to the default for the injected species ( ‘-eqs. B.31—35). Note: the present version of BALDUR requires the same value of A,, for all b Angles defining orientation of the line around which the distribution of beam neutrals has symmetry (called the “beamline” below). 8~,,is the poloidal angle of pivot point (‘-eqs. D.1340—1359); posi-
tive
(deg)
~1l,
is below the midplane (cf. fig. 4
of ref. [6].
0
84b
Mass of beam particles. If HABEAM(b)
82,,
is the angle between
beamline and the midplane of the torus; not °2 of fig. 4 of ref. [6].93b is what is usually called the “injection angle”. 03f, is
(deg)
the angle between the beamline (as projected onto the midplane of the torus) and the major radius vector at the pivot point; 0 is dead perpendicular injection (‘-eqs. D.1340—1359); ef. fig. 3 of ref. [6].84~,is
the angle between the vertical axis of the injector face and a line which lies in the plane of the injector face and is parallel to the torus centerline D.1119 —1128
wr°
D.1129 —1138
h~°”
D.1139 —1148
~
$$HAPER(10)
*
iO~
section
Width [diameter for Nt7~I =1
(cm)
2.9.2.1
D.1391—1400)1 of aperture normal to the beamline at pivot point (‘-eqs. D.1340—
(‘-eq.
IO~
section
(cm)
2.9.2.1
0
section
1359)
$$HAPERV(10)
$$HDIV(10)
*
*
(deg)
2.9.2.1
Height of aperture normal to the beamline at pivot point (‘- eq. D.1340—1351) ignored if N,f°’t= 1 ( ‘-eqs. D.1391— 1400) Beam divergence: probability of each partide 0H
leaving injector face at an angle
( 8v) in the Horizontal (Vertical) direc-
tion from the beamline is D.1149
~v
$$HDIVV(10)
W,,
$$HEBEAM(10)
0 (deg)
—1158
D.1159 —1168
*
0
(keV)
P(8H,8v)~exp([(OH/~H)2 +(9~/~~)2]/2+
section 2.9.2.1
Energy of full energy component of injected neutrals. If W’~,= 0., beam b is never turned on (‘-eqs. 3.1371—1380)
D.1169
h?~
$$HEIGHT(10)
*
—1178
D.1179 —1188 D.1189 —1198
*
$$HFOCL(10)
*
$$HFOCLV(10)
Present version of BALDUR requires b energy components, i.e., only 2 different
*
section
Height of injector face (ignored if ~
(cm)
0
2.9.2.1
1)
1010
section
Horizontal focal length of beam (cf. top
(cm)
2.9.2.1 section 2.9.2.1
view of fig. 2 of ref. [6]) Vertical focal length of beam (cf. side view in fig. 2 of ref. [6])
1010
(cm)
=
6. Present version of BALDUR also requires that there be no more than 6 distinct may be active at any time in the simulation.
Wb
No.
Symbol
Code (array size)
Default (logical value or units)
D.1199 —1228
HJ~R
$$HFRACT(3,10)
*
H~ H~,R
H~
Description
section 2.9.2.1
Current fractions of neutral atoms at full, half and one-third energy. Full energy fraction; Half energy fraction; One-third energy fraction
1 0 0
HII~R
D.1229
Used in eqs no.
$$HFUTZ(10)
1
“Approximation factors” for beam treatment
—1233e D.1229
H
1
1
2.9.2.2a
D.1230
H2
1
2.9.2.2e
D.1231
H3
0
2.9.2.2d
Factor multiplying correction for scattering of fast ions into banana-trapped orbits (for current drive)
D.1232
H4
0
2.9.2.2d
Factor multiplying averaging of parallel velocity Over fast ion orbit (for current drive). If H4 0., there is no averaging
D.1233
H5
1
section 2.9.2.2b
Factor multiplying beam slowing-down rate
D.1233a H6—H10 D.1234 —1243
of beam ions Factor multiplying electron collisionaiity in collisional toroidal electron current driven beams
Not used
$$HIBEAM(10)
**
0 (kA)
section 2.9.2.1
Total atomic current injected by a neutral beam (one atom counts as one electron). If I,,°= 0., beam b is never turned on (‘-eqs. D.1371—1380)
D.1244 —1253
$$HLENTH(10)
**
0 (cm)
section 2.9.2.1
Distance along the beamline from injector face to pivot point ( 4- eqs. D.1340—1359) (cf. figs. 3 and 4 of ref. [6]).
D.1254
$$HNCHEK
0.05
section 2.9.2.1
Maximum fractional change in n,, before recomputing beam deposition profile
D.1255 —1264
$$HPBEAM(10)
0
Not used
D.1265 —1274
$$HPROF(10)
0
Not used
D.1275 —1284
$$HPROFV(10)
0
Not used
D.1285 —1339
$$HR(55)
1
Not used in present version of BALDUR
D.1340 —1349 D.1350 —1359
I,,°
0
Multiplier on rate of charge-exchange loss
**
**
~
$$HRMAJ(10)
**
R~IN
$$HRMIN(10)
**
0 (cm) 0 (cm)
section 2.9.2.1
If HRMAJ(b) 0., we set R,,~= RMA.JOR(1) (‘-eqs. D.2027—2046). The major radius Rg.VO( and the vertical distance Z!VOI of the “pivot point” from the toroidal midplane (cf. side view in fig. 4 of ref. [6]) are defined as 1cos follows: ~1b, ~ = KRI~IN = sin R~’ 8lb +(e— R~°’ eqs. D.237, D.1079— 1118). If
R~IIN
0., we set R~~IN =
RMINOR(1), (‘-eqs. D.2047—2066) *
**
Present version of BALDUR requires b 6. Present version of BALDUR also requires that there be no more than 6 distinct energy components, i.e., only 2 different Wb may be active at any time in the simulation. Present version of BALDUR requires b less than or equal to 6.
358 No.
G.E. Singer eta!. Symbol
D.1360 D.1361
Code (array size)
Default (logical value or units)
$$HTCHEK
$$HTOFF(10)
1OFF
**
—1370 D.1371 —1380
z~
$$HTON(10)
/ One-dimensionalplasma transport code
* *
Used in eqs no.
Description
0.2
section 2.9.2.1
Maximum fractional change in 1’,, before recomputing deposition profile
0.
section 2.9.2.1
Time at which beam b is turned off
(s) 0 (s)
section 2.9.2.1
Time at which beam b is turned on. The species ( ‘-eqs. D.1401—1410), energy Wb (e— eqs. D.1159—1168) and current J,,~ ~—eqs. D.1234—1243) must all be specified in order for beam b to be included; if any of these are omitted, we set i,,°” = iOM so that the beam is never turned on
D.1381 —1390
~
D.1391 —1400
N,~’°’
1
$$HWIDTH(10)
**
$$NHAPER(10)
**
D.1401 —1410
$$NHBEAM(10)
D.1411
$NHE
**
0 (cm) 1 0
10
2.9.2.1 section
Width [diameter if ~ = 1 (‘- eqs. D.1424—1430)] of injector face 1 — circular port, 2 —e rectangular port (cf.
2.9.2.1
eqs. D.1119—1128, D.1129—1138)
section 2.9.2.1
Sets the injected species: If HABEAM(b) = 0. ( ‘- eqs. D.1069—1078), NHBEAM(b) = 0 —e injector b not active (cf. eqs. D.1371—1380); 1—H; —2—’D; —3 —e T; other values not allowed in present version of BALDUR. If HABEAM> 0., NHBEAM = 1— H (0
section
section 2.9.2.2
fast ions. Present version of BALDUR
requires NHE D.1412
$NHMU
D.1413 —1422
D.1423 (a—j) D.1424 —1433 D.1434
**
~
10
10
section 2.9.2
$$NHPRFV(10)
1
section 2.9.2.1
$$NHPROF(10)
1
section 2.9.2
Not used
$$NHSHAP(10)
1
section
1 —e circular injector face, 2 —e rectangular injector face (cf. eqs. D.1169—1178, D.1381—1390)
2.9.2.1
$NHSRC
1
Present version of BALDUR requires b less than or equal to 6.
section 2.9.2
Number of it (cosine of the pitch angle) groups in numerical solution of Fokker—Planck equation for fast ions. Present version of BALDUR requires NHMU 10 NHPRFV(b) = 1 —. beam has a uniform distribution at the injector face. NHPRFV(b)=2-.if N~~~=1 (‘-eqs. D.1424—1433), beam has a triangular distribution of intensity across the injector; otherwise, not used
Not used for input in present version of BALDUR (reset internally by subroutine DPOSIT)
G.E. Singer et al. No.
Symbol
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
359
D.1435
$$NIPART
iO~
section 2.9.2.1
Number of Monte Carlo particles used in each computation of beam deposition profile
D.1436
$$NIPROF
10
section 2.9.2.1
Maximum number of major time steps between computation of beam deposition profiles
D.1437
$$NIZONE
20
section 2.9.2.1
Number of zones used in beam deposition profile computation; NIZONE> 25 — we set NIZONE = 25;NIZONE> NZONES -e NIZONE = NZONES (e- eq. D.2366)
Alpha heating model D.1438
$RCWALS
0 (cm)
section 2.9.3.1
Major radius of the toroidal surface which limits a-particle orbits. If not set by input, is set in ALFINI to RMAJS
D.1439
$RDWALS
0 (cm)
section 2.9.3.1
Minor radius of the toroidal surface which limits a-particle orbits (cf. eq. D.1438). If not set by input, is set in ALFINI to RMINS
2
section 2.9.3
Type of a-particle model. 1 —e slow down where created but include first orbit losses; 2 —. slow down on first orbit; 3 — evolve orbit during slowdown; > 3 not allowed in present version of BALDUR. The wall position used in calculating orbit losses in all models is determined by eqs. D.1438—9
2.3i
Poloidal magnetic field at ~
D.14.40
N.~jp~
$NTYPE
Plasma model, timesiep control and neutral gas parameters D.1441 ~ t~) 0 —1460 8 0(t~) $$BPOID(20) (kG) D.1461
Bz
$$BZ
D.1462 —1481 D.1462
I(r~~,t~) ~
$$CURENT(20)
I(r~~,t~)
600
D.1463 —1481
~
(kA) 0 (kA)
D.1482 —1483 D.1484
np,,
$$DENGAO(2)
n~
$DENGA1(2)
—1485
t~) —e I(r~~,i~)
35 (kG)
(‘-eqs. D.2109—2128). If C1~> 0. then B9(r~~i, t~) — 8~(i~~) for f >10 table 1
2.3i
0 3) (cm 0 (cm3)
at time tB
Initial toroidal magnetic field at magnetic axis R(t~).Fort> t~,B~= 1/R (~eqs. D.2027—2046) 3 (‘-eqs. Total D.2104—2128), toroidal current ignoredatiftime BPOID(1) ij > 0. (‘— eq. D.1441). If C 1~> a, I(r~~i, :~)
4a
Initial a; ignored central if DENGAS density of(a, hydrogen 1) > 0 isotope
3b,4a
Density at all times in outer “dummy zone” for hydrogen isotope; ignored if DENGAS (a, 1) >0
G.E. Singer ci at
360 No.
Symbol
Code (array size)
D.1486 —1595
~
D.1596 —1597 D.1598
ni,,
$DENIMO(4)
nL
$DENIM1(4)
Default (logical value or units)
Used in eqs no.
Description
0 3) (cm
section 4
DENGAS(a, j — 1) = initial density of hydrogen isotope a in radial zone j = 2, 3 N + 2 (cf. fig. 1). If n~(r~~ 2, 0) is not specified, we set n~(r~~2, 0) = n~’(r~~1, 0). Boundary value for isotope a at any time is n~’(r,~~3, 0)
0 3) (cm 0 (cm3)
4a
a; Initial ignored central if DENIMP(a, density of impurity 1) > 0 species
4a
Density at all times in outermost radial zone for impurity species a; ignored if DENIMP(a,1)>0
$DENIMP(4,55)
0 (cm3)
section 4
$$DENMON(20)
0 (cm3)
section 3.2
DENIMP(a, j — 1) = initial density of impurity species a in radial zone ‘~, j 2,3 N+2 (cf. fig. 1). [Ifn~,(r~f 2,0) is not specified, we set n ~,( r~ 2’ 0)n (r~+ ~ 0).] Boundary value for species a at any time is ~ 0) Target density for feedback monitoring of n~at t~P(‘-eqs. D.1952—1971), j= 2, 3 20. DENMON(1) is used as a switch. If DENMON(1) 0., n~is held constant at n~’(t~) between times t~/.. 2and ~ j = 2, 3 10. If DENMON(1) = 1., n~ is linearly interpolated from n (~-1) to n(t~.2) between times ij~and t~.1, j = 2, 3 19. Do not set DENMON(1) to any other values
$DENS(55)
0 3) (cm
4section
Tabular input: used orwith FRACTH (‘eqs. D.1888—1889) FRACTI (‘-eqs. D.1890—1891) to specify initial densities which are not determined by eqs. D.1482—1709.
0,
$DENSO
5 x 3) 1013 (cm
4c
place DENS density (‘- eqs. Initial central for D.1730—1784); profiles to reignored if DENS(1) > 0
1,
$DENS1
iO’~3) (cm
4c
fig. 1) at value all times; ignored if DENS(1) Density in outer “dummy zone” >(cf.0 (‘-eq. D.1730)
10—6
section 6
Initial value of time step
0) $DENGAS(2,55)
—1599 D.1600 —1709
~
D.1710 —1729
n(t~
0)
9)
D.1730 —1784
D.1785
D.1786
/ One-dimensional plasma transport code
n
n
D.1787
$DTINIT
(s) D.1788
l~tma,,
$$DTMAX
2X103 (s)
6g
Maximum allowed time step
D.1789
i~:,,,,,,
$$DTMIN
iQ~ (s)
Minimum allowed time step
D.1790 D.1791
Yb
$EBFIT
1
Xb
$EEBFIT
0
section 6 4e 4e
Initial Jc1[1_(r/rwaiiyb]Yb. EBFIT is ignored if NBFIT *0 or if EEBFIT =0 (‘— eq. D.2332)
G.E. Singer et at
/
One-dimensional plasma transport code
361
No.
Symbol
Code (array size)
Default (logical value or units)
Used in eqs no.
Description
D.1792
x~
$EEFIT
3
‘Ic
Replace DENS (e— eqs. D.1730—1784) by (n
1, + (n01
—
n11)[1 _(r/r,,,~i)x]Y)
(~_
eqs. D.1785, 1786, 1799); ignored if DENS(l) >0 or if NNFIT *0 (‘-eq. D.2333) D.1793 —1794
x~
$EEFIT(2)
3
4a
Initial density of hydrogen isotope a is n~= n~ + (n~a— n~)[1 — (r/r~~,)~]’~ (e_ eqs. D.1482—1485 and D.1800—1801); ignored if DENGAS(a, 1) > 0 or if NNHFIT * 0 ( ‘-eqs. D.2334—2335)
D.1795 —1796
x~
$EEIFIT(2)
3
4a
Initial density of impurity species a is n’ = n~,,+(n~,,— n~)[1—(r/r~~i)~]~ (‘-eqs. D.1596—1599 and D.1802—1803); ignored if DENIMP(a, 1) > 0, or NNIFIT * 0 (‘-eqs. D.2336—2337)
D.1797 D.1798
XTa
xi.,
$EETEFT $EETIFT
2 2
4b 4b
Initial temperature profiles are fT11 j=i,e (e— eqs. D.1797—17987, D.1805—1807, D.2207—2208 and D.2249—2250); ignored if NTEFIT *0 or NTIFIT *0, respectively. See D.1792
D.1799
y~
$EFIT
1
‘Ic
see D.1792
D.1800 —1801
y~
$EHFIT(2)
1
4a
see D.1793
D.1802 —1803
y,’
$EIFIT(2)
1
4a
Outer exponents (‘-eqs. D.1795—1796)
D.1804
E10~,~
$$EIONIZ
0.04 (keV)
2.9.lb
Electron loss for each ionization of any neutral hydrogen isotope
~
D.1805
vTC
$ETEFIT
1
4b
Outer exponent (‘-eq. D.1797)
D.1807
y~
$ETIFIT
1
4b
Outer exponent (‘-eq. D.1798)
D.1808 —1847
I,(t~”)
0 2s1) (cm
3b
$$FLGAS(2,20)
Specifies hydrogen a gas puffing influx rate eqs. for neutral isotope a (e— D.2252—2253), linearly interpolated between times ij~ (‘-eqs. D.2229—2248). This is independent of density monitoring (‘-eqs. D.1710—1729) (cf. C 2®)
D.1848 —1887
Fa(tj)
0 2s1) (cm
3b, section
$$FLIMP(4,20)
Specified tral species theaflux across of impurity r = r~ ion or neu1,linearly interpolated between times tj (~— eqs. D.2256—2275) (unless C2®> 0)
2.9.13 D.1888 —1889 D.1888 D.1889 D.1890 —1891
f~ f~” f~ f,~
$FRACTH(2)
$FRACTI(4)
4c 1 0 0
4c
Used in conjunction with DENS, DENSO, DENS1 (‘- eqs. D.1730—1786) to get initial densities of hydrogen isotope im, purities that are not determined by eqs. D.1482—1709
362
G.E. Singer
ci at
/
One-dimensional plasma transport code
No.
Symbol
Code (array size)
Default (logical value or units)
Used in eqs no.
Description
D.1892 —1911
Fm~(tjG)
$$GFLMAX(20)
(cm
section 3.2
Maximum gas puffing rate which can be forced by density monitoring linearly interpolated between times ~ ( e— eqs. D.1952—1971)
D.1912 —1951
f,°(t~)
0
section 3.2
Fraction of hydrogen isotope i (‘-eqs. D.2252—2253), in gas puffing due to density monitoring, linearly interpolated between times tj( e— eqs. D.1952—1971). If we set f?(if)1—f~(t~)
D.1952 —1971
i~
$$GFTIME(20)
i0~ (s)
section 3.2
Times for density monitoring (‘-eqs. D.1710—1729, D.1892—1951, D.1912— 1951 and D.2360). Note that ~ is used as a switch; for t~°> 0, the initial density monitoring target is set equal to the initial line-average (NPUFF = 1) or volumeaverage (NPUFF = 2) electron density (— eq. D.2360)
D.1972 —2026
r]’
$RADIUS(55)
0 (cm)
section
Tabular input for real spatial zone locations (cf. fig. 1); zones determined instead by NRFIT ( ‘-eq. D.2361) if RADIUS(2)
D.2027 —2046 D.2027
R(t~)
R(t~)
130
D.2028 —2046
R(4)
(cm) 0 (cm)
D.2047 —2066 D.2047
r~~ii(t~)
D.2048 —2066
r~~i(i~) —r~~1i(t~)
D.2067 —2086
~
D.2087 —2106
r,,~,iat(t~’)
1016
2s
$$GFRACT(20,2)
$$RMAJOR(20)
$$RMINOR(20)
r~~,(t~)
D.2107
40 (cm) 0 (cm)
table 1
$$RPA(20)
0 (cm)
section 2.9.11
Minimum major radius along the flight path of pellet launched at time i)’ (e— eqs. D.2278—2287)
$$RPELA(20)
0 (cm)
section 2.9.11
Radius of pellet launched at time eqs. D.2278—2287)
appendix F
Generate major printouts every time interval SEDIT
0 (s)
D.2109 —2128
$$SPLOT t]~
Major radius of magnetic axis, linearly interpolated between times if( 4- eqs. D.2131—2150). If C 1~,> 0., R(t$) — f,11,(i~)for j >10 (‘-eqs. D.2119—2128) [33] Minor radius of wall, linearly interpolated between times t~ (e— eqs. D.2131 —2150)
$$SEDIT
D.2108
5.1.1
table 1 2.3i
$~~(i~B)
—R(t~)
1)
$$TBPOID(20)
table I
0 0. (s)
t1” (‘—
Generate plots every time interval SPLOT D.1441
Times for changing plasma current [or for changing variable ripple for j >10 if C146 > 0. (e— eqs. D.2037—2046)]. Note that if the combined total number of tj~plus t~ (‘— eqs. D.2131—2150) exceeds 19, the first 19 unique nonzero times will be used and the rest ignored. The if and t~ always are merged by the code into a single array of dimension 20 whose first location is set to zero
CE. Singer et at No.
Symbol
/
One-dimensional plasma transport code
363
Code (array size)
Default (logical value or units)
Used in eqs no.
Description
D.2129
$$TCOLD
0.003 (key)
section 2.9.1
Temperature of recycled neutral hydrogen [but cf. NLGLIM (‘-eq. D.1014)] and C 198
D.2130
$$TCOLDP
0 (keV)
section 3.2
[If TCOLDP 0, we set TCOLDP the originally prescribed value of TCOLD (‘-eq. D.2129) regardless of how NLGLIM (e— eq. D.1014) and C198 are set.] Energy of gas puffing neutrals
$$TCOMP(20)
0 (s)
D.2027 —2066
Times for changing ~ and R for adiabatic compression. See note at TBPOID (e— eqs. D.2119—2128)
$TE(55)
0 (keV)
section 4
TE(j —1) = initial electron temperature in radial zone ,~, j = 2, 3 N + 2 (cf. fig. 1). [If ~ 0) is not specified, we set 1~(r~~2) = 7~,(r~+1, 0)]
0.2 (keV) 0.01 (keV)
4b 4b
Initially, T,=Tie+(T~~Tie) [1_(r/r~~iyTh]YT~ (‘-eqs. D.1797 and D.1805), ignored if TE(1)> 0. (‘-eq. D.2151)
D.2131 —2150
t~
D.2151 —2206
~
D.2207
T,~
STEO
D.2208
T1,
$TE1
0)
D.2209 —2228
$$TEDIT(20)
0 (s)
appendix F
Major printouts at time steps immediately following times TEDIT
D.2229 —2248
$$TGAS(20)
0 (s)
D.1808 —1847
Times to change specified hydrogen gas influx
4b
Initially,
D.2249
7~
$TI0
0.2
(keV) D.2250
T0
D.2251 D.2252 —2253
N~AS
D.2252 D.2253
N~AS
D.2254
N,MP
7 = Th + (lb — T,1)[1 — (r/rw~i)~~T~])’ni (‘-eqs. D.1798 and D.1807); ignored if TI(1) > 0
$TI1
0.01 (keV)
4b
(4-eq. D.2472)
$$NEDIT
10
appendix F
Generate major printout every NEDIT time steps
section 2
Sets the hydrogen isotopes for species a. If WTGAS(a) = 0. (‘-eqs. D.2328—2329), NGAS = 1—H, NGAS=-2—D, NGAS=-3—T. If WTGAS>0, NGAS=1—H (0
$NGAS(2)
1 0
N2GAS
$NIMP(4)
0
section 2
5-’B, 2—’
10—. Ne,
6—C, 11—.
7-eN,
Na,
8-’O,
9-sF,
12 —‘Mg, 13 —‘Al,
14 —‘Si,
16—eS,
18-. A,
19—. K,
21 —. Sc, 28 —‘Ni,
22 —. Ti,
23 —. V,
24 —.
20
—‘Ca,
Cr, 26 —. Fe,
47—. Ag,
29 —‘Cu, 30—. Zn. 33 —‘As, 36—eKe, 40—.Zr, 41—Nb, 42—.Mo, 45—ellis, 50 —. Sn, 54 —. Xe, 55 —. Cs, 56 —. aa,
64
73 — Ta, 77—. Ir,
37—’Rb.
—‘ Gd,
83 — Bi,
79 -. Au, 80 -s Hg,
86 — Rn, 90 — Th, 92 — U,
others not allowed). Note: it is not necessary to include 4He to get heating from D : T fusion D.2256 —2275
tj
$$TIMP(20)
0 (s)
D.1848 —1887
Times to change specified impurity influx
364
G.E. Singer et at
No.
Symbol
D.2276
t
D.2277 D.2278 —2287
/ One-dimensional plasma transpori
code
Code (array size)
Default (logical value or units)
Used in eqs no.
Description
$TINIT
0 (s)
appendix F
Time before first time step
im~,,
$$TMAX
I (s)
appendix F
Computation is terminated if
iJ’
$$TPELA(20)
0 (s)
section 2.9.11
Times to launch pellets. For j >1, pellet is ignored if ~! 0
$$TPLOT(20)
0 (s) 0 (cm/s)
section 2.9.11
Velocity of pellet launched at time D.2278—2287)
Atomic weight of hydrogen isotope a. If WTGAS = 0, we set A,, according to the default for the species ( ‘-eqs. B.31_35) Atomic weight of impurity a. If WTIMP = 0, we set A,, according to the default for the species (cf. section 7.3 and appendix F of ref. [17])
0
D.2288 —2307
7)
Generate plots at times TPLOT
D.2308 —2327
Vi,eiiet(i
$$VPELA(20)
D.2328 —2329
A,,
$WTGAS(2)
0 (amu)
(throughout)
D.2330 —2331
A,,
$WTIMP(2)
0 (amu)
(throughout)
D.2332
Y,,
$NBFIT
0
t >
4e
x,
$NNFIT
0
‘Ic
If NNFIT
D.2334 —2335
x,,Ft
$NNHFIT(2)
0
4a
(‘-eqs. If NNHFIT(a) D.1793—1794) * 0, we set y,,~
D.2336 —2337
x~
$NNIFIT(2)
0
4a
If NNIFIT(a) * 0, we set y,,’ eqs. D.1795—1796)
D.2338
$$NPEL
1
D.2339
$$NPEL2
1
*
4-
eq. D.1790)
0, we set x1 = NNFIT1(=‘-eq. D.1792) NNHFIT(a)
D.2333
N~°~,(tJ’)
(‘-eqs.
If NBFIT * 0, we set Y 6 = NBFIT (
D.2340 —2359
t]’
NNIFIT(a) (‘-
Not used in present version of BALDUR section
Not used in present version of BALDUR
2.9.11
Hydrogen isotope (4- eqs. D.2252—2253) in solid pellet launched at time t]’ (— eqs. D.2278—2287). If NPELGA(J) does not follow the numbering conventions of NGAS (‘-eq. D.2252) the pellet does not contribute to the hydrogen density
$$NPELGA(20)
D.2360
$$NPUFF
0
section 3.2
Density averaging method used for density mombring. NPUFF = 1—s line average, NPUFF = 2 — volume average. cf. DENMON ~~eqs. D.1710—1729)
D.2361
$NRFIT
1
section 5.1.1
Plasma transport zones equally spaced in minor radius (NRFIT = 1) or volume (NRFIT = 2); ignored if RADIUS(2) 0. (‘-eqs. D.1972—2028)
D.2362
$$NSKIP
1
appendix F
Printout information on every NSKIPth zone in major printout =
NTEFIT (‘-eq.
YT =
NTIFIT (s— eq.
D.2363
Yr.
$NTEFIT
0
4b
If NTEFIT * 0, we set Yr. D.1805)
D.2364
Yr~
$NTIFIT
0
4b
If NTIFIT * 0, we set D.1807)
$$N’VFY
0
appendix F
Print short summary to teletype every N’TTY time steps, N’ITY = 0 —e suppress teletype output
fig. 1
Number of real spatial zones in plasma transport calculation; maximum = 50, section 5.1.1
D.2365 D.2366
N
$NZONES
20
G.E. Singer ci at No.
Symbol
Code (array size)
/
Default (logical value or units)
One-dimensional plasma transport code Used in eqs no.
365
Description
D.2367 —2416
$$NLXXX(50)
0
D.2417
$$NHSKIP
2
section 2.9.2; appendix F
Printout information on every NHSKIPth zone in beam printout
NFUSN
0
section 2.9.3
NFUSN = 1—s always include D : T fusion model. NFUSN * 1—’ include D : T fusion model unless D has been specified as a hydrogen isotope ( ‘-eqs. 3He as an impurity (‘-eqs. D.2252—2253) and D.2254—2255), in which case use D: D fusion model described in section 2.9.3.2
D.2419
$$NPELOU
1
section 2.9.11
Allow pellet printouts when every NPELOUth pdlet cluster is injected
D.2420
$$GWMIN
section 2.9.1
Minimum weight below which particles are no longer followed in Monte Carlo neutral algorithm
D.2421
*
section
Relative power to each lower hybrid mode; cf. n
2.9.6
(‘-eqs. D.439—488)
D.2418
*
RLPOWR(50)
1.E—6 0
Dummy input switches (not used)
11 —2470 D.2471 —2526
T~(i~, 0)
D.2527
f,’
D.2528 D.2529 D.2530
*
*TI(55)
$AFSLOW * *
C~’~ G~,,
NLGPIN
$CPVELC $CPVION
0 (key) 1 —1 (True)
Factor multiplying alpha particle slowdown D : T fusion model
section 2.9.1
.TRUE. -s include proton impact ionization in Monte Carlo neutral calculation If > 0 — multiply by T~,(for CPVELC) or l’~(for CPVION) to get energy per particle carried by convective terms in energy transport equation
section 2.1
D.2531
$$BZSTAR
D.2532
$$ELECDO
D.2533
$$ELECTE
1.0 (key)
D.2534
$$APRESR
8x1013 (keV/cm3)
D.2535
$$RASTAR
~
D.2536
$$QSTAR *
*DFUTZV
12 (kG) 13 (cm 4X103)
Normalization constants for semiempirical model B~
x40 (cm)
r*
2.2
q~ must be input through CFUTZ(377); before version BALDP22M
0
2.7b
0
2.1H
(6,25) * *DFUTZD (6,25)
rate in
section 2.9.3
1.5 1.5
D.2531 —2535
D.2537 —2616 D.2617 —2696
TI(j —1) = initial ion temperature in radial zone i~, j=2,3 N+2(cf. fig. 1). [If Tj(r,~~2, 0) is not specified, we set T,(r~~2, 0) = T1(r~~,÷1,0)]
eD,,,,
366 No.
C.E. Singer ci at Symbol
Code (array size)
Default (logical value or units)
/
One-dimensional plasma transport code Used in eqs no.
Description
D.2697 —2776 D.2777 —2856
* *DFUTZE (6,25) * *DFUTZI (6,25)
0
* *
0
2.11
e~,,k
D.2857 —2880
* *VXSEMI (6,6)
0
2.7b
v,~,a=1,2,t’,h,i,e
D.2881 —2904
* *DXSEMI (6,6)
0
2.1H
D,7,,, a =1,2,c”,h
D.2905 —2908
*
*XE5EMI(6)
0
2.11
x,~,
D.2909 —2912
*
*XI5EMI(6)
0
2.11
x~
*QMHD1
0
2.1J
MHD correction to 1X
QMHD2
0
2.1J
Singularity avoidance parameter for I; must also be input through CFUTZ (379) before version BALDP22M
*RRSTAR
140
2.1J
Normalization constant c~= r~/R~(‘-eq. D.2535). (Version BALDP21M will not read this variable in NAMELIST, so default values must be used)
D.2913
QMND1
*
D.2914
QMND2
* *
D.2915
R~
*
D.2916
* *
MEMPRT
0
2.1!
ee,k
91; must be input through CFUTZ (378) before version BALDP22M
Specifies term for which D~” and v~j,~” 5for first is printed term out. Default value gives v~”,D~”
Appendix E. Subroutines The main branches of the program are described. This list does not include OLYMPUS [7] and BALDUR utilities, dummy EXPERT calls, error sequences, nor the details of the neutral gas package, coronal radiation package, neutral beam package, alpha-particle package, pellet injection package or output routines. The order of events within routines is not always exact; e.g., the call to IMPRAD(1) by AUXVAL occurs somewhere in the middle of the routine, not at the end. Outer calling structure Program BALD UR 1. Set up I/O units (except for ripple input data file). 2. Call BLOCKDTA Store fundamental constants. 3. Call MASTER 1. Call BASIC Initialize OLYMPUS variables. Call MODIFY, which reads input deck header cord used for restart switches (not implemented in present revision of BALDUR). 2. Print date and time.
G.E. Singer et at
3. Call
/
One-dimensional plasma transport code
367
COTROL 1. Call LABR UN Store BALDUR version number. Read runs labels from input deck. Work revision number, labels, “PROGRAM BALDUR”. 2. Call CLEAR Clear common blocks. Includes calls to ACLEAR, GCLEAR, HCLEAR, ZCLEAR. 3. If run is not a restart Call PRESET Set default values for NAMELIST input variables (‘~—appendix D). 4. If run is a restart Call RESUME Restart capability is not implemented in present version of BALDUR. Do not use this branch. 5. Call DATA Read and write NAMELIST input. 6. Call AUXVAL 1. Call ERRCHK Check for obvious mistakes in input deck. 2. Call UNITS Computer units conversion factors.
3.
Set auxiliary values based on input data. This includes code flags, the radial grid, time dependent conditions, plasma conditions variables not defined in INITIAL or START, etc. (*— section 4). 4. If scrape-off model is on, output sheath limited current densities section 2.8). 5. Call IMPRAD(1) 1. Initialize neutral impurity influx parameters (*— section 2.9.13). 2. If scrape-off model is on Call PDX Initialize and compute scrape-off losses (~- section 2.9.5). 3. Initialize coronal radiation package. Read atomic data file FOR22 (‘c— section 7.3). 6. Call BEAMS(1) Initialize neutral beam package (+— section 7. If run is not a restart Call INITAL 1. Call IMPRAD(2) 1. If scrapeoff model is on Call PDX Initialize and compute scrapeoff losses (4— section (~-
2.9.2).
2.9.5).
Compute initial radiation terms, impurity charge states. Call coronal radiation package if selected (4— section 2.9.4). 2. Define physical initial conditions for x (4— section 5.2.3), B9. 2.
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Note that the electron density is redefined in START section 4). START 1. Set initial ~t, boundary conditions, volume integrals. Unlock neutral impurity influx flag. 2. Call GETCHI(2) 1. Call IMPRAD(2) Same as above, plus compute neutral impurity and He influxes. Includes call to PDX. 2. Compute boundary-centered densities and temperatures, zone-centered B 9 and J section 5.2.2). (~—
8.
Call
(~-
9.
10. 11.
12. 13.
14.
15.
Compute Z-effective, electron density. 3. Initialize packages for remaining source terms: call HEAT(1) (lower hybrid and input profile heating), ECRH(1) (electron cyclotron resonance heating), and either ALFINI(alphas) or HE3(1) (D—D fusion) as selected by input. Call OUTPUT(1) Initial alphanumeric output (call MPRINT, GPRINT, HPRINT, IPRINT, APRINT or FPRINT, RPRINT, SPRINT). Initial graphics output (call TGRAF(1), GRAFIX(1)). Call STEPON Do a time step of main computations. See ext. chart. Call OUTPUT(2) Control output flags. At times or time steps selected by input, do main alphanumeric output (call MPRINT, NCFPRT, GPRINT, HETPRT, HPRINT, IPRINT and either APRINT, FPRINT, as selected by input), short alphanumeric output (call SPRING, GPRINT), and/or graphics output (call TGRAF(2), GRAFIX(2)). CALL TSEND Test for completion of run. If run is not yet completed, do next time step: go to ~10. Call OUTPUT(3) Do main alphanumeric output for final time step if this has not been done already (same calls as in * 11). Do final graphics output and close graphics (call TGRAF(3), GRAFIX(3). Call ENDRUN Terminate run. Print final message, STOP.
Inner calling structure 10. Call
STEPON 1. Do short teletype output at appropriate time steps. Zero the iteration counter. CALL SAWMIX 2. Call RESOLV(1) Initialize flags and store values for predictor—corrector, extrapolation. 3. Unlock neutral impurity influx flag. 4. Call COEF
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1. Call GETCHI(2) 1. Call IMPRAD(2) 1. If scrapeoff model is on, Call PDX Compute scrapeoff losses Call DIVER. 2. Compute radiation terms, impurity (Z> and
5. Call 6. Call
Compute sources due to neutral beam injection. This includes Fokker—Planck calculation of beam fast ion distribution, and every so often includes call to Monte Carlo computation of beam deposition profile. 6. Call HEAT(2) Compute sources due to lower hybrid heating or input profile heating. 7. Call ECRH(J) Compute sources due to electron cyclotron resonance heating. 8. If D—T fusion has been selected by input, Call ALPHAS(2) Compute sources due to D—T fusion. This includes calculation of alpha-particle distribution, and includes call to Monte Carlo computation of alpha-particle orbits. 9. If D—D fusion has been selected by input, Call HE3(2) Compute sources due to D—D fusion. 10. Call CONVRT 1 .If nearly exact neoclassical transport model has been selected, compute transport coefficients for this model. This includes call to NCFLUX. 2.Compute A, B, C, D (4— appendix A). 11. Call CNVCOF Convert A, B, C, D (f— appendix A) from standard to interval units. SOLVEB Solve B9 equation. Compute source due to ohmic heating. BOUNDS Compute boundary condition coefficients.
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7. Call
RED UCE Compute P, 0, R, S (4— eq. 5.2.3g). Reduce equations to first order. 8. Call SOLVE Solve transport equations for new x9. Call RESOLV Time-step control. Set next ~t. Control flags and store values for predictor—corrector. extrapolation, time step repetition. 10. Increment iteration counter. Check that maximum number of iterations has not been exceeded. 11. If time step is to be repeated, or the predictor portion of a predictor—corrector time step has just finished, or the first i~t/2minor time step of an extrapolation time step has just finished, go to #4. 12. If the t~tminor time step of an extrapolation time step has just finished, go to #5. 13. Call CMPRES 1. Compress the plasma: adjust x~ B9 radial grid. 2. Call other routines which may need to compress: BEAMS(3), HEAT(3) and either ALPHAS(3) or HE3(3) as selected by input. 14. Call
PDRIVE Compute sources due to pellet injection.
15. Call
GETCHI(1) Update quantities based on x: densities in standard units, ion and electron temperatures. Also update B9, B~ in standard units. Note that electron density (but not electron temperature) is recomputed in GETCHI(2). 16. Lock neutral impurity influx flag. 17. Call GETCHI(2) Almost same as COEF*1, but new x values were used, and there is no neutral impurity or He influx computation when IMPRAD(2) is called. 18. Advance the time. Time step is completed. The calling sequence and nesting level for the major plasma source routines are given below. Neutral gas fueling 5. NEUGAS 6. MONTE 7. GCLEAR 7. SETVAR 7. SPLITS 7. MCARLO 8. XSECT 8. LOCATE 9. VELOC 9. REFLEC 8. VSOURC 8. VELOC 8. FOLLOW 9. TIME! 9. VELOC
G.E. Singer ci at
9. ESCAPE 10. SPUTER 10. REFLEC 6. MATRX1 6. MATRX2 Neutral beams 5. BEAMS 6. DPOSIT 7. ICLEAR 7. HRSET 7. HRCAL 8. HRSET 8. CXSNEU 8. INJECT 9. SORSPT 9. ROTATE 8. ROTATE 6. ENDRUN 7. DAYTIM 8. DATE Alpha particles 5. ALPHAS
6. ALFEND 7. ENDRUN 8. DAYTIM
9. DATE 9. CLOCK 8. RUNTIM 9. SECOND 6. ALFSET 7. ALFEND 8. ENDRUN 9. DAYTIM 10. DATE 9. RUNTIM 10. SECOND 7. XINTER 8. ALFEND 9. ENDRUN 10. DAYTIM 11. DATE 11. CLOCK 10. RUNTIM 11. SECOND
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One-dimensional plasma transport code
6. BOUNCE 6. ARLPUT 7. ALFEND 8. ENDRUN 9. DAYTIM 10. DATE 10. CLOCK 9. RUNTIM 10. SECOND 7. SQUEZE 8. ALFEND 9. ENDRUN 10. DAYTIM 11. DATE 11. CLOCK 10. RUNTIM 11. SECOND Pellet
4.
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[46]
PDRIVE 5. PELLET 6. ZINCON 6. INICON 6. PELRK4 7. PELRAT 8. PELABL 9. PELQE 9. PELQF
Appendix F. Sample nm Sample BALDUR run A number of test cases have been developed to benchmark the BALDUR code [36,52]. These include analytic test case comparisons for the energy, particle and poloidal magnetic field diffusion equations as well as machine specific test cases. The test cases simulate a variety of tokamak scenarios and are used to validate new versions of the code. They have been documented in refs. [36,52] and may be obtained from the MFECC by “FILEM read 4776 .sbvtst all”, see ref. [52] (this issue). Our sample BALDUR run is the neutral beam and compression heated TFTR test case from this series. The input deck for our sample run is shown in fig. 2. The top line of zeros begins in column 1. It is a holdover from old BALDUR versions which were restartable. The next four lines are comments used to label the run. The comments must begin in column 2 and may contain up to 48 characters per line. Below that are the two NAMELISTS, NURUN1 and NURUN2. NAMELIST input variables should be entered in the format shown. Column 1 should be blank. The end of each NAMELIST is indicated by a $END line. Appendix D described all the input variables for BALDUR. Here we call attention to several variables
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CC K))’) ++ 0 0 0.(0— Nb
50) V
It)
000 010.10 N (1)
NNNN 0000(0000000 I I I I I I I C — I I I I I I I I I I I I I I I I I I I I I I I I I I 4)S00000V00000000000000000000000000000000000V000 0001(O0)01(0NNOW*0)NU)K)010010(0U)(001.(D(O01*K)(ON..*NN*(0N01WU).-*N VV(0(0(0U)(ON(00)0.N0)*(O(001.~ONNU)**01N*00)0.-01*10NK)N(0*0*N(O0 4)VK)K)K)K)K)K)K)0)********(0U)(0(0**K)N.-K)N.01NID*.-01(0K)0(ON0K)*ONN 0 — (0(0U)(0U)(0(0U)U)U)(0U)(0(0U)(0U)U)U)(0U)U)(0(0(0U)U))fl*****0)K)K)K)NN.-.-0)(O*K) S
0. 0 —
0 0)(0~ 5) 5) C 5) — 0’ -0 (‘0-— 0 —
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0.
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N U)
N N N N N N N N N N N N N N C-I N N N N N N N N N N
s
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-
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—
(0 — — ‘-...
++
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00.
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00 00
-—
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11.
G.E. Singer e at
/ One-dimensional plasma transport
code
383
not referenced elsewhere. NRUN (t-.- eq. D.12) and TMAX (4— eq. D.2277) are used to control the duration of the run. TINIT (4— eq. D.2276) sets the time at the beginning of the run. NLOMT1, NLOMT2 and NLOMT3 (i’- eqs. D.86—235) can be used to omit various packages and subroutines. The frequency of printed output is controlled by NEDIT, NSEDIT, SEDIT and TEDIT (t— eqs. D2251, D.1046, D.2107 and D.2209—2228, respectively). NLPOMT (i’- eq. D.1048—1067) is used to omit certain output pages. NSKIP (4— eq. D.2362) and NHSKIP (.t_ eq. D.2417) control the number of radial zones printed out. NPELOU (4— eq. D.2419), CFUTZ(63), CFUTZ(70), CFUTZ(214) and CFUTZ(253) are special output controls; consult appendix D for details. NTTY (t— eq. D.2365) controls output to the teletype. NPLOT, SPLOT and TPLOT (-*-- eqs. D.1044, D.2108 and D.2288—2307, respectively) control the frequency of graphics output produced by BALDUR’s companion plotting codes. A number of variables associated with restarting capabilities are documented to facilitate future recovery of this capability. The printed output from the run is extensive. We reproduce it for one time step in fig.2. Explanatory comments follow. The printout begins with some OLYMPUS [7] printout followed by run labeling comments and the values of all input variables, whether or not these were set explicitly in the input deck. The array CFUTZ is also printed out in ten-colunm format for easy reading. In runs which include a scrapeoff model, a printout of sheath-limited current densities will appear immediately below the second NAMELIST. F. 3.1. First page The next few pages describe some of the major models used in the run and are self-explanatory. The zeroth time step printout gives a detailed breakdown of the initial conditions. A major printout is always generated for the first and for the last time step, and for intervening time steps as specified by input. Notice that the run label from the input deck and the BALDUR version label are printed out at the top of each major printout page. The quantities printed out on the first page are as follows: zone: radius: te: reg: ti:
j 1, where j fig. 1) is the BALDUR zone index radius of zone center rZ electron temperature, 1~( ,~z) two-letter mnemonic for dominant term affecting electron energy transport ion temperature, T1 (,~) —
(+-
reg: ne:
two-letter mnemonic for dominant term affecting ion energy transport electron density, n e ( FZ) ni: thermal ion density, n1(~)[4— eq. (2e)] ez: toroidal electric field at jz: toroidal current density, J(rZ) safety factor, q(i~) total beta: local thermal toroidal beta ~ ( Ee + E1 )/( B~/8’n) The total at the bottom of the page are the total electron and ion thermal energy in the plasma, the total numbers of thermal electrons and ions in the plasma, and the total plasma current. These totals are computed as volume-weighted sums over all physical zone. F. 3.2. Second page The quantities printed out on the second page are as follows: zone: j 1, where j fig. 1) is the BALDUR zone index radius: radius of zone center nu-el: electron collision frequency (t— eq. B75) —
(—
384
G.E. Singer et at
/
One-dimensional plasma transport code
nu-h: ion densities:
hydrogen ion collision frequency (~ eq. n0(i~)for each hydrogen and impurity species
zeff:
Zeff(I~jZ) (#—
B.74)
a
eq. B.64)
F. 3.3. Third page zone: j 1, where j (.1— fig. 1) is the BALDUR zone index radius: radius of outer boundary of BALDUR zone j. ~ 4-if 2R ~~I.iXe~ b electron conduct: —
(F.3.3a)
—
electron convect: ion conduct: ionconvect:
4R~2R~i{qe+ Xe~~ r r=rj*,b
—
4lT2RrbXB~
(F.3.3b)
j
(F.3.3c)
b
r
r=r~+,
4’r2Ri~ 1 qj+~j~T~ ~, r r—r,~,
(F.3.3d)
neutral losses:
—
radiative losses:
4~2RJ~*1(Q~)+ QHE+ Q1MP+
ohmic heating:
4~2Rj’)+IQoHr
alpha heating:
4if2RfrJ+I(QF
+
other heating:
4ir2Rf~*1(Q~
+
+
4)if2RfrJ*1(QN
Q~~)rdr Q~)rdr
(F.3.3e)
dr
(F.3.3f)
Q~)rdr
(F3.3g)
+ QLH + QLH + Q~C+ QEC)r
dr
(F.3.3h)
2R[—i~ total gain:
4-ii
1(q0 +
q1)
M1’)rdr] +
e—i coupling:
+ QEC
4)if2RjrJ*IQIEr
—
QN
—
QREC
—
QN
—
QRAD
—
QHe
—
(F.3.3i)
Q~
dr.
The following symbols used above have been defined elsewhere: Xe(4- eq. 2.6d), q~(4—eq. 2h), 2.6c), q 1(~—eq. 2h), Q~(—eq. 2.9.lc), Q~~(’— eq. 2.9.12b), Q)(~~~ eq. 2.9.4a), QHE(4- eq. 2.9.lOb), QIMP(4- eq. 2.9.13d), Q~(—eq. 2.9Jb), QOH(4- eq. 29.8a), Q~(—eq. 29.3.lh), Q~(~ eq. 2.9.3.li), Q~(~— eq. 2.92.2k), Qr(~—eq. 29.2.2e’), QLH(+... eq. 29.6a), Q~’~’(+— eq. 2.9.6b), Q~(~ eq. 2.9.7a), QEc(4- eq. 2.9.7b), QIE(4- eq. 2.9.9.a).
x~(— eq.
F. 3.4. Fourth and fifth pages The fourth and fifth pages list the transport coefficients calculated at the outer boundary of each physical zone. The first two columns of each page list the zone numbers and the radii of the outer boundaries of the zones at which the transport coefficients are calculated. On page four: k-e tot!: total electron thermal diffusivity k-i totl: total ion thermal diffusivity
CE. Singer et at / One-dimensional plasma transport code
385
neoclassical Ware pinch on ions v~1e: neoclassical Ware pinch on electrons d-h tot!: tota! diffusion coefficients for hydrogen ions by species d-i tot!: total diffusion coefficients for impurity ions by species
v~iate:
On page five are printed the semiempirical transport coefficients specified through the DFUTZE, XESEMI, etc., variables Xe(1)... xe(5): 1st five terms in the semiempirical electron thermal diffusivity x1(1): 1st term in the semiempirical ion thermal diffusivity
v (0):
total semiempirical inward pinch total semiempirical diffusion coefficient
d(0):
Setting input variable MEMPRT = i, (i = 1,... ,6), causes the ith term in the semiempirical pinch and diffusion coefficients to be listed in the last two columns. Setting MEMPRT — 0 causes the total semiempirical pinch and diffusion coefficients to be listed. F. 3.5. Sixth page The confinement times printed at the top of the sixth page are computed for each species, for the electron and ion energy, and for the total energy: particles:
fonardr
a=1,2,e,h,
electron energy:
f~E0r’dr’ rqe+J~(Q~”+Q~+QIMP+QN)r~dr/
ion energy:
f~E~r’ dr’ , rq1+J~ç(Q~+ QREC)r~drF fr(E
total energy:
r(q~
n-tau is computed as:
+
-
q1)
+ Jr(QRAD
(F.3.5a) (F.3.5b)
,
(F.3.5c) +
E.)r’ dr’
+Q~+ QIMP + Q~+ Q~+ QREC),.~~dr’’
(total energy confinement time).
(F.3.5d) (F.3.5e)
In these expression, r is the minor radius at which the confinement time is computed. The four values of r are chosen so as to be roughly evenly spaced over the range 0(4- eq. F.3.4h). The experimental confinement times are: thermal/ohmic:
f(EJr)SrQoHr~ + E1)r’ dr’dr’
(F.3.50 J”s~r(E+E+nB)r’
total:
Jr~or(QOH+QF+
Q.~+Q~+Q~+
QLH~
dr’ QLH~
QEC+
The following symbols are defined elsewhere: Q~(4- eq. 2.9.8a), 2.9.3.li), eq. 2.9.2.2k), eq. 2.9.2.21), Q~(-—eq. 2.9.6.a), 2.9.7a), QEC(4- eq. 2.9.7b). The mean (volume-averaged) densities are: Q~(~
=
~f~’°’ti~~ dr,
Q~(~
j = e,i.
.
QEC)r~dr Q~(I—
Q~’(---
eq. 2.9.3.lh), eq. 2.9.6b),
(F.3.5g)
Q~(~ eq. Q~(+—
eq.
(F.3.5h)
386
G.E. Singer et al.
/
One-dimensional plasma transport code
The mean (volume-averaged) temperatures are: = SCSEIr dr/ Th 1r dr, j = e,i.
f
j
(F.3.5i)
The line-average electron density is:
~ —f 1
r~~ii
~
o
‘~edr.
(F.3.5j)
The betas are computed as follows: .~
electron:
/~e =
ion:
f3X =
(F.3.5k)
,
[B~/8-if]
beam ion: /3~=
alpha:
(2/3)[(2/rs~r)Jo”~Eerdr]
~
=
(2/3)[(2/ç~r)f~.’rEjrdr] [B~/8~] (2/3)[(2/r2
)JrscrnBr
scr
(2/3)[(2/
(F.3.51)
drj
(F.3.5m)
,
r)Jø~’.’flEa’
dr]
(F.3.5n)
,
[B~/8~I
total:
(F.3.5o)
~
Here x stands for 0 (poloidal) or z (toroidal). For poloidal-beta, B9 is evaluated at r expressions are defined elsewhere: n~ (~ eq. F.3.7e), nE0(— eq. F.3.9b). The loop voltage is:
=
~scr
The following
(F.3.5p)
21rRq~(J—J~~0)~) Ir=r.-,’
where ~h is the resistivity (+— eq. 2.8a), J is the toroidal current density (4— eq. B.63), and beam-driven current (— eq. 2.9.2.2e). The rates of neutron production by D—D reactions are as follows (neutrons/second): thermonuclear:
NDDn
beam-plasma:
NDDn,
total:
ND~ = NDDfl +
thermonuclear:
N~Dfl=
beam-plasma: total:
N~Dfl=
j N~, dt’, t NI~Dfldt’, j
NDDfl
N~Dfl+ N~Dfl.
=
=
~beam
is the
(F.3.5q)
~fl~(aV>DDfl,
(F.3.5r)
“~DDn’
(F.3.5s)
(F.3.5t) (F.3.5u)
D is the density of deuterium. ~ ) DDn (-f-- eq. B.220) is the thermonuclear D(D, n)3 He reactivity. N~Dfl is defined in appendix B (— eq. B.320). The upper limit of integration, t, is the time at which the printout occurs. The lower limit of integration, timt’ is the time at the start of the run (+— eq. D.2276). Qtech is defined as:
Qt~h=
~b
[i— (total gain)
I
r=r~/(ot~
heating)
I
,~,.-,]
(F.3.5v)
G.E. Singer et at
/
One-dimensional plasma transport code
387
where Pf is the fusion power (+- eq. B.322), ~b is the input beam heating power (4-. eq. B.321), and the total gain (~ eq. F.3.3i) and other heating (~— eq. F.3.3h) are as defined for the third output page. Q~~h is defined and printed only when neutral beam heating is active. Note that for most cases of interest, “other heating” reduces to the absorbed beam power. The maximum change over one time step describes what is limiting the time step. It prints out ô~ (1— eq. 6b) in %, prints the physical quantity (density, energy density or B 9) which exhibited the maximum change in eq. 6b, and prints the number of the zone in which change was greatest. The conservation checks are (c.c. compression correction): for hydrogen species:
f~”[n~ — f,~,,(8n0/at’ S~)dt’] r dr + (c.c.)a
=
—
—
1,
a
=
1,2,
(F.3.5w)
,rwall
J
tla1~~1
0
for impurity species: (0na/8t’) dt’}rdr+(c.c.)a
t
jrw.u[fl a
CaCOflS =
J~
‘.‘
r
—
E1~~‘11~~a’ dr]
a
1,
=
(F.3.5x)
~
~
for electron and ion energy: =
c~cons
—
frW.’II
f
[(Ee + E~)—
~
aE
+
[(aEe/0t’
1/at’ [J~w5II(Ee+Ei)r
—
—
Q)
dt’] r dr + (c.c.)e
+
drj t~tmi)
(c.c.)~ —1,
(F.3 .5y) for B-poloidal: I
r
(ITR/2)f
=
“>
II
B~(r)rdr
0 ~ —
+ ~=InIt
f
c’rrRaEz(r = a)B9(r
=
1) dt
tm,
Q~Hrdr} dr + c.c.s}/[(~R/2)f0
rWII
2(r)r drj
—
1.
(F.3.5z)
B0
In these expressions, t is the time at which the printout occurs. The numerators represent values at the present time and integrals up to the present, while the denominators are taken at the start of the run, t,~ (+—eq. D.2276). The following expressions used here are defined elsewhere: (3fla/8t) (~eq.2a), S (~eq. 2.9.lla), (aE~/at),j= e,i (~eq.2b), Q~(~eq. 2.9.llb). If zones are added or removed during compression, then the compression corrections above account for the particles and energy which are added or removed. Compression heating and poloidal field energy changes are also included in the compression corrections.
The particle and energy conservation checks are generally of the order of 10-10 or less. The B-poloidal check is generally of the order of iO~or less. During neutral impurity injection (~— section 2.9.13), the conservation check value for the injected species may be somewhat higher. In cases where maximum values of density and/or temperature are set in the scrapeoff (+— C 129—C131), the particle and energy conservation checks are bypassed.
G.E. Singer et al. / One-dimensional plasma transport code
388
F 3.6. Scrapeoff summaty When a scrapeoff layer is included in the numerical model, page 7 of each large edit consists of a list of special quantities in the scrapeoff region: power (kW):
—
~
2j~f waIIQs
+
dr,
Q~)r
(F.3.6a)
the total power flowing to the divertor or limiter through the scrapeoff. ion fluxes into divertor (per second):
—
4if2Rf
maCSr dr,
(F.3.6b)
for each hydrogen and impurity ion. total ion flux into the divertor (per second): total of the ions in (F.3.6b). ion fluxes into the scrapeoff (per second): for each hydrogen and impurity ion. ion fluxes onto the wall (per second):
2RF~crFa(i~cr),
4TT
4¶R2rwall Fa ( rwaii),
(F.3.6c)
(F. 3.6d)
for each hydrogen and impurity ion. line den (ions/cm2):
f~’~’~e(1) dr.
separatrix radius:
r~
(F.3.6e)
(F.3.6f)
1.
zone:
j—
1,
r (cm):
rz
zone centered radius.
power (kW):
—
where j is the BALDUR zone index.
45rT2Rf‘~*1(QS +
ions/second:
—
4~2RJrJ+I~Ssr
the total ion flux for zone
(F.3.6h)
Q~)rdr,
the power following to the divertor or limiter in zone
(F.3.6g)
(F.3.6i)
j.
dr,
(F.3.6j)
j.
T~(eV):
7,(I~),
electron temperature.
(F.3.6k)
T~(eV):
Ti(rjz),
ion temperature.
(F.3.61)
ne:
n0(r~z), electron density,
(F.3.6m)
~‘~a(’~)’
a
total ion density.
(F.3.6n)
The remaining quantities are computed from the divertor model of section 2.9.5 and are evaluated at zone centers. ul (cm/s):
plasma flow velocity into the divertor.
(F.3.6o)
amachi:
the match number of plasma flow into the divertor.
(F.3.6p)
tau2:
divertor plasma opacity to returning neutrals.
(F.3.6q)
t2 (eV):
plasma temperature at the divertor plate.
(F.3.6r)
d2:
plasma density at the divertor plate.
(F.3.6s)
G.E. Singer et at
/ One-dimensional plasma transport code
389
F. 3.7. Neutral beam fast ion distribution summary The characteristics of the neutral-beam-injected fast-ion distribution are written by subroutine HPRINT. The definitions of the output quantities are given below. zone:
j
radius:
zone-centered radius,
—
1;
j is the BALDUR zone index.
where
,~.
average energy/particle:
parallel: local average of the parallel energy per particle Er(r)
=
fE~~ dEf
f~(ii,
E, r)~2d~i/n5,
(F.3.7a)
total: local average of the total energy per particle (see B.201) EB(r)
=
fE’~’,’~dEf
~
E,
r) d~/n8.
(F.3.7b)
taus: thermalization time; i.e., the time to slow from the local maximum fast-ion energy, Em~,to the local minimum fast in energy, E~ (see B.194) =
(3v~ 0)’log[(7~1+
1~)/(y~ + 1
v~0E1
v~eE~)j.
(F.3.7c)
beam dens: local density of fast ions (see B.195) 8(r) = dEff(~t,E, r) d~t, n
(F.3.7d)
j~m
ener dens: local energy density of fast ions (see B.202) n~(r)
(F.3.7e)
=nB(r)EB(r).
cos pitch ang: local average of ~ = v
mean:
1/v (see B.203) ,LB(r)
=
f~dEf
8, f(ti,
E,
(F.3.7f)
r)it d~/n
std.dev.: local standard deviation of the distribution in p~(see B.204) 1/2
I~E
a(r)=~f
2
f(ji,E,r)(~L—~) d~t/n~
mdEI
I~JEmi,
-
(F.3.7g)
~
cx power over heat + cx power: the fast-ion charge-exchange power loss divided by the sum of the beam heating power and the charge-exchange power (see B.197) f~~(r)
=
Q~(r)/(Q~(r)+ Qr(r)
+
Q~(r)).
(F.3.7h)
part sources: local rate at which fast ions slow down below E~ and enter the thermal ion population. ~a,th where a is the index for the beam injected species (see eq. (2.9.2.21) and B.150 and B.198). fusions: local rate at which beam—target D—T fusion reactions are occurring (see B.196) flDT(T) =n?J
cYDT(E)[2E/(AbmP)1
dEff(t&, E, r)~sdii,
(F.3.7i)
G.E. Singer etat
390
where n?
~ D for tritium beams, and n9 = 0 for hydrogen beams. ion heat fraction: fraction of local beam heating power which heat ions (see B.199) =
f.B(r)
fl~ffor
=
deuterium beams
/ One-dimensional plasma transport code
+
Q~(r)/(Q~(r)
~?
=
Q~(r)).
(F.3.7j)
The “totals” row entries are: avg. energy/particle: parallel: global average of the parallel energy per particle (see B.190) (Er)
U11B/NB,
=
(F.3.7k)
total: global average of the energy per particle (see B.192) (EB) =
UB/NB.
(F.3.7~)
beam dens: total number of fast ions (see B.187) NB = (2~rr)2RofOrdrf’E
dEf’f(r,
~s, E) d~L,
(F.3.7m)
ener dens: total energy in fast ions (see B.193)
~
=
cos pitch angle: mean: global average of
drfE~~.’~E dEf
,s
=
f~(ji,
E, r) dEL.
(F.3.7n)
v1/v (see B.191)
(,~)= (2~rr)~Rof°r drfEmxE dEJ’f~(~L,E, r)~dIL/NB, std. dev.: local standard deviation of the distribution in (aB)
=
(2~rr)2Rof°rdrf~’E dEf’f~(,i,
,i
(F.3.7o)
(see B.205)
E, r)(~i—
(~B)2
d~/NB.
(F.3.7p)
cx power over heat + cx power: volume integrated fast-ion charge-exchange power loss divided by the sum of the integrated beam heating power and the integrated charge-exchange power (see B.206) =
P~/(P~+ P~+ Pg).
(F.3.7q)
part source: volume integrated rate at which fast ions slow down below population (see B.207) =
(2~)2Rof °~B~h(r)r
~
and enter the thermal ion
dr.
(F.3.7r)
fusions: total rate of beam target D—T fusion reactions (see B.208) NDT
where n?
(2rr)2R0fñ~yr(r)
= liT
dr,
for deuterium beams,
(F.3.7s) n?
=
~ I) for tritium beams, and
~?
=
0 for hydrogen beams.
ion heat fract: fraction of total beam-heating power which heats ions (see B.209) FB =
P~/(P~+ p,B).
(F.3.7t)
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The totals at the bottom of page 6 are: total perp. energy: total perpendicular energy in the fast-ion distribution (see B.189) =
(2ir )2Rf r
drf5”E
dEf’J?(ti~ E, r)(1
—
~2)
d~,
(F.3.7u)
total parallel energy: total parallel energy in the fast-ion distribution (see B.188) 2dii,
(j~B=(2.~)2R 0frdrfEn~~xEdEJlfB(,j,
E,
(F.3.7v)
r)~
total beam energy: total energy in the fast-ion distribution (see B.193) UB
=
(F.3.7w)
U~B+ L~,
total fast-ion charge-exchange power loss (see B.210) =
(2rr)2Rof
n
drf ~~acx(E)[2E/(Abmp)1~/21
0(r)r
other energy loss rates: not presently used
(=
f’fB(,1
E,
r) d~,
(F.3.7x)
0.) (see B.215),
plasma heating rate: total beam heating power (see B.216)
2R
pB
+
=
=
(2’rr)
0f (Q~(r)+ Q~(r))rdr,
(F.3.7y)
energy check: fractional energy nonconservation (see B.212) F~= 1
—
(Ui~~ ~ —
—
~as +
mp)/~,
(F.3.7z)
particle check: fractional particle nonconservation (see B.214) F~=1 —(N~~ ~
(F.3.7A)
beam target rate: total rate of beam target D—T fusion reactions NDT (see F.3.7s above and B.217). F.3.8. Neutral beam profile printout The neutral beam deposition profiles are written by subroutine IPRINT. The first section characterizes the neutral beam injectors which are “on”. active injector no.: energy Wb:
beam index of each “active” beamline; beam voltage of the injector (see D.1159); power in energy compon.: injected power in each of the energy components (see B.128) pNB = (Wb/j)4°Hj~. The neutral beam deposition profile for each energy component is given in the next section.
zone: in-surface out: NB
,/
NB. ~‘)±i~
index of FREYA zone; minor radius of the inner and outer surface of the FREYA zone (see B.129)
(F.3.8a)
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h (r) at energy levels: neutral beam deposition function H( r) for each energy component (see B. 121a)
H~(r)=
f’H~(~, r) d~
percent loss: fractional shinethrough for each energy component (see B.131) L~B= I
—
(F.3.8b)
(2/a2)j°Hn(r)r dr.
The total neutral beam deposition results, summed over all energy components, are presented in the bottom section. zone: avg. rad.: total h(r):
cos of pitch ang: mean: std. dev.: deposition:
index of FREYA zone; zone centered radius of the FREYA zone; total neutral beam deposition function, H(r), summed over all energy components (see B.132).
local average of ~ = v11/v over deposition function (see B.133) local standard deviation of ~i distribution (see B.134) aNB(r); local particle deposition rate (see B.124 and B.135) ñNB(r)
=
n~1
NB/~tasmHn(~
I2NB(r);
(F.3.8c)
sink rate of thermal hydrogen due to charge exchange between neutral beam atoms and plasma ions (see B.126) n~(r); % for total h (r): fractional shinethrough of total neutral beam injected power (see B.1 37) 2) jH(r)r dr. (F.3.8d) LNB = 1 — (2/a Average for: cos of pitch ang: mean: global average of s = v 11/v over deposition function (see B.138)
(2/a2)f
=
charge ex.:
ñ
(r)r dr;
(F.3.8e)
volume average of ñ~~(r)(see B.141) =
(2/a2)f°n~0)(r)rdr.
(F.3.8f)
Total particles/second for: deposition: integrated particle deposition rate (see B.142) ~rNB
charge ex.:
=
(2rr)2Rof
h~(r)r dr;
integrated source of neutrals (see B.143) 2Rofñ~X(r)rdr. ‘a = (2~r)
(F.3.8g)
(F.3.8h)
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Lastly, for the full energy component of each active beam the “opacity” of the plasma integrated along the beam axis is given. beam no.: index of each active beam. optical depth to minimum: minor rad: the opacity to the point along the beam axis which has the minimum minor radius (see B.122) ~ major rad.: the opacity to the point along the beam axis which has the minimum major radius and is inside the vacuum vessel (see B.121b) ~
F. 3.9. Alpha particle page Characteristics of the D—T fusion product alpha-particle distribution are written by subroutine APRINT. The definition for the quantities listed under the column headings are: zone: radius: hi-e alphas: energy; slowdown t.:
j 1; where j is the BALDUR zone index; zone-centered radius, i~ local particle density of suprathermal alpha particles (see B.233) n a( r); local density of suprathermal alpha particles (see B.234) nEO(r); thermalization time; i.e., the time to slow down from the alpha particle’s initial energy, E0, to thermal energy (see B.232) ;0(r);
therm-fusion:
local thermonuclear D—T fusion reaction rate [see eq. (2.9.3.lb) and B.221]
—
nDT(r)
rate-total:
=
local total D—T fusion reaction rate [see eq. (2.9.2.2g) and B.168] ñDT(r)
1st-orb loss: qe:
=
17.6 MeV
=
+
n~T(r);
f
nDT(r)
dy/f
(
+
Q~+ ~
dv;
(F.3.9a)
time-integrated D—T fusion reaction rate (see B.241) nDT(r)
ion fract:
nDT(r)
local prompt-loss fraction due to alphas born on unconfined orbits (see B.226) L0(r); The fusion Q for the volume inside the present radius (see B.238) Qe
total fusion:
nDnT
=
f
‘~DT(’)
dt;
(F.3.9b)
fraction of alpha heating which heats ions (see B.242) =
f10)(r)
Q~(r)/(Q~(r)+ Q~(r)).
(F.3.9c)
The column averages are: hi-e alphas: volume average of the particle density of suprathermal alpha particles (see B.243) =
energy:
(2/a2)j’~n0)(r)rdr;
(F.3.9d)
volume average of the energy density of suprathermal alpha particles (see B.244) =
(2/a2)f°nn5(r)r dr;
(F.3.9e)
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/ One-dimensional plasma transport code
volume average thermonuclear D—T fusion reaction rate (see B.245) (n~~) = (2/a2)fñ~T(r)r dr;
rate-total:
(F.3.9f)
volume average total D—T fusion reaction rate (see B.246)
1st orb loss:
= (2/a2)f
ñDT(r)r dr;
(F.3.9g)
integrated prompt loss fraction due to alphas born on unconfined orbits (see B.247) L10~=
total fusion:
f~
dr/~DT;
(F.3.9h)
volume average of the time-integrated D—T fusion reaction rate (see B.248) (F.3.9i)
ion fract:
the fraction of total alpha heating which heats ions (see B.249) Fia
=
f
QE(r)r dr/{f (Q~(r)+ Qr(r))r
dr}.
(F.3.9j)
The column totals are: high-e alphas: total number of suprathermal alphas (see B.250) N0 energy:
=
(2~TT)2Rof
dr;
(F.3.9k)
total energy of suprathermal alphas (see B.251) 2R U,, = (2.rr) 0f nEa(r)r dr;
therm-fusion: volume integrated thermonuclear D—T fusion reaction rate (see B.252) 2Rof ñ~~r dr; volume integrated total D—T fusion reaction rate (see B.253) NDT
=
(21T)2R
(F.3.9n)
0JOañDT(r)r dr;
1st orb loss:
(F.3.9m)
(2~)
~
rate-total:
(F.3.9tf)
volume integrated alpha-particle loss rate due to alphas born on unconfined orbits (see B.254) NDTIoss =
(2~)2RofañDT(r)L,,(r)r
dr;
G.E. Singer et a!.
total fusion:
395
volume integral of the time integrated D—T fusion reaction rate; i.e., the total number of 3.5 MeV alphas and 14.6 MeV neutrons produced during the simulation (see B.258) NDT =
ion fract:
/ One-dimensional plasma transport code
(2.~r)2Rof’~nDT(r)r dr;
(F.3.9p)
total power for alpha heating of ions (see B.259) =
(2~)2RofQ~(r)r
dr.
(F.3.9q)
The lower hybrid heating, pellet injection and Hawryluk—Hirschman transport models each have associated printout pages which are activated by the models themselves and not by NLPOMTs. The He-influx model produces an additional line in the SPRINT short printout. The input-file ripple model produces a printout of values read from the file at time step 0; further ripple printout must be triggered by NLPOMT(7).
Appendix G. Units BALDUR uses three sets of units, called “external”, “internal” and “standard”. External units are the units used for input and printout. They may be changed without having to rewrite the whole program, but are fixed for a given version of the code. Standard units are cgs, and are used for formulae which one wants to be in the same form as “the literature”. For example, transport coefficients are computed in standard units. It is not expected that the choice of standard units will be changed during the life of BALDUR. Internal units are used when the transport equations are solved for a time step. They are supposed to be chosen so that quantities such as density and energy density are near 1. At present, they are fixed, but they may be changed from one version of the code to another. In all three sets of units, certain relations are assumed among the various units: Particle (and energy, etc.) densities are per cubic length unit; Current density (j) is in current units per square length unit; E-field is resistivity times current density; Power is energy per time; Flux is particles (energy, etc.) per square length unit per time unit; Area is square length, volume is cubic length, etc. In internal units, B, energy, length and time units may be chosen; the rest are determined by the previous relations and the following: energy = mv2/2, energy density = temperature X density. Maxwell’s equations in internal units are: ~‘x
B= 2ITJ+
(dE/dt),
2-iivx E= —dB/dt.
The last two equations determine E and J (and with the current density and resistivity relations, i~and I) from the four independent internal units. There are conversion factors to convert from one set of units to another. For external units, the names of the units are kept in variables. The conversion factors and names are set in subroutine UNITS.
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transport code
Unit
External
Standard
Internal
magnetic field (B) density energy (E) temperature (H) current (I) current densities (j) length (1) mass(m) power (P) resistivity (ii) time(t) electric field (E)
kG particles/cm3 J keV kA kA/cm2 cm g W m~2-cm s V/cm
G particles/cm3 erg erg stat-A stat-A/cm2 cm g erg/s s s stat-v/cm
10 G particles/p.m3 nerg (about 2/3 keV) 2/3 nerg 5 mA 5 mA/p.m2 p.m 10p.g 100 nerg/s (21T/S) x 10~’ m~l cm lOms 2~nV/cm
Conversion factors: To convert from standard units to internal, multiply: magnetic field density energy temperature current current density length mass power resistivity time electric field
by by by by by by by by by by by by
USIB USID USIE USIH USII USIJ USIL USIM USIP USIR USIT USIV
To convert from internal units to standard, multiply: magnetic field by UISB density, etc. by UISD, etc. To convert from standard units to external, multiply: magnetic field, etc. by USED, etc. To convert from external units to standard, multiply: magnetic field, etc. by UESB, etc. To convert from internal units to external, multiply: magnetic field, etc. by UIEB, etc. To convert from external units to internal, multiply: magnetic field, etc. by UEIB, etc.
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References [1] R.D. Richtmyer and K.W. Morton, Difference Methods for Initial Value Problems (Interscience—Wiley, New York, 1967). [21J. Dahlquist and A. Bjorck, Numerical Methods (Prentice Hall, Englewood Cliffs, New York, 1974). [31J. Chang, Lawrence Livermore Laboratory Report UCID-15992 (1972). [41J. Chang and G. Cooper, J. Comput. Phys. 6 (1970) 1. [5]M. Hughes and D. Post, J. Comput. Phys. 28 (1978) 43. [6]C.G. Lister, D.E. Post and R. Goldston, Third Symp. on Plasma Heating in Toroidal Devices, Varenna, Italy (1976). [7]J.P. Christiansen and K.V. Roberts, Comput. Phys. Commun. 7 (1974) 245. [8]D.F. DBchs, D.E. Post and PH. Rutherford, Nucl. Fusion 17 (1977) 3. [9]C.E. Singer, L.-P. Ku and G. Bateman, Princeton University Plasma Physics Laboratory Report PPPL-2414 (1987), to appear in Fusion Technol. (1988). [1015. Hirschman and D. Sigmar, NucI. Fusion 21 (1981) 1079. [illJ. Mandrekas, C.E. Singer and D.N. Ruzic, J. Vac. Sci./Technol. A 5 (1987) 2315. [121 W.W. Pfeiffer, RH. Davidson, R.L. Miller and R.E. Waltz, General Atomic Report GA-A-16178 (1980). [13] H.C. Howe, Oak Ridge National Laboratory Report ORNL/TM-9537 (1985). [14] J. Blum, Association Euratom Report EUR-CEA-FC-1120 (1981). [15] G. Bateman, Princeton Plasma Physics Laboratory, private communication (1987). [16] R. Wunderlich and K. Lackner, Max Planck Institut für Plasmaphysik, Garching, private communication (1986). [171 A. Silverman, D.E. Post, CE. Singer and D.R. Mikkelsen, PPPL Transport Group, Princeton Plasma Physics Laboratory, Applied Physics Division Report #23 (1983). [181 CE. Singer, M.H. Redi, D.A. Boyd, A.J. Cavallo, B. Grek, D.B. Heifetz, R.A. Hulse, D.W. Johnson, W.D. Langer, B. Leblanc, DR. Mildcelsen, F.G.P. Seidl, A. Eberhagen, 0. Gehre, F. Karger, M. Keilhacker, S. Kissel, 0. Kiüber, D. Meisel, H.D. Murman, H. Niedermeyer, H. Rapp, H. Rohr, A. Stabler, K-H. Stever and F. Wagner, Nucl. Fusion 25 (1985) 1555. [19] M.H. Redi, W.M. Tang, P.C. Efthimion, DR. Mikkelsen and G.L. Schmidt, Nucl. Fusion 27 (1987) 2001. [20] A.H. Boozer, Phys. Fluids 23 (1980) 11. [21] RJ. Goldston and H.H. Towner, Princeton Plasma Physics Laboratory Report, PPPL-1638 (1980). [22]N.A. Uckan, K.T. Tsang and J.D. Callan, Oak Ridge National Laboratory Report, ORNL/TM-5438 (1976). [23] P. Rutherford, private communication. [24]F. Hinton and R. Hazeltine, Rev. Mod. Phys. 48 (1976) 239. [25] R. Hawryluk, S. Suckewer and S. Hirshman, Nuci. Fusion 19 (1979) 607. [26] S. Hirshman, Phys. Fluids 20 (1977) 589. [27]C. Bolton and A. Ware, DOE/ET/53086-32 (October 1981). Note that table 2 of this reference should read d~ = —0.352 and b~ —3.42. [28] CS. Chang and F.L. Hinton, Phys. Fluids 25 (1982) 1493. [29]R. Hawryluk and S. Hirshman, private communication (1977). [30] C. Kieras-Phillips, Princeton University Plasma Physics Laboratory, private communication (1987). [31] W.M. Stacey, Jr. and Di. Sigmar, NucI. Fusion 19 (1979) 12. [32] S.P. Hirshman, R.J. Hawryluk and B. Birge, Nucl. Fusion 17 (1977) 611. [33]J. Ogden, G.E. Singer, D.E. Post, R.V. Jensen and F.G.P. Seidl, IEEE Trans. Plasma Sci. Eng. PS-9 (1981) 274. [34]L. Johnson and E. Hinnov, IQSRT 13 (1973) 333. [35] D.C. McCune and H.H. Towner, Bull. Am. Phys. Soc. 25 (1980) 966. [36] M.H. Redi and DR. Mikkelsen, Applied Physics Division Report No. 32 (January 1985) 21 pp. [37] D.F.H. Start and J.C. Cordey, Phys. Fluids 23 (1980) 1477. [38] D.R. Mikkelsen, G.E. Singer and Ri. Goldston, Princeton Plasma Physics Laboratory Report PPPL-TM-344 (1981). [39] W. Langer and C. Singer, IEEE Trans. on Plasma Sci. Eng. PS-13 (1985) 163. [40]R.L. Freeman and E.M. Jones, Cutham Laboratory Report, CLM-R-137 (May 1974). [41] J. Ogden and S. Bernabei, Princeton Plasma Physics Laboratory Report, PPPL-1615 (1979). [42] A. Kritz, private communication (1985). [43] L.M. Hively, Nucl. Fusion 17 (1977) 873. [44] S.L. Milora and C.A. Foster, Oak Ridge National Laboratory Report, ORNL-TM-5576 (1977). [45] W.A. Houlberg, private communication (1985). [46] W.A. Houlberg, M.A. Iskra, H.C. Howe and S.E. Attenberger, Oak Ridge National Laboratory Report, ORNL/TM-6549 (1979). [47] J. Behvich, J. Phys. Colloq. 3 38 C3-43-C3-52 (1977). [48] C. Mercier, J.P. Boujot and F. Werkoff, Comput. Phys. Coinmun. 12 (1976) 109. [49] WA. Houlberg, Oak Ridge National Laboratory Report ORNL/TM-8193 (1982).
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[501 GB. Tarter, J. Spectros. Radiation Trans. 10 (1977) 531. [51] D.E. Post and R.V. Jensen, At. Data Nuci. Data Tables 20 (1977) 5. [52] M.H. Redi, Comput. Phys. Comsnun. 49 (1988) 399.
275
BALDUR: A ONE-DIMENSIONAL PLASMA TRANSPORT CODE C.E. SINGER D.E. POST 1, D.R. MIKKELSEN ‘,M.H. REDI 1, A. McKENNEY 2 A. SILVERMAN F.G.P. SEIDL P.H. RUTHERFORD 1, R.J. HAWRYLUK 1, W.D. LANGER1, L. FOOTE ~‘, D.B. HEIFETZ ~, W.A. HOULBERG ~, M.H. HUGHES6 R.V. JENSEN 1, G. LISTER7 and J. OGDEN 8 ~,
~,
~,
‘Plasma Physics Laboratory, Princeton University, Princeton, NJ 08545, USA 2New York University, New York, NY 10012, USA ~ Corporation, Los Angeles, CA 90007, USA 4Los Alamos National Laboratory, Los Alamos, NM 87545, USA ~ Ridge National Laboratory, Oak Ridge, TN 37830, USA ~Grumman Aerospace, Princeton, NJ 08540, USA 7Max-Planck-Institut fur Plasmaphysik, Garching bei München, Fed Rep. Germany 85ch~,lof Engineering and Applied Science, Princeton University, Princeton, NJ 08544, USA Received 10 September 1984; in revised form 1 July 1986; in final form 18 September 1987
A version of the BALDUR plasma transport code which calculates the evolution of plasma parameters is documented. This version uses an MHD equilibrium which can be approximated by concentric circular flux surfaces. Transport of up to six species of ionized particles, of electron and ion energy, and of poloidal magnetic field is computed. A wide variety of source terms are calculated including those due to neutral gas, fusion and auxiliary heating. The code is primarily designed for modelling tokamak plasmas.
PROGRAM SUMMARY Tine ofprogram: BALDUR version BALDP47M
Keywords: plasma physics, thermonuclear fusion, tokainak, one-dimensional, transport
Catalogue number: ABBS Program obtainable from: CPC Program Library, Queen’s Urnversity of Belfast, N. Ireland (see application form in this issue) Computer: CRAY-i; Installation: Magnetic Fusion Energy Computer Center (MFECC), Lawrence Livermore Laboratory, Livermore, California Operating system: CRAY Timesharing System
Nature ofphysical problem The purpose of the version of the BALDUR code documented here is to calculate the evolution of plasma parameters in an MHD equilibrium which can be approximated by concentric circular flux surfaces. Transport of up to six species of ionized particles, of electron and ion energy, and of poloidal magnetic field is computed. A wide variety of source terms are calculated including those due to neutral gas, fusion and auxiliary heating. The code is primarily designed for modeling tokamak plasmas but could be adapted to other toroidal confinement systems.
Programming language used: FORTRAN High speed storage required: 721000 (octal) Number of bits in a word: 64 Peripheral used: disk No. of lines in combined program and test deck: 45271 (decimal)
Method of solution The plasma and the poloidal magnetic field encircling the plasma column are described by up to nine dependent variables, including up to six particle densities, ion energy, electron energy and poloidal magnetic field. These variables are functions of time t and radius r. The code was designed for a circular poloidal cross section but can be applied to constant effipticity ellipses with neutral beam injection.
OO1O-4655/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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The basic equations form a set of parabolic initial value equations with nonlinear coefficients and source terms. The equations are differenced with a conservative Crank—Nicholson scheme with an adjustable degree of implicitness [1]. Time centering of the source terms and transport coefficients is accomplished using a predictor—corrector scheme [21or by extrapolation methods [3]. Convective terms are handled with a variety of up-stream and down-stream weighting schemes [41. Monte Carlo techniques are used to compute (1) a-particle heating, (2) neutral gas transport [5] and (3) the ionization of neutral beams [6]. The evolution of the fast ion distribution, resulting from neutral beam injection, is computed as a function of energy and pitch angle using a simple multigroup Fokker—Planck scheme. Restrictions on the complexity of the problem BALDUR has been applied to a wide variety of tokamak simulations, and contains a variety of checks which terminate the computation if it goes outside the range of validity of the model. The most commonly encountered of these gives a message that the thermal velocity of the ions exceeds the neutral beam injection energy. Thermal collapse of the plasma edge will often drive the time step below a prescribed minimum (cf. section 6). However, care must be taken to provide physically reasonable initial and boundary conditions and to examine carefully the assumptions used in modeling a given device, especially when operating outside the range of parameters covered by published applications. For example, line radiation losses assume coronal equilibrium, and users must use the output to support their own calculations to assess the validity of this assumption. A number of common input errors are checked at the beginning of each computation.
Typical running time Execution time is typically but not exclusively between three and ten minutes on the MFECC CRAY, depending on the complexity and length of the discharge simulation, and the sophistication of the physical models taken from among the various choices available in BALDUR.
Unusual features of the program The majority of BALDUR is written in OLYMPUS [7] Fortran. NAMELIST is used for specifying input parameters. A choice among a variety of graphical and pnnted outputs is provided. The code also accesses US Magnetic Energy Fusion Computer Center Library subroutines such as their random number generator, which must be used to exactly reproduce the test case given below.
References [1] RD. Richtmyer and K.W. Morton, Difference Methods for Initial Value Problems (Interscience—Wiley, New York, 1967). [2] J. Dahiquist and A. Bjorck, Numerical Methods (Prentice Hall, Englewood Cliffs, New Jersey, 1974). [3] J. Chang, Lawrence Livermore Laboratory Report UCID15992 (1972). [4] J. Chang and G. Cooper, J. Comput. Phys. 6 (1970) 1. [5] M. Hughes and D. Post, J. Comput. Phys. 28 (1978) 43. [6] C.G. Lister, D.E. Post and R. Goldston, Third Symp. on Plasma Heating in Toroidal Devices, Varenna, Italy (1976). [7] J.P. Christiansen and Ky. Roberts, Comput. Phys. Cornmun. 7 (1974) 237, 245.
CONTENTS 3. Introduction 2. Diffusion equations 2.1. Diagonal anomalous diffusivities 2.2. Off-diagonal anomalous coefficients 2.3. Ripple diffusivities 2.4. Neoclassical transport 2.4.1. Simplified neoclassical diffusivities 2.4.2. Nearly exact neoclassical particle fluxes 2.5. Summed diffusivities and limits on diffusivities 2.5.1. Summation 25.2. Linear interpolation for scrape-off limits 2.6. Final diffusivities 2.7. Ware pinches 2.8. Resistivity 2.9. Sources 2.9.1. Sources due to neutral hydrogen isotopes 2.9.2. Neutral beam injection and fast-ion thermalization
277 277 281 282 282 283 283 285 285 285 286 286 287 288 289 289 290
2.9.2.1. Neutral beam injection 2.9.2.2. Fast-ion thermalization 2.9.3. Thermonuclear fusion power and fusion-product heating 2.9.3.1. D—T fusion 2.9.3.2. Catalyzed D—D fusion 2.9.4. Radiative losses 2.9.5. Scrape-off losses 2.9.6. Lower hybrid and arbitrarily specified auxiliary heating 2.9.7. Electron cyclotron resonance heating (ECRH) 2.9.8. Ohmic heating 2.9.9. Collisional energy interchange 2.9.10. Cold helium source 2.9.11. Pellet fueling 2.9.12. Recombination 2.9.13. Neutral impurity influxes 2.10. Minimum densities and temperatures 2.11. Compression
291 291 293 294 295 296 296 297 298 298 298 298 299 299 299 301 301
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3. Boundary conditions and density control 3.1. Boundary conditions 3.2. Density control 4. Initial conditions 5. Solutions of the diffusion equations 5.1. Space and time mesh 5.1.1. Space grid 5.1.2. Time grid 5.1.3. Variables used in code 5.2. Differencing 0CN 5.2.1. Crank—Nicholson parameter 5.2.2. Poloidal field equation 5.2.3. Particle and energy diffusion equations 5.2.4. Predictor—corrector and extrapolation 5.2.5. Numerical form for sources 5.2.6. Convection 5.3. Compression 6. Time-step control
302 302 303 303 304 304 304 305 305 305 305 306 307 308 309 310 310 311
7. Structure of the code and test run 7.1. Structure of the FORTRAN program 7.2. COMMON variables 7.3. Atomic data file 7.4. Ripple amplitude input array 7.5. Plotting 7.6. NAMELIST input description and test case sample run 8. Code version nomenclature Appendices A. Definition of matrices B. Formulae C. COMMON blocks D. Input variables E. Subroutines F. Sample run G. Units
277 312 312 312 312 312 312 312 313 313 316 328 329 366 372 395
LONG WRITE-UP 1. Introduction BALDUR is a one-dimensional (1-d) transport code designed to simulate a wide variety of plasma conditions in tokamaks. It solves equations similar to those outlined by Düchs, Post and Rutherford [8], but includes many additional physics effects and more flexibility in programming. BALDUR concentrates on a detailed treatment of neutral hydrogen transport, auxiliary heating, multispecies effects (including an extensive atomic physics package), orbits of alpha particles, plasma compression, ripple transport and edge processes such as “scrape-off’ losses. BALDUR is therefore complimentary to other similar transport codes which have more extensive treatments of other effects such as magnetic topology, MHD fluctuations, off-diagonal terms in the transport coefficients (BALDUR has a limited number of off-diagonal terms), etc. [12—16,45—49]. The BALDUR model has been used to aid interpretation of experimental data in smaller tokamaks and for evaluating designs of larger fusion devices. In this paper, we first outline the equations solved in BALDUR, then describe the numerical methods used, the structure of the program, the form of the input and output, and the results of a standard test case. The present work is limited to an overview of the code and its structure and a complete list of the input variables. More detailed information about the details of the code and its implementation on the US Magnetic Fusion Energy Computer Center are available from the authors of the present paper [17].
2. Diffusion equations This is the first of three sections of this paper which give a mathematical formulation of the physical problem solved in BALDUR. The formulae presented give a description of the physics used, except for corrections which are included merely to avoid singularities in the innermost computational zones. For clarity, a backward-pointing arrow in the form (4— eq(s). *) following a symbol or formula denotes that the equation(s) indicated by “*“ define that symbol or formula. We use Gaussian units except where indicated by appropriate subscripts. In the actual computations, inputs are given in the most commonly used engineering and scientific units and converted to a computationally convenient set of internal units (cf. appendix G).
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The diffusion equations solved in BALDUR are —--~-(rI~)+S
~
0,
8E
1
aB0
~2
a
a=1,2,te,h,
(2a)
j=i,e,
(2b)
a ~ a(rB9) ~
a~
a (2c)
~C~(TlJbeam),
where ~a is the number density of species a, E1 is the energy density of thermal ions or of electrons, and B~is the magnetic field along the (poloidal) direction encircling the plasma column. T~[*— eq. (2h)], q1 [4— eq. (2h)], S~[÷— eqs. (2.9a—b)] and Q~[*— eqs. 2.9c—d)] are the fluxes and sources defined below. i~ [— eq. (2.8c)] is the parallel resistivity, and ~beam [*— eq. 2.9.2.2e)] is the net current density driven by circulating fast ions. Consistent with the neglect of finite pressure effects on the magnetic geometry, the bootstrap current is also neglected. A version of the code which includes these effects has been developed by Bateman [15]. Theimpurities subscriptswith 1 and 2 refer to hydrogen andnotation the subscripts t° and h refer to 3He, 4He or heavier a given average atomic isotopes mass. The é,h refers to any of the four impurity species allowed in BALDUR version BALDP47M-PO9A. (In some places, we replace the subscripts I or 2 by D or T to indicate specific hydrogen isotopes.) The electron density is = ~B +
faKZ)aFn~n,
~
(2d)
a=1,21,h
where n 8 [~- eq. (2.9.2.2j)] is the density of fast hydrogen isotope ions due to neutral beams and (D.995) is a user supplied factor used to simulate hydrogen dilution. The contribution to ~ and Ee of electrons associated with fast a-particle is typically small and is ignored.2)~,J. WeThe use average the symbol KZ)O for ( )~ is the the meanover charge of species anda given denotespecies the mean by flux (Z surface. KZ)a and KZ2) average all charge statesa of on asquare given charge magnetic 0 are determined as follows: (~—eq.
KZ) a
2\c~r0~
~
a=
\Za~ /0 Z~, /Z\c0~~
‘
\/Z
Ia
‘
N = 2 ATOMC NATOMC=l,
(2d’)
where NATOMC (.— eq. D. 1023) is an input variable. K Z )coronal and K Z2 )coronal are the average values in coronal equilibrium, which are functions of electron temperature determined as described elsewhere [17]. This model is designed primarily for fully-ionized or high-Z impurities and should be supplemented by coronal nonequilibrium calculations when radiation from carbon or oxygen is significant. A computational package which does this is under development by Wunderlich and Lackner [161. The total thermal ion density (2e) a
defines a locally common thermal ion temperature
T~= (2E 1)/(3n1).
(2f)
The local electron temperature is T~=(2Ee)/(3fle),
where
tie
~
eq. (2d)] is defined above.
(2g)
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Table 1 Symbols not defined in text Symbol defined by Dependent variables in diffusion equations n,,, a =1,2 [~- eq. (2a)] n~,a — e’,h [4— eq. (2a)] E~,j e,i [~-eq. (2b)] B 0 [~- eq. (2c)] Independent variables r
3) density of hydrogen isotope a (cm density of impurity species a (cm3) energy density of electrons or thermal ions (erg/cm3) poloidal magnetic field (G)
minor radius (cm) eq. D.2276) time (s)
t
(‘—
Plasma geometry r,or
(‘— (4--
R
meaning (units)
(4—
C120, appendix D) inner boundary of plasma scrapeoff (cm) eqs. D.2047—D.2066) outer boundary of plasma (cm) eqs. D.2027—D.2046) major radius of magnetic axis (cm)
Temperatures and electron density [(.— eq. (20]
temperature of thermal ions (erg) electron temperature 3) (erg) [(b-eq. (2d)] electron density (cm
T
[(~- eq. (2g)]
Constants e
eq. eq. (~— eq. (4— eq. (+- eq. (i—
c
(f—
m,,
Standard tokamak variables Bz q = rBz/RB 0 C = r/R
(-
B.12) B.8) B.19) B.21) B.23)
electron or proton charge (statcoul) speed of light (cm/s) electron mass (g) proton mass (g) 3.1415926536
eq. D.1461) toroidal magnetic field at r = 0. (G)
(~eq. B.62) (~—eq. B.76)
safety factor inverse aspect ratio
Table 2 Mesh variables N 2 N+1 N+2
(,~ ~ (,~ + ~7)/r~ —
0.5[(r]~i)2
(rI~)2]/r2
2r~~/[(r)~.i)2 _(rb)2] r~~1 R n+1 t~±
(‘~‘),,÷~ = t,,~1— t,, —
e
r,,..~
NZONES LCENTER LEDGE MZONES XZONI(j) XBOUNI(j) DXBOUI(j) DXZONI(j) DX2I(j) DX2INV(j) RMINS (centimeters) RMINI (internal units) RMAJS (centimeters) RMAJI (internal units) NSTEP TAI (internal units) TBI (internal units) DTI (internal units) DTOLDI (internal units) ThETA=THETAI
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We now describe the fluxes T~and q~,and the sources Sa and in eqs. (2a) and (2b), and the terms in the magnetic diffusion eq. (2c). BALDUR draws from an extensive literature on the derivation of 1-d transport processes in tokamak plasmas. We therefore restrict ourselves to a survey of the formulae used and the appropriate references. The particle and energy fluxes for the most general case treated BALDUR are =
F~’°~ + F~ +
F~°’ — n
1t,1,
F~=F~”~ + F~+ F,~ —
—
2~ + F~°’ — flhVh,
Fh
~
qe=q~+~+O_q~,
r2 = F~’~+ F~+ F~ol =
(2h)
F~ + F~
q~=q~lex+qz+q~oI_q~f,
where
an D
Fttex
11
D12
D11
Dlh
0
0
D2~
D22
D21
D2h
0
0
D11
D12
D11
Deh
0
0
an2 F7X
(2i)
Dhl
Dh2
qfIex
Dei1~
De2T~ De~KZ)i1~ DehKZ)hl~
~eXe
0 0
aTe
qflex
D11T
D12T1
0
n~1
-~
Dhe
0
Dhh
D11T1
DIhTI
and the fluxes dependent on gradients of mean charge and fast ion density are: 2~
n1D1
pZ
rz =
q~ Z
n
1D?~
0
r~z
0
tl2L~~2~
ti2L.2h
n~D~
n~D~
0
nhDM
nhDl~Jl
0
ar _____
.
(2i’)
ar
fl~I~De~nh1~D~ XB~ pr~Z TnZ 0
q1 ~I~i~~I( ‘ih 1~-’Ih The particular form chosen for eqs. (2h) and (2i) is the result of the historical evolution of the code as progressively more extensive transport models were included. The models include a flexible combination of anomalous and neoclassical transport to be described next, contributions due to gradients in
+~
j =
$=1,2,e,h,i; except
(2j)
ts=$=i,
e,i
(2k)
of terms which can include anomalous diffusivities D~°m[— eq. (2.la)] and ~ eq. (2.lb)], thermal ion energy diffusivity x~’~ [~— eq. (2.3a)] due to toroidal magnetic field ripple (cf. section 2.3), and an [~—
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approximate treatment of neoclassical diffusivities D~°[4— eqs. (2.4.la—d)], and x7°[~— eqs. (2.4.le—f), and some of (C1,..., C5®)]. The elements of an input array are described in appendix D. This flexible approach to transport is useful in view of the current lack of an experimentally verified “first principles” transport model. We shall now describe each contribution to these diffusivities and then show the precise formulae [~— eqs. (2.5.la—c)] we use to prescribe the combinations, eqs. (2.6a—d). 2.1. Diagonal anomalous diffusivities The contribution from anomalous transport (i.e., transport due to plasma turbulence) to the diagonal elements of the matrix of transport coefficient in eq. (2i) is given by m =D~ + D~mfh0mp10c~~+D~, ~ = 1,2,~,h, (2.la) .D:o =
X~ +
+
j
~
=
i,e.
(2.lb)
Here we have separated anomalous transport into empirical models used to fit experiment, models motivated by simple theories of turbulence and semiempirical transport models. Minor modifications to the theories used by Düchs, Post and Rutherford [8], and a variety of ad hoc prescriptions to fit various experiments are described in appendix D. The use of ad hoc empirical prescriptions has largely been superseded by applications documented elsewhere [9,18,19] using the above semiempirical formulae, so the older empirical formulae are not described in detail here. The semiempirical diagonal diffusivities are =
a
~
,,
=
1,2,i,h,
(2.lc)
k
x7m~’~= ~x*flx~.s n
(2.ld)
j=i,e.
k
The scaling exponents eDflk and efflk are specified by NAMELIST input arrays (i— eqs. D.2537—2904). In the present version of BALDUR, the allowed dimensionless scaling variables are X~,,k= KZ)a, A~,K, B/Be, r/r~,~
ne/n, p/p,
A~,Zett, I Xne I,
A~, X
IXh~,(T
,
I
T~/T~,
q/q~,IA~H,1~~
1/7~); a=1,2,t~’,h,i,e,
(2.le)
for k = 1,. ,20, respectively. The exponent g = v’ (1 + 1/C295). Here Aa is the atomic mass of species a [for a = 1,2,ë~h;we set A, =Ae = =
i)tnei =r/Lne,
IX~i=r/L~, IA~eI=C293+(t~/Lre), ~
(2.lf)
where L~ =
Xa/I
axe/ar I
(2.lg)
is the scale height of variable Xa. Binomial averaging over five radial zones avoids numerical problems with these profile factors. The profile factor f,. (e.g., used to adjust edge transport) is given by (1, =
1 + (C~ — 1)[(r/a) ~C~,
r/aC291, —
C291]/(C292
—
C291),
C29~< r/a C292, r/a> C292.
(2.lg’)
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The circular cylinder approximation used to determine the safety factor neglects a significant singularity in the shear for tokamaks with a separatrix, so we add a correction to obtain for the shear
IXqi
QMHDI
i q—
q ar
i+
QMHD2
+c296,
(2.lh)
where q1~~ = q(r~~) is the “cylindrical q” as calculated in BALDUR at the separatrix. The diffusivities D,7a (4—eqs. (D.2881—2904), x~(*— eqs. D.2904—2912) and QMHD1 and QMHD2 (4—eqs. D.2913—2914) are given by NAMELIST input. To allow an option to keep semiempirical formulae from exceeding Bohm diffusion especially in the plasma periphery, D:~m1emP1~~~t and are limited to a maximum of C51DB (4— 2.lj) when C51 >0, where D’~= cTe/l6eBz.
(2.li)
The factors C2~—C296are typically used to prevent singularities (e.g., C293, C294, C295), or to omit terms such as the anomalous pinch in the scrapeoff (e.g., C2~—C292). 2.2. Off-diagonal anomalous coefficients The off-diagonal anomalous coefficients in eq. (2i) include only so-called thermal convection processes [7]:
D anom = C~D ea
pv
31
D anom = C’ D ~V
a=l,2,e,h,
(2.2a)
00
where ~ and ~ (4— eqs. D.2529—2530) are input constants. In particular, energy transport due to density gradients in the presence of trapped ion or electron modes [8] is ignored. 2.3. Ripple diffusivities Toroidal field ripple affects only the ion thermal diffusivity in BALDUR:
:~i:’
~
~
(2.3a)
where the Ripple-Trapping, Ripple-Plateau and Banana-Drift contributions are 2T~ [Gn(8fjX~IOO,C, =
32.880[(cTj)/(eRBz)]
=
(c1~5(~~)9 + Cl~(6JanabIe)8)DR,
X~’ (C~(~ec~)8+~
4l~~/2)3/2
+ GZ(~va~abIeI9=O,C,~=~/2)3/2],
(2.3b) (2.3c) (2.3d)
where DR ~3(~r/2) v1p91]/R, (2.3e) where iY, and ~ are the ion thermal velocity and poloidal gyroradius. In addition to the coefficients C141 and C1~,there is an input switch C1~to allow a choice of various combinations of X~T, x~and X~D. A proper choice of ripple transport model is particularly important for simulation where the form of the banana-drift coefficient X?D given here may be inappropriate [20]. A variety of methods are available for
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specifying the ripple amplitude, 6 [17]. The G factors in eq. (2.3b) are computed either as described by Goldston and Tower [21], or by Uckan et al. [22] depending on the input switch C149.
2.4. Neoclassical transport Two options are available for computing neoclassical particle transport coefficients. Most published applications of BALDUR use a treatment which follows a summary due to Rutherford [23] (similar to that of Hinton and Hazeltine [24] but with a more complete treatment of impurity transport). There are also three options for computing neoclassical ion thermal diffusivities. Should a more complete and exact treatment of neoclassical particle diffusion be desired, there is an option to include a nearly exact neoclassical treatment of particle fluxes [25,26] which is appropriate in BALDUR for cases with fully-stripped ions. 2.4.1. Simplified neoclassical diffusivities The diagonal neoclassical diffusivities for particle transport in eq. (2i) are: 2XeA~1(Te+ T,)/(7~3~/2B~)], a = 1,2,e,h, = [D~a + CHneq and
D,~ 0=
( (
~
D~n~~)+(~
b=1,2
~
D~n,,) +
b=1,2
(
D~~aKZ2)bnb),
b=C,h ~
(2.4.la)
(2.4.lb)
2) D~
0n~,
a
=
where DI~b= D~b=
/(CHIq2AjA~,/2)/(B~rI1/2),a = é,h; b (0,
f(C1Ajq2,L~2)/(B~r~1/2), a,b=i,h;
~
=
1,2;
C110 C110>0,
C1100,
(2.4.lc)
(2.4.ld)
(0, C110>0. Primarily to allow comparisons with codes that assume fixed isotope ratios, we have included an option which scales like neoclassical coefficients and tends to force similar profiles for both hydrogen isotopes: D~= fC356(1 ~ (0,
C1100, C110>0.
The remaining diffusivities, D~’a[4— eq. (2.4.lj)] and The diagonal thermal diffusivities are
{
X~
=
CEqSA n1~2[i B~T~
~
=
X~°” + X~°~”,
+
0.1(1.6 ++ 2Zeff)pe*] 0.73(1.6 2Z ) ~3/2
DIOb
[4—
+
1.13
(2.4.ld’)
(eq. (2.4.1k)], are defined below. + O.5Ze~~ +
0.59 + ZCff 0.55Z
(2.4.le) (2.4.lf)
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x~’ C~81=0, C12
X?OI1~I=
~
C281
=
1,
~
C281
=
2,
B~T~ (1+o.74e3/2v~)[1+1.o3(v~)1/2+o.31v~I 3”2v~+ [i + l.03(v~)1”2 + 0.31v~](1.76645e3v~)}, (24 if’) (O.661/2C~I) (1 + 0.74 where x~’ (~— C 281) and X~H are the ion thermal conductivities given by Bolton and Ware [27], and Chang and Hinton [28]modified for Zeti # 1 as described in the appendix of ref. [18]. Here HH =
for for
qs={~2
1
qI,
(2.4.lg)
q<1,
1”2(ec)2/3, C~ 1”2/e, AH = ~a1,2Aafa/~a=l.2fla, C CE and CH = 8(2rrm0) 1= C(2AHmP) 1 = C111 = 1”2, A~ = [Z~~ 2.67+ Zeff)]/[3.4(l.l3 + Zeit)], and P~ab= (A CH(mC/mP) 1( 0A b)/(Aa + A,,). Zett is the usual effective charge, A. and A0 the Coulomb logarithms, and p0” and v j!~ the electron and mean hydrogen collisionality. x?°~ is defined below. The off-diagonal diffusivities in this simplified neoclassical treatment are: particle density-gradient interactions involving at least one impurity, anomalous hydrogen mixing which scales like neoclassical transport, the pinch or screening of particle fluxes due to the ion temperature gradient, and thermal convection. There are also neoclassical contributions due to gradients in mean charge. The off-diagonal coefficients affecting particle fluxes due to density gradients are ~D~’aKZ)bfla, a = 1,2; —D~j~’(Z~0n0,a=i,h;
D~° ab =
D,~°=D,~°=
bb=l,2, = ~,h,
)aK)~a’
~DHana,
D~=
a, b
~
(2.4.Ih)
(a=t~’,b=h)
and
(a=h, b=e’),
(a=1, b=2)
and
(a=2, b=1),
=
(2.4.li)
i~,h.
(2.4.1k)
Particle transport due to an electron temperature gradient is not included. Since the particle transport is dominated by anomalous processes in most applications, this is thought not to be a significant omission. The particle transport coefficients due to an ion temperature gradient has only neoclassical contributions. Thus,
a!
D
ai
=
D”
0°=
—
~ nbDH0(KZ)h+KZ~b/2), ~b=t’,h nbD(1 + KZ)a/2) + ~ nbDia(KZ)b
—
a=1,2,
KZ) 0KZ),,),
a
(2.4.i()
=
b=1,2 The particle transport coefficients due to a gradient in mean impurity charge are D,~= —D~’0n,,, a=1,2; b=e,h,
(2.4.lm)
D,~=—D~KZ)ana, a,b=i!’,h.
(2.4.ln)
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The so-called thermal convection coefficients are defined below in section 2.6. To define these for the ions, we need the following definitions: D~°=1.5~ =
(D~~a
1.51
(D~’a”— D~b
D~=~1.5[
a=1,2,
(2.4.1o~
(D~
=
b=t’,h
~
D~4’bnb+
b=1,2
~
D~
(2.4.lq)
b=t,h
The part of x~° which depends on particle diffusion coefficients is Cony = DT!e~ X~ pv ai na, (‘~
(~)4 .-~.
1
r
a= 1,2,t,h
where ~
(~—
eq. D.2530) is an input constant.
2.4.2. Nearly exact neoclassical particle fluxes Here we outline the option which calculates all neoclassical contributions to particle diffusion in a thermal plasma with up to four fully-stripped ion species [25,26]. The only approximation made for this case is an interpolation between the banana and plateau regimes. For partially stripped impurities, we make the additional approximation that each isotope can be represented by a single species with charge
~~‘=
a
CI12I?”+C1III~
for C1100) C110>0\ for
a=1,2 t~’h
(2.4.2a)
the classical, Banana-Plateau and Pfirsch—Schlllter contributions. The present version of BALDUR assumes the “strong temperatures equilibration” approximation [25] in applying eq. (2.4.2a); the validity of this approximation for a particular simulation should be checked before using this model. 2.5. Summed diffusivities and limits on diffusivities 2.5.1. Summation First, the terms in the diffusivity matrix of eq. (2i) are summed. The diagonal terms in the neoclassical contribution can also be multiplied by constant factors. Thus, m+ D~°, a, fi = 1,2,e,h,e,i, (2.5.la) = .D:;° = X~m+ Cjx:00, (2.5.lb) =
~anorn +
X~~0
+
(2.5.lc)
~
where C 9,
Ca,Cj= C10, C11, C12,
=
=
a=1,2, a=I,h, j—e,
(2.5.ld)
j=i,
D:r[1 + (Ca — C-5~°, j
=
i,e.
1)6ap],
a$ = 1,2,t’,h,
(2.5.le) (2.5.lf)
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The Kronecker delta symbol 6a~ is used to compact the notation, and we have already defined the diffusivities D~°m[4— eqs. (2.la), (2.2a)], Da7’ [~— eqs. (2.4.la), (2.4.lh), (2.4.li), (2.4.U’), (2.4.10) and (2.4.lp)], x~°m [4— eq. (2.lb)], X~PPk[4— eq. (2.3a)], X~° [4— eq. (2.4.le)] and x~° [+— eq (2.4.lf)]. .
2.5.2. Linear interpolation to scrape-off limits Next, a constant empirical limit can be set on the diagonal particle and electron energy diffusivities in the scrape-off region r> r~~ 1 1a, a=l,2, DaaC53+D0
C
1~aa=
54
a = (,h, r> ~
+ D,~a,
C120> 0, (2.5.2a)
~eC55,
where D~0is defined by eq. (2.4.lb). Using this model sets similar limits on the respective contributions of D11 and Daa of eq. (2.2a) to the electron and ion thermal convection. Continuity in Daa and Xe i5 maintained by linear interpolation to the values (computed as above) at a specified fitting point = C5,, rscr:
=D:~mI~C
1~aa
56~ + [(r
(C~= C53 Xe
for
a
C56r1~1)/(r1~. — Cs,,r~r)] (c~— D,~mI0~550
1,2; C~= C54 for a
=
X~~~mir=Cs6r, +
—
[(r
—
C56rsCr)/(rscr —
=
e,h); a
=
~s,,rscr)I (q~~m
i,2,e’,h, —
)
+
C56r~~~ < r <
‘scr’
(2.5.2b)
~eir=cs6r,~,) + D~a, (2.5.2c)
It is also possible to override other computations of diagonal scrape-off diffusivities by specifying C83 giving D00 = CS3DB + 8
Xi
=
D~0 for r> ‘scr’ a = i,2,e,h, C83 for r> rscr, j = e,i.
>
0, C120> 0,
>
0,
(2.5.2d) (2.5.2e)
C811.5D
Thus for r < C mand ~e = X”m~These limits can be helpful when the user has an a we have D,,~ = D,~ priori empirical56i0~, estimate of the magnitude of the diffusivities in the plasma periphery. 2.6. Final diffusivities To summarize, the diffusivities are sums which can include anomalous, ripple and neoclassical contributions. In addition to the scrape-off limits on Daa and Xe given in eqs. (2.5.2b)—(2.5.2e), a maximum can be placed on the diagonal diffusivities. Stated symbolically, the final diffusivities are Daa
=
min(C
a
43,D00),
=
l,2,é’,h; r
< ‘scr’
(a=i,2; T>TScr~ C830 ~a=l,h; ~ C830 m+D~°~ a#$; a,/3= i,2,t’°,h,e,i, —
Daa=flMn(C43,Daa),
and C530, and C540,
(2.6a) (2.6b)
Da~=D~°
1.5D
11, Dea=
i.5[DIl+(1_l.5(rfl/rT)DT1},
—
i.5D11,
—
1.5[D11
+
(i
—
1.5(~/r~))DTuj,
a=1,2,i,h, C910, a=1,2,e,h, C91>0, a a
= =
l,h, C91 0, i,2,~,h, c91 >0,
(2.6b)
(2.6.b
)
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(C520, ~C >0
287
(2.6c)
52—
r< Xemin(C
{r>r~r,Cg3O
44,Xe), XB
where
=
DTI
Xe
and
C550,
1.5D11,
—
(2.6d) (2.6d’)
is a transport coefficient based on the theory of trapped ion instabilities [7,33].
2.7. Warepinches The total pinch flux for particles of species a is I
~
+ navware, H .
I a nv=c . .. a a ~psem,emplneal
a a
=
=
1 ,2 , I,h,
where rem1emp!~aI =
na~ ~ ,,
fl
(2.7b)
~
k
where v1~’ (4— eqs. D.2857—2880) and eVflk (4— eqs. D.2537—2616) are input constants, and Xak is the dimensionless scaling variables defined in section 2.1. The inward energy fluxes accompanying these particle pinches are qve
=
C~’VTe
~
na
(2.7c)
a=1,2,e,h
q”
=
nava + qware
CJ~~TI ~
(2.7d)
a1,2,t,h
where C~(i— eq. D.2529) and ~ (i— eq. D. 2530) are input constants. The values of the Ware pinch velocities in eq. (2.7a) which describe the flow of hydrogen isotopes due to the presence of a toroidal electric field E,, are given by a formula [29] very similar to that of Hirshman and Sigmar [10], U~iare=
C19L13(Zeti, vC*,)(cl,hJ/B4).
(2.7e)
Here L13 (4—eq. B.115) and L23 (4—eq. B.116) are dimensionless functions with values often of order unity, and flh (~- eq. 2.8a) and J (4—eq. B.63) are defined elsewhere. The analogous impurity pinch is neglected [10,29]. The Ware electron energy pinch is of the following form: ~
~ Ware
T ~
nav~are
a=1,2 flaV~jare
if
C3650,
if
C365 > 0,
a1,2
where V~jare =
C365C19L23 (Z~, ~ )( c’qhj,/B0).
(2.7f)
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The Ware ion energy pinch is i~’i ~r L~pylj
n0vHWare
qware=
0
~ç 1
‘-365 —
if
C365>0.
If the beam-driven current in included as described below, then E. i~Jand the present treatment of the Ware pinch is incorrect. A more complete treatment of the neoclassical effect of neutral injection on particle transport [31,32] has not been included. 2.8. Resistivity The plasma resistivity used to calculate the ohmic heating
[4—
eq. (2.9.8a)] is [17,25]
(2.8a)
C,j/(neTeAeF),
flh
where F—
C,~ fTrap
—
if if
C,~1>0, C~1 0,
(28b)
(see appendix B for C~(4— eq. B.46), i-~ (+—eq. B.73), AE (*—eq. B.65) and fTrap (4—eq. B.81)). The resistivity used in the poloidal magnetic field diffusion equation (2.c) is =
C399~,
(2.8c)
unless: (1) C478 is used to limit the current density in the central region when q( r) < 1 (this simple ‘sawtooth model’ is intended to be used as a substitute for — and not in conjunction with — the more elaborate sawtooth model described below), or (2) a scrape-off region is included in the simulation. (Note that in these cases the equilibrium toroidal electric field will not be constant as a function of minor radius and the B~01conservation check may not be as small as usual.) These simplified models should be avoided in cases where the associated poloidal field energy is significant. If C478> 0, the resistivity is modified to limit the current density inside the q = 1 radius: = C3~i1/(i
+ 6[i
(r/rl)21
—
where r1 is the outermost q
=
},
r
1 radius, ~
< =
r1, ~h(r1) and 6
(2.8d) =
min(2, C478).
If a scrape-off region is included in the simulation the resistivity is modified in the scrape-off region [19] r>r1~~,
(2.8e)
where = S
31Jh, ih1lim~ (10
i~ti i~i ~max, <~s,max,
(2.8f)
where ~1im =
(7~/LeJ)ln(A),
[3, A=\(~,max+~)/(J~,max_~),
=
(eL/4)(ne
—
(2.8g) Jt<0, ~
n~)(1/’rii+ 1/~~).
J~[4— eq. (2.9.8b)] is the toroidal current density, and L, ~ and ~ are defined in section 2.9.5.
(2.8h) (2.8i)
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An option is available to flatten temperature and density profiles periodically inside a sawtooth mixing region defined by helical flux conservation (.i— C~2—C~9).
2.9. Sources The sources or sinks in the diffusion eqs. (2a—b) are computed with a variety of algorithms. These algorithms (denoted by a mnemonic superscript) include neutrals (N), neutral beam injection (B), fusion (F), radiative losses (RAD), scrape-off losses (S), lower hybrid heating (LH), electron cyclotron resonance heating (EC), ohmic heating (OH), collisional ion—electron energy interchange (IE), a volume source of cold helium from the wall or limiter (HE), recombination (REC) and neutral impurity influxes (IMP). In particular, Sa=S~+S+SaF+S+Sa~, Sa Qe
=
S+ S~+ SaHE+
=
Q~+ Q~+
S~MP,
a
a=1,2, =
(2.9a)
i,h,
(2.9b)
Q~+ Q~ + Q~+ Q~H+ Q~+ Q~+ ~
+
Q~+ Q~,
(2.9c)
~
(2.9d)
For each source we now give a brief description of the nature of the model (including any references to more complete descriptions), formulae for the contributions of each model to the sources in the diffusion equations, and a list of the options available for implementation of the model. 2.9.1. Sources due to neutral hydrogen isotopes The neutral gas source terms are computed using a Monte Carlo algorithm [5]. The calculation involves computing the trajectories of a specified number of sample particles (.— eq. D.1010), and then using these trajectories to calculate the source rates to be used by the energy and particle balance equations. Up to three types of neutral sources are computed for each hydrogen species: edge sources due to the recycling of plasma ions and neutrals, edge sources due to gas puffing and volume sources due to charge exchange between injected beam neutrals and plasma ions, and due to radiative recombination of plasma ions and electrons. Thus a total of up to six neutral source types may be included for a run with two hydrogen species. The volume sources due to recombination and the edge sources due to gas puffing may be omitted (4— eq. D.1030, D. 1710—1729 and D.1808—1847, cf. also sections 2.9.12, 3.1 and 3.2). Because the Monte Carlo computations are expensive, they are only done every ten time steps or so (o— eq. D.lOlOa). Separate computations are done for each source type, using a dimensionless neutral source rate of 1.0 for each type. The neutral temperature and density profiles and the ion and energy source terms computed based on these normalized neutral source rates are then linearly rescaled to physical units each time step. The source terms computed are (cf. refs. [17,34])
5:4=
~ Yk{ ~ nenak
+
~
[flb~~a0k<0t~akb>p
+ (1
—
8ab)(fl°akflb
b =1,2 =
—
~ k1,N5
Yk{
~ a=1,2
[t1et~k<0t1>e
+
~
h1b1~k
b=~1,2
E~01~}
a
=
1,2,
(2.9.la) (2.9.lb)
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and Q,N =
~
Yk{
k=1,N
~
+
a=1.2
5
~
+
{fle’~kKGu’)e
~
nb~kKa0akh)pj~
b=1.2
flbnak(aVakb)cx(Ea8TI)+(beams)}.
(2.9.ic)
a=1,2 b=1,2
Here Yk is the scaling factor for neutral source type k [17]. N8 is the number of source types, n~8is the unscaled density of species a neutrals originating from neutral source type k, and 6~,, is the Kronecker delta. ~~ v )e is the rate coefficient for electron impact ionization, and K ev0 kb )~,and K “ Va kb )~are the rate coefficients for proton impact ionization and charge exchange, respectively, of species a neutrals originating from source type k interacting with particles of species b. Proton ionization may be omitted via an input switch (~eq. D.2528). The input parameter E~00~1(4—eq. D.1804) representing the average energy lost to the plasma per ionization is typically set to 40 eV. E,~8 is the temperature of species a neutral originating from source type k. E0°is the local temperature of species a neutrals, computed as Ea~
~ k=1.N~
yknakEak/
/
~
‘fk’~ak~
(2.9.ld)
k=1,N5
The density of neutral species a is (2.9.le) k
1, N5
We should note that the Monte Carlo algorithm in the present version of BALDUR includes the approximation that neutrals are launched after every charge-exchange event from a Maxwellian distribution characterized by the local thermal ion temperature T1. Under some circumstances, this can cause noticeable error in the ion energy balance [35], a fact which should be kept in mind when interpreting the results from simulations where charge exchange is an important energy transfer mechanism. The total neutral hydrogen density (2.9.lf)
is also used in equations for slowing down of fast ions and in computing losses due to parallel flow in the scrape-off region, as described below. 2.9.2. Neutral beam injection and fast-ion thermalization The neutral beam heating model calculates the neutral beam deposition and the thermalization of the beam-injected fast ions. Monte Carlo methods are used to model the penetration of a finite-size beam of neutral hydrogen atoms (H, D or T) into a torus with concentric elliptical magnetic surfaces. When these neutral atoms lose an electron in a collision with a background plasma particle, they become part of the suprathermal fast-ion population in the plasma and an electron is added to the plasma electron population [4— eq. (2d)]. Coulomb collisions with background plasma ions slow the fast ions and heat the plasma. When fast ions have slowed to the average energy of the thermal plasma ions, they appear as a source of plasma ions in the ion particle diffusion equation. Coulomb collisions also change the pitch angle of the fast ion; i.e., v11/v is changed. Charge-exchange reactions between fast ions and the thermal neutral gas which is present remove particles from the fast ion population. The beam—target fusion reaction rate [43] between fast ions and background D and T ions are computed. Background ions are assumed to be thermal with zero average velocity and beam—beam reactions are neglected. Velocity space effects of major radius compression of the plasma are modeled by relocating the bins in v11/v space and redistributing the
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291
fast-ions among new energy groups in a manner which conserves the magnetic moments and canonical angular momenta of the fast ions. 2.9.2.1. Neutral beam injection. The Monte Carlo code FREYA is used to model neutral beam penetration in a torus [6]. The results are: (1) a radial array of the volume source-rate of thermal neutrals S,~BCX, created by charge-exchange reactions between the incoming neutral beam atoms and the thermal plasma ions; and (2) the source of fast ions (given as a function of radius, ~s= v11/v, and energy) created by ionization of the incoming neutral beam atoms. The Monte Carlo sample neutral-beam particles are initialized with an energy of W,, (4— eqs. D.1159—1168), W,,/2 or W0/3 at a randomly chosen location on the rectangular (or circular) face of the neutral beam’s ion source The sample particle’s initial velocity includes components for focussing in the horizontal and vertical directions and randomly chosen components which model Gaussian beam divergence in both the vertical and horizontal directions. The neutral beam injection is then oriented in the tokamak coordinate system by a series of two translations and four rotations. The sample particles are then tracked in a straight line until they: (1) do not pass through the beam port at the vacuum vessel (in this case the particle is initialized again and there is no power loss), (2) are ionized in the plasma at a location determined by Monte Carlo techniques, or (3) strike the interior of the vacuum vessel after passing through the plasma region without being ionized. If the sample particle is ionized by a charge-exchange collision with a plasma ion it contributes to the volume neutral-atom source array which is used by the neutral gas transport code. When any kind of ionization of the sample particle takes place, the value of p. = v1/v is calculated and the source array for the fast-ion distribution is incremented. 2.9.2.2. Fast-ion thermalization. The fast ion population is represented by a nonthermal distribution which is a function of radial zone, energy E, and the cosine of pitch angle, ~z= v11/v. The equation for the time evolution of this distribution is a version of the Fokker—Planck equation which includes the effects of average drag and pitch-angle diffusion caused by collisions with the background plasma, and the loss of fast ions caused by D—T fusion reactions with the background plasma ions and by charge-exchange reactions with the neutral gas. The output of the fast-ion model is calculated by integrating over the distribution function to obtain the sinks and sources to both energy and particles which are used by other parts of BALDUR. The equation for the time evolution on the fast ion distribution, f,,~(p.,E, r) (i— eq. B.147) is
—r~~-
a
=S:4BI_(r~X+VDT)faB+
~
a
ala 2)~~_ , (1~~.p.
where the average-drag energy loss rate is —W=2H 5(v~E+y~1/f~).
(2.9.2.2b)
The source term SBI (f— eq. B.125) is calculated by the neutral beam injection code (see section 2.9.2.1). ~ (~- eq. B.163) is the charge-exchange loss rate, and ~DT (f— eq. B.167) is the fusion reaction rate, v~ and ~ (4— eqs. B.160 and B.161) are the energy loss rate coefficients for collisions with plasma electrons and ions, respectively: y5~(4— eq. B. 159) is the pitch angle scattering rate coefficient. The charge-exchange loss rate, ~ includes the “fudge reactor” H1 (4— eq. D.1229) to correct for the neglect of reionization of the escaping neutral fast ions. The slowing down rate, W, includes the factor H5 (4— eq. D.1233). The effects of major radius compression are modeled by redistributing the fast-ion distribution in E and p. space as dictated by conservation of magnetic moment and canonical angular momentum. Under the conditions of adiabatic compression of the major radius from R1 to R2, 2E the perpendicular energy of a fast ion scales as E~—‘ CE~ and the parallel energy scales —* C 11, where C = R1/R2 is the
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compression ratio. The radial zone of the fast ions is unaffected by compression, the absolute radius of each zone scales with the change in the plasma column minor radius, i.e., r c~a cr C°5. The density of fast ions scales inversely as the zone volume, i.e., cr C2. The beam-driven current, ~beam’ in eq. (2c) includes terms for toroidal effects on ion orbits, pitch-angle scattering of ions into trapped orbits and the neoclassical electron return current. The cylindrical approximation to the circulating fast-ion current, 1~eam (~—eq. B.178) is ~beam
Jf
=
‘f(p., E, r)p.[2E/(A,,m~)~1/2 du,
(2.9.2.2c)
where Emax (4— eq. B.144) and Emin (+— eq. B.145) define the fast ion energy range, and A,, (~—eqs. D.1069—1078) is the atomic mass of the beam injected fast ions. Toroidal corrections can be applied to 1~eam to produce ~i~eam ~r.)’~eam(1
—
H
1(m)),
(2.9.2.2d)
3Zeft~)~TT/(2K
where
1(m)=
K(m),
H
c~,
H
4>0,0
K ir/2,
m=2t,/[(1+f)p.~,];
H40;
where (— eq. B.179) is the aspect ratio r/R0, and K(m) (f— eq. B.183) is the complete elliptic integral of the second kind. p.1, = Rtang/(r + R0) is an estimate of the average initial value of v1/v of the injected fast ions where Rtang (+~. B.180) is the tangency radius of the neutral beamline. The user supplied factor H3 (f— eq. D.i231) controls the approximate corrections for pitch-angle scattering of fast ions onto bananatrapped orbits. The user sets H4 (+—eq. D.i232) to include (H4> 0) or exclude (H4 0) a correction for the toroidal variation of v~1. The total beam-driven current, ~beam (*— eq. B.184) includes the electron return current (complete with a trapped-electron correction): ~beam
=
CjJ~eam(1—
(1/Zett)E1
—
((1.55
+ 0.85/Zeft)\/~
—
(0.20 +
1.55/Zett))/(1
+ H2pe*
)1 },
(2.9.2
.2e)
where e = r/R0, C1 is a user prescribed factor (— eq. D.1068), and the user prescribed factor H2 (— eq. D.1230) controls the (typically small) effect of electron collisionality, i.~” (— eq. B.75). Eq. (2.9.2.2e) represents a fit to the results of Start and Cordey [37]; the term H2 i~’~is a generalization of their results to finite Ve*. The net hydrogen particle source of thermalized fast ions, h~th (~— eq. B.150) is given by the rate at which fast ions join the thermal ion population minus the beam—target fusion rate: 4a,th =
6aa’f1 d~i[
—
14/(Emin)1 f~’~(p., E,,,,,,, r)
—
6a?~DT’
(2.9.2.2f)
‘
where a’ is the injected hydrogenic species, and ? = T if the fast ions are deuterium, or ? = D if the fast ions are tritium. The beam—target fusion term is the rate at which beam—target fusion reactions remove thermal hydrogen ions from the background plasma: ~DT
=
j
dEfn~aDT(E)fa(~, E,
r)[2E/(A,,mp)I1~’2 dp.,
(2.9.2.2g)
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293
where 0DT (~—eq. B.164) is the cross section for D—T reactions between a fast ion and cold background ions. The net hydrogen ion particle source-rate in eq. (2.9a), ç’B...
-B
.B
— ~NB
nacx, is the sum of the source of thermalized fast-ions, ~~th [4— eq. (2.9.2.21’)], the source of thermal hydrogen ions due to fast-ion charge-exchange reactions with thermal neutral hydrogen (4— eq. B.169), —
nath + nacx
L,,,~,1_1
=
E,
r) dp.,
(2.9.2.2i)
and the loss of hydrogen ions which charge exchange with injected neutral beam atoms, ~ (see section 2.9.2.1 and B.126). We note that the electron density ne (- eq. 2d) responds instantly to changes in the fast ion density (4—eq. B.175) nB =
J
dEffa(I.L, E, r)
dp..
(2.9.2.2j)
Thus, beam injection may cause a brief decline in T~when cold electrons from the incoming beam neutrals dilute the available electron energy before the beam ions have appreciably heated the electrons. The beam heating of electrons, Q~(4— eq. B.151) in eq. (2.9c), and beam heating of ions, (4— eq. B.152) in eq. (2.9d), are given by integrating each energy loss rate [eq.(2.9.2.2b)] over the fast ion distribution:
Qr
Q~=
=
H5J~”2y~EdEf f~(p.,E, r) dp., HSJ Em~(2’,sj/v~)
dEf f,~(p.,E, r) dp. + fdp. ~
(2.9.2.2k) —
J~’(E,p~n)1 f(p.,
~
r). (2.9.2.21)
When deuterium beams are injected into a plasma containing deuterium, the neutron production rate for the reaction 3He+n(2.5 MeV), (2.9.2.2m) D+D—. due to beam—target reactions between fast ions and background ions is calculated for comparison with experimental diagnostics. These reactions do not affect the plasma particle or energy sources or the fast ion distribution. The beam—target neutron production rates is (4— eq. B.173), ‘~DDn = ‘E,~~~1-l
where
GDDn
(2.9.2.2n)
nDGDDn(E)[2E/(Abmp)lfa,
(~ eq. B.172) is the cross section for the reaction in eq.
(2.9.2.2m).
2.9.3. Thermonuclear fusion power and fusion-product heating There are several features which are common to the various thermonuclear fusion models available in BALDUR. These are the calculation of “sink” rates of reacting fuel species, source rates of charged fusion products that have thermalized, and the heating of plasma ions and electrons by charged fusion products. The compression heating of suprathermal fusion products is not modeled, and the distribution in scaled
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minor radius, r/a, is unchanged by compression; the compression modeling simply moves the fast ions with the plasma as it compresses in minor and major radius. If NTYPE = 1, the density of alphas in each zone is unaffected by compression; i.e., the number of alphas is not conserved. 2.9.3.1. D— T fusion.
D
+
The fusion reaction rate for
T —*4He (3.52 MeV)
+
n (14.1 Mev)
(2.9.3.la)
is calculated for reactions between tritium and deuterium ions which are in either the neutral beam injected fast-ion population or the thermal population [37]. The beam—target reaction rate is calculated in the fast ion code [see eq. (2.9.2.2g)] and the thermonuclear reaction rate is calculated in the D—T alpha thermalization code. The sum of these reaction rates is the rate at which 3.5 MeV fusion-product alpha particles are produced in the plasma and enter the suprathermal population. The prompt-loss fraction is calculated by determining the fraction of alpha particles which are created on unconfined trajectories and promptly collide with a toroidal surface of major radius RCWALS (4— eq. D.1438) and minor radius RDWALS (— eq. D.1439). Coulomb collisions with plasma ions and electrons slow down the confined alphas over a finite time and heat the plasma. When the alphas have thermalized, they enter the source rate of 4He impurity ions. The sink rate of thermal D and T due to the D—T fusion reaction, S (~—eq. B.222) is given by S,,F=h~T=nDnT(av)DT, a=D,T, where an analytic approximation to reaction is given by 1~DDn=
KGV)DT
(2.9.3.lb) (~—
eq. B.2i9) is used. The reaction rate for the D(D,n)3He
0.5n~/av)DDfl,
(2.9.3.lc)
and KOV~DDfl (~— eq. B.220) is the D—D fusion rate coefficient. This latter reaction rate is used only to calculate the neutron production for diagnostic purposes and is not used to calculate a sink of D, a source of 3He or plasma heating. The different versions of the D—T plasma heating model are based on the same alpha-particle energy-loss model. Coulomb collisions with plasma electrons and ions remove energy from suprathermal alphas at the rate _E,y=fs0[CaeXe3/(1+0.75Xe3)+Cai1/Va,
(2.9.3.ld)
and Caj ~ eqs. B.227 and B.229) are the coefficients of the electron and ion heating terms, is the ratio of the speed of the alpha particle, Va~ and the thermal electron speed Vthe~ f~’5 (~— eq. D.2527) is a factor which varies the slowing down rate. In the limit of small the fraction of the initial energy which is deposited in the ions over the course of the thermalization process is where
Cae
= V,,,/V~
Xe,
2 c~_udu h~(x~)=—~i 3 x~’0
1+u
= (2/x~){[arctanE(2x~-1)/V~I+1T/6I/~_ln[(1
+x~)2/(1
-x~+xflj/6}, (2.9.3.le)
where x~is the ratio of the initial energy, En 0 = 3.52 MeV, and the critical energy E~(4— eq. B.230) is x~= Eao/Ec. By setting the input variable NTYPE = 3 (4— eq. D.1440), the user selects the most elaborate nonlocal D—T plasma heating model in BALDUR. The suprathermal alpha distribution is presented by a group of sample alpha particles which are selected by Monte Carlo techniques from the confined alpha population.
given
by
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295
Each sample particle’s trajectory in the poloidal plane is calculated, the plasma heating is calculated along the trajectory, and the collisional perturbations of the trajectory are also included. The model with NTYPE = 2 is very similar to the above; the only difference is that collisional perturbations of the sample trajectories are not included, and the alpha energy is deposited along the initial trajectory. A very fast local plasma heating model is selected with NTYPE = 1. The confined-alpha production contributed to a local density of fast alphas and the slowing down process removes alphas from this population. More specifically, the time evolution of the fast alpha density, n a (*— eq. B.233), is governed by
dna/dt= —na/;a+ñDT, where
;a
~DT
(2.9.3.11)
eq. B.232) is the thermalization time for alphas and
(4—
(n~,T+ ~lDT)(1— L(r))
=
(2.9.3.lg)
is the confined-alpha source rate, and L(r) rates [see eqs. (2.9c—d)] are given by
eq. B.226) is the prompt-loss fraction. The plasma heating
(~—
Q~’=naEa(1—hj(xc))/;a,
(2.9.3.lh)
Q~’= naEahj(xc)/;a,
(2.9.3.li)
(see eqs. B.236—237) and the particle source of thermalized 4He SaF~~la/Tsa’
[4—
eqs. B.235 and (2.9b)] is
a=4He.
(2.9.3.lj)
2.9.3.2. Catalyzed D—D fusion. The catalyzed D—D reactions are D +D
—~
T(1.01 MeV)
+
D + D —*3He(0.82 MeV) D
+
T
D
+ 3He
p(3.03
+
MeV),
n(2.45 MeV),
He(3.52 MeV) + n(14.06 MeV),
—~‘~
—~
He(3.67 MeV) + p(14.67 MeV).
(2.9.3.2a) (2.9.3.2b) (2.9.3.2c) (2.9.3.2d)
This model is called “catalyzed” D—D fusion because we assume that tritium produced in the first reaction is quickly and completely consumed in the third reaction because of the much larger D—T fusion cross section. The particle sinks and sources are calculated for the chief ion species, D and 3He, and the incidental species H and 4He, if they are present in the simulation. All charged fusion products are assumed to be confined in the zone of their birth, and in the model they all slow down instantly and deposit their initial energy in the plasma electrons and ions. The reaction rates are (*— eqs. B.262—264) =
0.5n~Kav)DDP,
(2.9.3.2e)
1~DDn=
0.5n~
(2.9.3.2f)
~DDp
11D’He =
nDnl~KcJv)D3H~.
(2.9.3.2g)
The particle sources of H, D, 3He and 4He in eqs. (2.9a) and (2.9b) (see eqs. B.268—270) are: S,~”=+(~DDp+~D3He), a=H,
S,’= —(2ñDDP+2ñDDfl+ñD3HC),
(2.9.3.2h) a=D,
(2.9.3.2i)
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G.E. Singer ci at. t4D3He+flDDn~
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/
One-dimensional plasma transport code
3~j a— rae, 4r1
oF . . ~a =flDDP+nD3He,
a— rae.
.
The plasma heating due to the six charged reactions products is (see eqs.
B.271—273)
6 Q~’=
~ñ,E 01(1—h(x1)),
(2.9.3.21)
i-I
6 (2.9.3.2m)
~ñ1E0,h(X,),
QF=
i= I
4He), E where ñ, is the reaction rate for the ith reaction product (ñDDP is the rate for 3.52 MeV 0, is the initial energy given in eqs. (2.9.3.2a—d), h(x) is given by eq. (2.9.3.le), and x7 is the ratio of initial energy to the critical energy, E~,(4—eq. B.274): x,~= EOI/EC,. 2.9.4. Radiative losses Hydrogenic bremsstrah.lung is always included as a loss in the electron energy balance unless NATOMC = 0. Options also exist to include impurity bremsstrahlung, line radiation and synchrotron radiation. (Additional losses due to ionization of neutral isotopes of hydrogen, helium or impurities are described in sections 2.9.1, 2.9.10 and 2.9.13, respectively.) The following formula for the radiative source terms illustrates how these options are2R(T controlled: \ —6 ( j. \ Qhydrogen(y — 1,NATOMC~ eff ~ eJ 2,NATOMC~e\~1 ~ ~RAD — —~ (z — 1~ 2J
e
coronal (
‘~2,NATOMC~e
~a
a
e
a=e’,h
2.5
X 10_25Cioo(1 +
-~4~)(i +
____
1
~t~b0tk~~ev
(2.9.4a)
Note that setting the input switch NA TOMC = 1 (4— eq. D.1023) also forces impurities to be fully ionized, and therefore also directly affects neoclassical and other transport coefficients. C1~is a (usually factor 3tot = ~ small) 7~+ n~T correcting the synchrotron volume emission for reabsorption and reflection. Here / 1= (2/3)n~]/(B~/8rr).RI~~em is the bremsstrahlung radiation, and ~ is the coronal radiation for species a and is given by tabular input (ci. section 7.3). 2.9.5. Scrape-off losses Losses due to flow of plasma along field lines to a material plate in the scrape-off region are approximated by the order-of-magnitude estimates [39] S,~=—C~°,
Q~= =
L=
=
—
where C~=~4’
~
~nJ1/(TH + ;~)— 61~q~’/L, j = e,i,
for for
(2.9.5b) (2.9.5c)
L/(Mv~),
I( C127 C128,
(2.9.5a)
r < min(r~~,r~5i1), r,~i
( (Lv~)/(21/2v5) ~ 0,
T11,
C123 ~123
0, ~ 0.
(2.9.5e)
>
(2.9.5f)
C.E. Singer ci at
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297
In the appendices we define the mean ion mass ~ (~— eq. B.309), the inner boundary of the scrape-off region ~ as determined by the input parameter C120, the inner boundary of a “second scrapeoff” r~ras determined by the input parameter C121, and the charge-exchange frequency v.~ (4— eq. B.316). The connection length L [‘f— eq. (2.9.5e)] is typically set to half the average length of a magnetic field line between intersections of a material boundary. The parallel Mach number, M, can be determined by a ‘two-chamber” model (~— C381), which describes the continuity of particles, momentum and energy between a dense, sonic flow at the material boundary and a subsonic flow in the BALDUR scrapeoff [39]. The resulting Mach number is with
:~=~,‘
7..{C384~ (cf. ref.
(2.9.5g)
[52]).
T
ad
M=—~
1/2
~
(2.9.5h)
,
~-
div
where = 1
Adjv
=
ndIV—
—
(1
— C381)(1
—
exp( —
(2.9.5i)
C38o/AdI~)),
vO/(ndIV
(2.9.5j)
neTe/2Tdjv,
(2.9.5k)
ex~(~AOFJ ln~(k_11iivev)).
(2.9.51)
A~ are the electron impact ionization coefficients of Freeman and Jones [40], v0 = (2 E0/m0 )1/2 (*— eq. C385) is the neutral energy, m0 is the average neutral mass, and Td1~is the solution of 1”2TdIV —
5ne(~v)
2(7e + yI)ndIV(2TthV/~i~)
E0
1/2 a
q~’ =
41~+
C382(1
—
(2.9.5m) q~’=
min[C386n0v4h0(7~ — T41~),K~(7~, — TdIV)/LII],
where Vthe = (2[(1 — C387)7~+ C3g7Tav]/me }1/2 with C387 = to approximate a so-called thermal barrier.
(2.9.5n) 0
for conventional heat flux limit and C387
5~2 = (12.5
—
9.34Z
~/2)(3m1/2)/[4(21T)h/2Ztre4AI
= 1
(2.9.5o)
~
where A 0 is the Coulomb logarithm and Tay = (1 — C388)1 + C388Td1V. Note that the model of charge-exchange friction used here [eqs. (2.95a—b)] differs slightly from ref. [39]. The value Ye = 2.9 [4— eq. (2.95g)] is chosen to approximate the effects of a plasma sheath at the material boundary assuming no secondary electron emission. 2.9.6. Lower hybrid and arbitrarily specified auxiliary heating In the lower hybrid (LH) heating model, a spectrum of radio frequency waves is launched from the edge of the plasma and propagates inward along a radial line perpendicular to the cylindrical cross section of the plasma. Absorption by linear electron Landau damping heats electrons until a lower hybrid resonance point rLH is reached. At TLH, the remaining power is deposited in the ions. If there is no lower hybrid
298
G.E. Singer
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at. / One-dimensional plasma transport code
resonance point in the plasma, electron Landau damping continues as the waves propagate across the magnetic axis and outward along a radial line [41]. The implementation of the LH model in BALDUR allows options to neglect ion heating or electron Landau damping, or to substitute in place of LH heating an arbitrary specified heating source for ions and electrons, as described in appendix D (~— eqs. D.322—D.379, D.381—D.438, D.489—D.492). 2.9.7. Electron cyclotron resonance heating (ECRH) The standard ECRH model adds a specified power p~C(4—eqs. D.238—D.257) to the electrons and pEC (~— eqs. D.258—D.277) to the thermal ions. The power is deposited in a torus with a specified rectangular cross section (— eqs. D.318—D.321). The power added to each radial zone is proportional to the electron density multiplied by the volume of the zone lying within the given box. Thus, QEC =
n~ dv,
PECnfb(r)/J
j =
(2.9.7a)
e,i,
VECRH
where f~( r) is the volume of that portion of the heating region lying in a given radial zone and VECRH is the volume of the entire heating region. Versions of BALDUR are also available which contain ray-tracing models of ECRH [42]and detailed modelling of ICRH [33]. 2.9.8. Ohmic heating Electron heating by current dissipation is (2.9.8a)
QoHnhJz(JzJb)
where q~,is given in eq. (2.8a),
is given in eq. (2.9.2.2e) and
~b
c a(rB9)
ar
(2.9.8b)
2.9.9. Collisional energy interchange Collisions between ions and electrons produce the sources QIE
3CA(
QIE
~ a=1,2,,~,h
2>a [~eq.
(2.9.9a)
fla
Aa
(4-eqs. D.2328—D.2331) and ; (4—eq. B.73) are defined
where Ac (*—eq. in appendix B. B.4),
arccos( C
75) from the normal to the cylindrical surface of C77) with a specified energy (4— C76). This results in a volume source
C~[c7, max(0,F~) + C72F~]ne
C
ii
,-‘
~
‘-70
U
70> 0 c’He
(4—
=
n —t’
and (rHe or
f
—
r)
C73 rwall, otherwise
—
r1
~73r~511,
/
Hedr /(C7SAHe)1,
3He or 4He a=
(2.9.lOa)
where ~ is a normalization constant (.~—eq. B.60a), AH (~— eq. B.60) is the local mean free path for ionization of helium by electron impact, F~ is the parallel helium loss in the scrapeoff in particles/ second,
C.E. Singer et at /
One-dimensional plasma transport code
299
and F~~i1s the outfiux to the wall in particles/ second. The electron energy source due to the ionization of these helium neutrals is 3He or 4He, (29 lOb) or 4He. QHe —(iO~eV)C74S~, a=3He a=
J
2.9.11. Pellet fueling Spherical solid hydrogen pellets are described by their hydrogen species (4— eqs. D.2340—D.2359), velocity (4— eqs. D.2308—D.2327), pellet radius (4— eqs. D.2087—D.2106) and tangency radius (4— eqs. D.2067—D.2086). Pellets can be injected in the tokamak midplane at up to twenty specified times (4— eqs. D.2278—D.2287). The ablation of the pellets is calculated according to an extension of the model of Milora and Foster [44]. The pellet affects the plasma in a direct fashion by changing the particle density and electron energy density (pellet terms do not appear in the diffusion equations): i~n 0(r)= S~~dt(r), a
= 1
or 2,
(2.9.lla)
i~E~(r)=—(3OeV)An0(r),
(2.9.llb)
where the determination of i~n0(r)is described in detail elsewhere [46], and 1~Ee(r)accounts for electron energy loss during ionization of pellet neutrals. Apart from the self-limiting features of the ablation calculation plasma parameters are assumed to be held fixed during pellet ablation. The contribution of fast ions and alphas to pellet ablation is included. An essentially unlimited number of pellets may be injected by using the pellet-cluster model (+— C230). In this model, a cluster of pellets, rather than a single pellet, is injected at up to twenty specified times (— eq. D.2278—2287). A cluster consists of a stream of identical pellets whose firings are separated by a specified time interval (f— C231—C250). Pellet properties can be changed only from cluster to cluster; strictly speaking, when the cluster model is used, one should read “pellets in cluster” for “pellet” in the definitions of these properties (~— eqs. D.2067—2106, D.2087—2106, D.2278—2287, D.2308—2327, D.2340—2359). Within a cluster, pellet firings continue until the next cluster begins or until a specified maximum number of pellets has been fired (~— C252). 2.9.12. Recombination The particle sink due to recombination of hydrogen ions is —nane
5REC
where
3/
\‘r
/
(2.9.12a) (~—
eq. B.57). The corresponding ion energy loss
\
Q —
~2~n1+n2)1ine\ave/r~COm.
(Note that the code combines Q~EC with the charge-exchange power in subroutine NEUGAS.; cf. also section 5.2.5.) Electron energy losses due to recombination are typically small and neglected. Recombination contributes a volume source of neutrals (cf. section 2.9.1). The input switch NLRCOM (4— eq. D.1030) is used to omit all recombination effects. 2.9.13. Neutral impurity influxes Influxes of neutrals of one or two impurity species (*— eqs. D.2254—2255) may be prescribed by input (C2~—C226eqs. D.1024, D.1848—1887, D.2256—2275) for impurity species with atomic numbers 36. The neutral impurity atoms are subject to ionization, and thus contribute an impurity ion source term SMP
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One-dimensional plasma transport code
(a = 1,h) and an electron-energy loss term due to initial ionization QIMP In addition, the coronal radiation produced by each influxing impurity species [cf. eq. (2.9.4)] can be multiplied by an empirical nonequilibrium enhancement factor (C205—C206). The physical model assumes that the neutral impurity density falls off exponentially with increasing penetration depth. This approximates a steady-state condition where most of the neutral-impurity influx involves recycling impurity-ion outfiux off walls, and is correct for any portion of the influx due to controlled gas puffing. Using realistic ionization rates, neutral impurity atoms ionize within a few centimeters of their injection point. Hence, slab-geometry conditions are a good approximation, and the source rate becomes n°(s) n~(r~Ml’) exp[ —A(s)/v], A(s)
= fSfle(X)KGV>iz
(2.9.13a)
dx,
(2.9.13b)
where Koz>1~ is the electron impact ionization rate coefficient [51] which is correct in the case of a directional stream of neutral gas directed normal to the wall. The local source rate of impurity ions due to the neutral influx is then 51MP =
(2.9.13c)
C~MI~F]MISneKov)~nO(s), a = 1,h,
where C~MP is a normalization constant, and FIMP will be defined shortly. This production of impurity ions, due largely to electron-impact ionization of impurity neutrals, involves energy losses which affect the electron energy QIMP =
C203 SLMP e’ 5e
,
aa—~t ,
(2.9.13d)
204
The total neutral influx
FIMP
of species may be determined
by
either of two methods
(~—C
2~).In the first method, the flux is prescribed directly by linear interpolation between input values F10 (~— eqs. D.1024, D.1848—1887, D.2256—2275). As generally accepted models of the influxes are not available for realistic wall materials, the user usually has to make an externally supplied estimate of these influxes (cf., however, section 3.1). In the second method, the flux is determined by a feedback algorithm which attempts (not exactly, or always successfully if this requires excessive influxes) to achieve prescribed total number of impurity ions at specified times MP=max{O.,(4~2R,.walI)_1[Na~No
+L~]}.
(2.9.13e)
1) of neutral impurity a. Na4 is the prescribed total number of Here FOIMI’ the in influx rate (cm 2 scenter of the present time step. N~° impurity a ions the plasma at the is the calculated total number of impurity a ions at the beginning of the time step Na =
4~2Rf~ W,,l (r)r dr.
(2.9.13f)
L~is the estimated total loss rate (s1) for impurity a. ~t” is the present time step (cf. section 6). Na* is computed by linear interpolation from input arrays (f— eqs. D.1848—1887, D.2256—2275). This method is appropriate when an experimental measure of the average impurity content is available. Besides the initial ionization losses, it is possible to include impurity-radiation enhancement due to nonequilibrium excitation and deexcitation. The present treatment of radiation enhancement is quite
G.E. Singer ciat
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One-dimensional plasma transport code
primitive. For each impurity influx a, there are certain prescribed energies
301
(~—
C205, C2~)which are used
ad hoc to enhance the coronal-equilibrium radiation as follows min[C225,1
C2osf~~S~MI’rdr/f~nenaR~0~~a1].a =
+
0
min[C226~1 +
IMP
0
1,
(2.9.13g)
IMP
C2o~j”~S~MI’rdr/f’~ nenaR~01~j,
a
= h,
where R~ot~~ is discussed in section 2.9.4. Note that this enhancement affects the coronal radiation only, and does not affect QIMP or noncoronal contributions to Q~D (~— eq. (2.9.4a)). 2.10. Minimum densities and temperatures Prescribed minimum values of density ne,,.~n= C129 and temperatures 1~ = C130 and T1T~= C131 can be set in the scrape-off region to avoid numerical overflows if the scrape-off region chosen is too large. The effect on the diffusion equations is equivalent to adding local source terms of the appropriate value to maintain these minimum values. A diagnostic for the activation of one of these equivalent source terms is the energy conservation check listed on the printout. Since these equivalent source terms are not included in the conservation checks, it will appear as if conservation is violated when they are activated. 2.11. Compression The purpose of the compression model is to calculate the effects of adiabatic compression, and the effects of changing the limiter radius with or without compression. Adiabatic compression effects are computed based on the major radius R specified as a function of time. The degree of compression from a time t1 to a later time t2 is described by the compression factor c c=R(t1)/R(t2). (2.lla) When the major radius is compressed, the flux surfaces 1”2r. are compressed in minorare radius, so that a flux (The flux surfaces assumed to remain surface which was at minor radius is remappedIf to concentric circular-cross section tori rthroughout.) thec mirror radius of the limiter varies so that 2rwaii(ti),
Twaii(t 2)
(2.llb)
=c~”
then the flux surface next to the limiter at time
t 1
remains next to the limiter at time
t2.
moves with respect to the flux surfaces. We define the “limiter compression” factor as 2rwaii(ti)/rwa cL = c~” 11(t2).
If not, the limiter (2.llc)
If CL < 1, the limiter is moving outward relative to the flux surfaces, and cold low-density plasma will fill the space between the limiter and the flux surface which used to be next to the limiter. If cL> 1, the limiter is moving inward relative to the flux surfaces, and some plasma will be “scraped off”. R (t) and rwaii(t) are linearly interpolated from input values (- eqs. D.2027—2066 and D.2131—2150). Under cL < 1 compression, the number of particles is conserved, while the energy of electrons and ions is increased to account for the work done in decreasing the volume. The toroidal and poloidal magnetic fields increase, with poloidal flux being conserved. These conditions lead to the following transformations under CL = 1 compression of plasma quantities 2, a=1,2,”,h,b, B 3”2, n0cxc 9crc E.ccc10”3, j=e,i, Jccc2, 4 3 . . (2.lld) j=e,1, B~ccc,
qc(c°.
302
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One-dimensional plasma transport code
Thermalization of the thermal compression energy is assumed fast enough that the usual transport coefficients determine the subsequent effects on confinement. Compression is discussed further in section 5.3. Effects of compression on fast ions due to neutral beams and on fast alpha particles are discussed in sections 2.9.2 and 2.9.3, respectively.
3. Boundary conditions and density control 3.1.
Boundary conditions
The inner boundary conditions on the diffusion eq. (1) are Bn
~E.
ar
Br
—~=--—-~=B
9=0 at
r=0,
a=1,2,e’,h; j=i,e.
(3a)
At the outer boundary, the value or radial gradient of each variable is specified. The following combinations are available for na and E~. Pedestal boundary conditions: NBOUND=O:
~lania
and
7~=T11, a=1,2,1,h; j=i,e.
Pedestal unless there are abnormal gradients at the wall: NBOUND=1:
na=nla
BE1
an0 —
Br
= 0
—
Br
=
and
T~=T1~, a=1,2,i,h;
0 for each a or
.
j
Bfla
where
—
Br
r,~,11
j=i,e. 0, or
ai~ Br ~
0.
Specified impurity influxes (with pedestals for hydrogen isotopes and energy): NBOUND
= 2:
na = nla
and
F0 = F10(t) +
As with NBOUND
=
NBOUND=3:
~
T’sputter(t),
a
= 1,2;
j= i,e,
= I,h.
2, unless there are abnormal gradients at the wall: n0=n10 Fa
except
T~=T1~, a
=
and
F10(t)
~
a=1,2; j=i,e,
+ T’~~~4101(t), a
=0,~.L=0foreach a orj where
=
~!H>o,~ or
(3b)
“Pedestal” boundary conditions are replaced by “zero flux” boundary conditions when the gradients at the wall are in the “wrong” direction. n10 (4—eqs. D.1888—1891), T1~(4—eqs. D.2208, D.2250) and Fia(t) eqs. D. 1808—D.1887) are prescribed boundary densities, temperatures and fluxes. Tputter(t) can be set to zero or to the flux of sputtered iron calculated from the Monte Carlo generated neutral spectrum incident on a stainless steel vessel at r = r~~1 using the sputtering coefficients of ref. [50]. The options NBOUND = 1 and NBOUND = 3 can be used to avoid unphysical conductive or convective energy influxes. (4—
G.E. Singer et at
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303
The following combinations are available to determine the outer boundary condition on the poloidal field eq. (2c). The evolution of the total toroidal current can be fixed i~=i~(t) at
(3c)
r=rwall,
or the poloidal field can be given B9=B9(t)
at
(3d)
r=rwall,
as described in appendix D (4—eqs. D.1441—D.1460 and D.1462—D.1481). Alternatively, the surface voltage can be specified C396). (4—
3.2. Density control The flux of neutral hydrogen atoms entering across the plasma boundary can be altered during the computation. First, the fraction ~ (~eq. D. 997) of outgoing neutrals returned to the plasma with an energy ~ (~— eq. D.2129) can be changed to a new prescribed value C102 at a prescribed time ~ = C89. Second, influx of gas puffing neutrals entering the plasma with energy TCOLDP eq. D.2130) can be prescribed as a function 1a .La~tj ~ at r—rwall, a—x,.~., rO_rO/ H\ — (4—
....1
‘,
/~i
where r~(tr) are input arrays eqs. D.18O8—1847, D.2224—2248). Finally, an additional neutral influx can be included for feedback control (“density monitoring”) of the volume-average electron density
4. Initial conditions The initial particle densities and temperatures can be prescribed in three ways. First, any profile can be given as an input numerical array. Second, profiles of the form n 0
=
nla +
(flOa
—
n10)[1
—
(r/rwaii)~’]“,
a
=
1,2,1,h,
(4a)
j=e,i
7.=Tij+(Toj_Tij)[1_(r/rwaii)xTuJ)~TJ,
(4b)
can be specified. Third, some or all of the starting densities can be specified as a given fraction P~,of the initial total ion density =
Fa(nij + (nt,1
—
n11)[1
—
~),
(r/rwaii)~]
a
(4c)
= 1,2,e’,h, 1~i =
fa/>fa,
where
f~,are input values
or n0eqs. = D.1888—1891). n where n1 is set by tabular input. In these expressions, The initial poloidal magnetic field profile can be specified either from an initial toroidal current profile (4-.-
Jcc [1— (r/rwaii)~]~’,
(4d)
jçg7~3/2,
(4e)
or as
with the normalization of B 9 determined by the boundary condition discussed in section 3.1. These profiles will be modified in the scrape-off region. The method of specifying these options is detailed in appendix D, starting at (eq. D.1482) for densities, (eq. D.2151) for temperatures and (eq. D.1441) for B9.
304 5.
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One-dimensional plasma transport code
Solutions of the diffusion equations
Here, we give a summary of the numerical methods and then write the difference equations used. The differential equations are first transformed to conservative difference equations of the Crank—Nicholson type with a variable degree of implicitness. In a given time step, the B9 equation is first advanced using the previous values of na and E1, and then the n0 and E~equations are advanced simultaneously using the old values of B9. The whole procedure can be done in a simple predictor—corrector sequence or in an extrapolation scheme. As the extrapolation scheme has not been found particularly useful, it is not discussed in detail here. The difference equations solved are as described in sections 5.1 and 5.2. 5.1. Space and time mesh 5.1.1. Space grid The code is one-dimensional in the minor radius r. The plasma is divided into N concentric cylindrical zones about the magnetic axis (r = 0). The plasma zones are numbered, in order of increasing r, from 2 to N + 1. In addition to these so-called real zones, there are two dummy zones, one inside the origin (r = 0) and one outside of r = rwal,, making N + 2 zones in all. The values of variables in the dummy zones are used to impose the boundary conditions on the var~.ablesin the real zones (cf. section 3). Note that rb refers to the inner boundary of zone j. The space grid for a particular run is set by input variables described in appendix D (~— eqs. D.1972—2026, D.2361 and D. 2366). r
0
Inner dummy zone BALDUR zone:
4
1
Main plasma region 3
LCENTR
Physical zone:
1
rb 1
r~
z
=
r r
z
—r
r
=
b
Z
b
r
,
ISEP
LEDGE—i
LEDGE
M
N—i
N
[1,,
rb 4
=
r
wall
rM+i
rb M
rb M+i
(r
+
wall
—r
,
b r 1
z
(r, J
+
= r
b —r 3
z
)/2
,
,
j—l
Number of physical zones Physical
b r N+2
=
b r N+3
=
=
N
zone number of first
=
Iz
z
rM
rb N
j
r
S
Outer dummy zone
wall
MZONES
z
r N+1 rb N+i
RADIUS(J)
=
Iz
rN
rb M+2
=
RNINS
rN+2 rb N+2
*
z N+l
j
radius of inner boundary of BALDUR zone
0
=
z
N+2
2 r, J
Scrapeoff
Iz
3
rb 3
2
b
4
—*
radius of zone center of BALDUR zone
1 r
r
r scr
M—i
I5
rZ
r
ISEPM1
2
I
i
rZ
=
r
=
r wall r
+
wall
=
RMINS
*
XBOUNI(J)
P.MINS
(r
—
wall
r N+i
NZONES
scrapeoff
zone
M
Fig. 1. BALDUR spatial grid.
CFUTZ(120)—1
rb N+3
XZONI(J)
CE. Singer ci at.
/ One-dimensional plasma transport
305
code
Physical quantities are described at zones and boundaries in a typical region of the spatial grid (cf. fig. 1). Densities n0, temperatures T1 and 7~,and energy densities E, and Ee are defined at zone centers. They are then averaged to get values at boundary points. For example,
T0b,= ~
~ 7~+ 1—
(5.1.la)
~ ‘:~ ‘:,-~
‘:~-~
‘)
r.Z_ri~
where j is the zone index and the superscript b or z indicates quantities defined at zones or boundaries. Conversely, B9 is defined at zone boundaries and interpolated to get values at the zone centers, assuming that the toroidal current density is constant across a zone. Thus, B in zone j is 2 b I b \2 — ~ ~2 b ~ ~~,÷1i ~ /2 ‘:~ ( z\2 — ~‘ ( b\-‘ B — ‘‘.. 2 rZ B r~B 91 — b \2 / b\ 91~1+ / b \2 / b\ 91. (5.1.lb) .‘
—
~‘i
)
~
—
~
)
~
Boundary-centered quantities (derived from the ofzone variables n0, in1~,T~,Ec, 2 ~a) are computed using averages the centered values of fundamental the fundamental variables the center E1, on (Z ),~) and
3r~
n~~— flZ;.i_1
(5.1.lc)
.
ijrj_i
5.1.2. Time grid The time grid is determined by a variable time step, chosen before a cycle: (5.1.2a) Quantities defined as “time-centered’ are appropriately interpolated between the values of depending on the solution method described below (cf. also section 6).
t,,
and
t” +
5.1.3. Variables used in code The variables used to describe the space—time mesh in the actual coding differ in trivial ways from those described in sections 5.1.1 and 5.1.2, as summarized elsewhere [17]. 5.2.
Differencing
We now describe the differencing scheme used for the diffusion equations. Crank—Nicholson parameter 0CN We solve a set of diffusion equations of the form
5.2.1.
BX =
131 ax —~--r~A -~j---+B
x) +Cx+D+~9(f),
where x is a vector of independent variables and A’, B’, C are matrices of nonlinear coefficients, and D is a source vector. When solving such equations numerically, two problems generally emerge, namely, stability and accuracy.
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A finite difference scheme can be made stable by making the difference equations implicit. The difference equations then have the form —OCN~(X)
+
(1 —OcN)~(x~),
(5.2.lb)
where f is the value of the variables at the new time n +1 x” is the value of the variables at the old ~ is the differenced form of e’(x) for time t~, ~(~‘) is the differenced form of t9(~)for t = t’2, = ~ ‘~+1, and the Crank—Nicholson parameter 0CN is a factor between 0 and 1 which controls the degree of implicitness. 0CN (4—eq. D.1020) is specified by input to the code. Setting 1/2 < <1 usually guarantees stability. A discussion of the second problem, accuracy, will be found in section 5.2.4, along with a further discussion of stability. +
Poloidal field equation The poloidal field equation used is
5.2.2.
3B
2 B 9
[4—
eq. (2c)]
~ 3(rB
c
9) ~
B
Br
(5.2.2a)
This is differenced conservatively to obey Ampere’s law JJ. dA = (c/4ii)fB9. dl exactly. This is done by defining B9 on the boundary and the toroidal component of J in the zone center. The integral from of Ampere’s law is
JJdA
=
~—JB.dl.
(5.2.2b)
In difference form, the toroidal contribution to this equation is —
rb2) =
~—(2iii~1B91+1
—
2ru~B91),
(5.2.2c)
or ~
—
(5.2 .2d)
(~1~~9,j+1 — ,,bB9~) 1’2 _rb2 j+1
I
r where we denote the toroidal component of J The central boundary condition is that B
by
J. The differencing of the other terms is straightforward.
9(O) = 0, see section 3.1 for the outer boundary conditions. The equation is set up as a variable implicitness Crank—Nicholson diffusion equation and solved by a tridiagonal solution technique. Ohmic heating is computed with a conservative technique developed by Roberts. The basic constraint is that
2~ f IJ.E+——IdV= 1 BB 81T
Btj
—f--~--ExB.dA.
(5.2.2e)
A4~
Taking the toroidal component of Ohm’s law to be E =
~1h [~
—
~beam]
and differencing in time, we write
(B~~±1)2 ~B~2 ~
2~t
~
n+1
=_c{V.~h[OcNJ”~’+(1_OcN)J~_J~eam]( ~
2
B~ o)
+4~11h[OCNJfl+1+(l_OCN)Jfl_J~eam](J 2+J)},
(5.2.2f)
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for the change in magnetic field energy. The first term on the r.h.s. is the Poynting flux and the second is the ohmic heating, which we can write as Q~3H=71h[OCNJfl+1+(1_OCN)Jn_J~am](J
(5.2.2g)
2+J).
When J is expressed in the conservative difference form in space, the above term is conservative in space and time. A conservation check can be computed by summing eq. (5.2.2g) over the plasma.1 There a slight + (1 — is0CN)~ difficulty with the Poynting flux in eq. (5.2.21) since (E x B)toroidal = h1[0cN1’~ ~beam]( B + + B )/2 and, while B 9 is naturally defined on the outside boundary, J and ~ are naturally defined at the zone center. While a scheme which conserves poloidal field energy to machine accuracy was not found, interpolation schemes are used to adequately minimize the errors in computing the energy conservation check. —
5.2.3. Particle and energy diffusion equations The other dependent variables besides B9 are the particle and energy densities. The equations for their time development [eqs. (2a,b)] form a set of nonlinear parabolic initial value problems. The equations are conservatively differenced in space, and set in a variable implicitness Crank—Nicholson form. The dependent variables are the hydrogen isotope densities (2 allowed), impurity densities (4 allowed), and the electron and ion energy density (cf. section 2). We write these variables in a vector x = transpose(nl,n2,n(,nh,ne,Ej). The particle and energy transport equations are
_v~.TP~ocN+C1x~ecN+D~.,
~
(5.2.3a)
is the divergence of the flux through zone j, C, is a matrix containing the coefficients of where terms T” which are to be differenced implicitly in ~ and source is a vector of explicit source terms. C and are defined in appendix A. The n + 9CN refers to the variable degree of implicitness: i.e., 9CN
+
X”~°~ (1
—
(5.2.3b)
OcN)x~+ ~CNX~’~
The fluxes F are defined as
~
(1 —OcN)(Bfx~+AJx~...l) +OCN(BfXJ
+A
1X~t~),
(5.2.3b’)
where the matrices A1 and B~are defined in appendix A. This approach was chosen so that we could difference implicitly n~Z~ — n~_ ~ as the difference of the gradient term (B
2 BE~. 3nj Br n7~Bn1 1 Br
—=—————,
j=i,e,
(5.2.3c)
from the chain rule (see also appendix A). The coefficients, 2/(3n~),7~/n1,are nonlinear, of course, and have to be time-centered. The gradients are differenced in the usual way eq. (5.1.lc)] [f—
~
n~—n~_~ r
(5.2.3d)
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The divergence is handled in a conservative scheme: ~!~rF)+CX+D,
(5.2.3e)
or in difference form =
a~+1F1+1—a~F1 +C1~1+D1,
(5.2.3f)
2 2]2.,TR] which will where a~ is the area of boundary j (21ri~2TrR) a nd J’~is the volume of zone j [(rb+ — ,.b 1 automatically conserve particles and energy [4]. The matrices A, B, C and the vector D are set up in subroutine CONVRT. They are then recast in subroutine REDUCE into the form
P
1x~+~ +
+
+
S~,=
0.
The boundary conditions are expressed in the form at r
3x = y,
(5.2.3g)
= 0
and r
=
a (5.2.3h)
aB~/Br+ 1 which translates into
(A
1
+
C1)~1+ B1~2=D1,
(5.2.3i)
at the center and
AN+2xN+1 +(BN+2+CN+2)xN+2=DN+2,
(5.2.3j)
at the edge. The boundary conditions described in section 3 are translated into A, B, C and D ‘s in subroutine BOUNDS, and cast into the P, 0, R, S framework in REDUCE. This gives us the usual tridiagonal matrix equations which are solved in the usual way [17]. The only complication is that P, 0, R, S are matrices, and multiplication and division in the usual method are replaced by matrix multiplication and matrix inversion. One ends up with solving a 3 x 3 up to 8 x 8 linear system in each zone by Gaussian elimination with partial pivoting. 5.2.4. Predictor—corrector and extrapolation For (1/2) ~ O~ 1, a linear set of equations, such as eq. (5.2.lb), would be unconditionally 0CN = 1/2 stable. is not However, eq. (5.2.3a) has nonlinear coefficients which will change during a time step, and necessarily the best choice for stability and accuracy. Stability often then requires 0CN to be close to one. Second order accuracy in time should then be obtained by time-centering the nonlinear coefficients. This is done in two different ways in the code. The most commonly used method is a two-step predictor—corrector scheme. First (predictor), a trial solution x~1is computed, using A, B, C, D evaluated at the old time t0. Next (corrector), A, B, most of C and pieces of D (cf. section 5.2.5) and the linear combination = ~“(l — 9~) + x~’~ 1, are computed, where 0 is a 1 time-centering parameter between 0 and 1A,setB,byC,input to the code D.1021). is computed using the Or-time-centered D and using x (—*~ eq. rather than The then final solution x”~ for use in the next time step. Typical problems are run with O~,and 0CN = 1 to x0 is stored make the problem as stable (implicit) as possible. It has been our experience with real problems that stability has been a much more severe problem than accuracy. Note that the computation of B 9 is done using the same time-centering scheme used in the computation of x. ~
+
~“~‘
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309
Numerical form for sources As various source terms present different stability and computational overhead, different numerical strategies have been adopted for different sources. Source terms are associated with the matrix C, the vector D, or both, and their time-centering depends in part on this association. The C-source is C~5 and thus is time-centered to the extent that ~ ~9CN is. D on the other hand, does not multiply x~On the corrector portion of a predictor—corrector time step, C is recomputed based on x~1 except for the WICHXS term discussed below. The only terms recomputed in D are S,~ [4— eq. (2.9.lOa)], Q~~E[~— eq. (2.9.lOb)], QIMP [. eq. (2.9.13j)], QRAD [- eq. (2.9.4a)], Q~H [4— eq. (2.9.8a)], and the WEIRS term discussed below (cf. appendix A). Note that 5,~EC[4— eq. (2.9.12a)], the particle sink due to recombination, is associated with C and is time-centered, while the corresponding neutral source rate is in D and is not time-centered. Also note that the corresponding ion energy sink Q~EC [~— eq. (2.9.12b)] is treated differently, as described below. Sources due to pellets (cf. section 2.9.11) are computed “instantaneously” as direct additions to or subtractions from x. They do not appear in C or D. They are computed only after all iterations due to time step repetition and predictor—corrector and extrapolation have been completed, and they bypass the conservation checks. In the pellet cluster model (~— C230), if the prescribed interval between successive pellet firings in a cluster is less than the current itt, the firing interval used by the code is set equal to ~t, and to compensate the ion deposition due to pellet ablation is multiplied by an enhancement factor equal to the minimum of i~t/(prescribedinterval) and C252. The charge-exchange loss term can be written as n0 n H (~ — T1 )( at’ )~. (n0 is the neutral density, n H is the hydrogen ion density, 7, is the neutral temperature, T1 is the ion temperature, and (av)~5the charge-exchange collision rate.) If n0 n H T~( av )~ is computed explicitly and 7~ T~as is normally the case, small changes in T1 can make large changes in this term, which in turn limits the time step. This can be especially “painful” since the neutral gas calculation is expensive and called only once every NGPROF (~— eq. D.lOlOa) time steps. Physically, 1~,/T1 changes much more slowly, so rewriting the equation as 5.2.5.
+ 9CN,
~
—
1)T1,
(5.2.5a)
allows one to make the equation implicit in T1, to compute the slowly varying (at’ )~[(7~/T — 1] term from the neutrals package, and to compute nOnH from the previous time step. QREC and the ion energy sink due to charge exchange of thermal hydrogen ions with fast beam neutrals are also formulated linearly in T1. The sum of these two terms plus the charge-exchange loss term is 0+ called WICHXS in the code. WICHXS is time-centered by linear interpolation in T1 as (WICHXS) [(B/BT 1)(WICHXS)]0[(E,’~°cN/n~ 7”)]. Note that WICHXS is associated with both C and D [8]. Under predictor—corrector or extrapolation, only the factor [E°~/n~ — 7”)] is recomputed: e.g., for the corrector portion of a predictor—corrector time step, we have —
9cN/fln+Op —
(WICHXS)~+ [a(WICHxS)/aT1] ~[(E
where
o
—0 ‘~E~ + 0 E~hl + 0 E~~1 p1 i p i J CN i The impurity radiation loss rate depends strongly on T~and can limit the time step, especially in problems with large radiation losses at the edge. The impurity radiation rate evaluated from fits to the coronal radiation rates (cf. section 2.9.4) is summed with Q~IE[~— eq. (2.9.lOb)] and Q~MP[4- eq. (2.9.13j)] into the code variable WEIRS. WEIRS is time-centered by linear interpolation of 7~as (WEIRS)5 + [(a/a7~)(wEIRs)1”[(E:~/n: — T~”)].Like WICHXS, WEIRS is associated with both C and D. Under predictor—corrector or extrapolation, all factors are recomputed; e.g., for the corrector portion of a predictor—correction time step, we have =
—
~.
~
CNJ
p~i4.O~ +
i
o
Op+
‘~“~
CN
i
=
(1 —0 ‘~
CNI[\
(WEIRS)~~°P + [((a/ai~)(WEIRs))”~°P[(E:+9P+6cN/n:+9P —
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In addition to time centering, WEIRS may be modified by the smoothing of sharp peaks in its spatial variation. This smoothing is activated by setting C199 = 1 or 2. 5.2.6. Convection Another standard difficulty is that, near the plasma edge, the character of the heat conduction equations often changes. The equations are of the form (5.2.6a) is the conduction (or diffusion) and B is a flow velocity. e represents Ee or E, (cf. section 2). When A(Be/Br) >> Be, the equation is clearly second order, and second order boundary conditions apply. When & >> A (Be/Br) convection dominates conduction, the equation is close to first order, and applying second order boundary conditions can give difficulties. Numerically, the boundary-centered flux is differenced as A
(e~+,—
— —
—AJ~1/2
L~r
—
BJ~1/2J±1/2.
(5.
.
)
The problem is that ~j+1/2 is not well defined in terms of the zone-centered quantities e~and Since B1 + 1/2 has the dimension of velocity, we can think of it as a “convection” velocity. If we average + 1/2 = ~(e~+~ + ~), the self-compensating nature of the diffusion equation is lost. For example, if t’j+I/2 = <0 and e 0, the “convective” flux does not go to zero, but to eJ+l/2v1+1/2, and the density in zone j can go negative. Several methods have been used to treat this problem [4]. The one we use is to construct the centering variable j+1/2 = ‘~~+1/2~J+ (1 — ~+1/2)eJ+1, where 61+1/2 is always bounded between 0 and 1 and 6, = 1/2 is the default condition. The actual 6 used is ~
j
=
—
signEt’] min{~, v~L~t/aL~r},
(5.2.6c)
,,
with i.~r = + — i~, where v is an appropriate velocity defined in subroutine CONVRT for each element of X and a can be fixed by input as a = C81. The usual value of a varies from 2 to 4. If a is 0, 6 is fixed at 1/2. The most important implementation of this scheme is for ion heat transport, where “convection” at the edge dominates conduction. The algorithm is also applied to electron “convection” and particle “convection” (primarily, due to the neoclassical “Ware” pinching). 5.3. Compression
Compression is modeled as a series of “instantaneous” compressions. The transport equations are used to compute the plasma state at t”~ from the state at 1” ignoring compression. Then the effects of compression in major radius are computed as if the compression which is supposed to have happened over the center time step occurred instantaneously at t~~ Then any effects due to limiter motion relative to the plasma are computed, as if the “limiter compression” had also occurred instantaneously at ~‘i ~ Note that the compression computations are done only after all iterations due to time-step repetition, predictor—corrector, and extrapolation have been completed. If cL < 1 [4— eq. (2.llc)], then cold, low-density plasma must be added. BALDUR does this by adding more zones all with cold plasma. Zones are added by the same method as was originally used to compute the mesh (~— eqs. D.1972—2026, D.2361). Zones are added until a zone is reached with an outer boundary radius r~greater than or equal to rwall. If r,~> rwall, we set r,~= rwall. Then we check to see that this “chopping off” has not resulted in too small a zone width. If the ratio of the width of the last zones to the width of the zone immediately inside is less than 0.2, this can mean that the time step must be extremely small in order to keep the finite-time-step errors reasonable. In this case, the small zone is discarded, and
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3~I
rwall is set equal to the outer boundary radius of the next zone ion. In the newly added zones, x is set to its edge value [~— eq. (3b)]. B9 is extended by assuming the current density is zero in the new plasma bold
~ ~r1— —
.!.±i~01dI r ~ ~rN+2), bold\
where the superscript “old” means after effects dependent on major radius compression have been computed, but before effects of limiter compression have been computed.
6. Time-step control
The initial value of i~tis prescribed by input (~ eq. D.1787). At each new time step, i~tis adjusted as follows. If, at the end of a time step, it is found that z~ t was too large, the plasma parameters are restored to their old values, and the time step is repeated with a reduced i~t.We describe here the checks used for time-step repetition and the method used to compute ~t. A time step is repeated whenever ~a’
E1’
or
6err/6max
>
B9<0, a1,2,e,h; x~,where
(6a)
j=i,e,
or when
n~1_n~I Efl+1_EP?~ B~1—B~ 6e~=max
n~
‘
E~
B
‘
(6b)
is the maximum change in any zone not bypassed by C 8max 126, isC161.-164 andtime x4 (4— and (e— eq. D.1018) are input constants. When extrapolation used, the stepeq.is D.1036) also repeated if Ceii.> Em~Xl where x 3 (4—eq. D.1035) and m~ (4—eq. D.1019) are prescribed constants and n?~+1_n~~ LEJ~+1_ (err
max
0
~
__________
0
E~ 1=2n~~—n~,a=1,2,e’,h,
n
E7~’=2E)—E~,
~ a=1,2t~’,h;j=e,i,
B90
(6c) (6d)
j=e,i,
(6e)
B’=2B~—B.
(6f)
Here the values labeled “1” are evaluated by calculating for two successive minor time steps at i~t/2and “0” labeled values computed with minor time step i~t. ~ is an estimate of the i~t error. Since extrapolation applies a correction to i~f and to f 0n+1 computed with minor time step i~t (i.e., ~ n +1 = f0n +1 + ~f = 2ff’ + 1 — ~0n + I and ~f = 2[ f~’ + 1 — f0n + 1]), the relative correction is ~f/f~. By requiring this correction to be small (e.g. 5%), the time integration error should be, e.g., (0.05)2. The E~tto be used for a new or a repeated time step is computed as follows. If the time step is to be repeated due to negative n5, E~or B9, the E~~t used is the old t~tmultiplied by x5. If the time step is repeated due to an excessive extrapolation error, the 8t used is the old ~t multiplied by x0 if x6> 0. For all other cases, the ~t used is the old i~t multiplied by a factor C— f1/cei.~ if ceu=max(8ej.~/m~,ee,.r/cmj >1, \min[x2,~tmax/2~t,xi+(1_xi)/cerr]
if
(6 ) g
Ce~
Here x5 and x6 are prescribed constants (4— eqs. D.1037—1038),
4~tm~ (4—
eq. D.1788) is a maximum time
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step, x2 (f— eq. D.1034) is a specified maximum rate of the time-step restriction and x1 (*— eq. D.1033) is a constant which damps the error controlled relaxation of the time-step restriction. (We note that l/Emax = 0 when extrapolation is not used). In order to avoid running problems where the time step has dropped to the point where the computation is not preceding at a reasonable rate, the run is terminated if z~tdrops below a prescribed minimum value ~tmjfl (~ eq. D.1789).
7. Structure of the code and test run The structure of BALDUR is largely based on the OLYMPUS system [7]. 7.1. Structure of the FOR TRAN program The structure of the program is outlined in appendix E. The function of the main program BALDUR is primarily to call the OLYMPUS subroutine MASTER. Then MASTER calls BASIC and CONTROL, BASIC calls MODIFY, etc., as outlined in appendix E. The bulk of the computation is controlled by the subroutine STEPON which advances time steps. The structure of subroutines called by STEPON is shown also in appendix E. 7.2. COMMON variables An alphabetic list of COMMON blocks and a description of their use is given in appendix C. 7.3. Atomic data file A file (called F0R22 in the present version of BALDUR) contains polynomial fits to data used to compute the mean charge, mean-square charge and radiative loss of impurities, assuming coronal equilibrium. The data for He, C, N, 0, Ne, Mg, Al, Si, 5, Ca and Fe are taken from Tarter [50]. Data for other impurities were produced from the program XSNQ as described in Post and Jensen [51]. Detailed information on the data in file FOR22 is given elsewhere [17]. 7.4. Ripple amplitude input array An input file (called RIPPLE in the present version of BALDUR) is required if diffusivity from toroidal field ripple effects is included [17]. 7.5. Plotting
BALDUR writes output files which can be used by postprocessor programs on CRAY and VAX computers. 7.6. NAMELIST input description and test case sample run A sample NAMELIST input is given in appendix F. An example of some of the output from this run is given in appendix F. A complete list of the NAMELIST variables and their use is given in appendix D.
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8. Code version nomenclature A system for naming new versions of the BALDUR code will now be described. All codes which use a substantial fraction of the BALDUR coding should be named BALDXNNY
-
X’N’N’Y’- X”N”N”Y”
where X indicates the institution where the version was made or modified, Y is an arbitrary code letter for the person responsible for providing information about the version and NN is a two digit number assigned sequentially to versions with a given X and Y. Whenever X or Y is changed in creating a new version, then a suffix containing the previous value of XNNY should be appended. For example, the version documented in this paper is BALDP47M
- PO9A
prepared at Princeton by Martha Redi, based on an earlier version prepared at Princeton by Alex Silverman. An earlier version in the BALDPXYA series, BALDPO1A, was supplied to R. Wunderlich at the Institute für Plasmaphysik. The standard version of the code made at the Institut für Plasmaphysik has been called BALDIO9R
-
PO1A.
(The code letter R denoted the person responsible for providing information about said modification.) Each institution should contact one of the first four authors of this paper to ensure they choose a unique institution code and to clarify any other questions concerning use of BALDUR. Acknowledgements Comments from G. Bateman and D. Stotler are gratefully acknowledged. This work was supported by the US Department of Energy Contract No. DE-ACO2-76-CHO-3073. Appendix A. Definition of A~,B~,C1, D1 The basic plasma transport eqs. (2a—b) and (5.2.3a—j) are written in terms of the dependent variables na and E’.. These equations are in the form actually solved by BALDUR. In contrast, eqs. (2h—i) are written in terms of (Bn0/Br). To go from one representation to the other, it is necessary to define A1, B~,C~,D1 in terms of the transport coefficients of section 2. 1. A1 and B1
The matrices A1 and B1 together act as a finite-difference operator on the dependent variables vector, x eq. (5.2.3b’)]. Let a be the analogous operator. a candefine be written as the sumB~.Then of zeroth-order 1: ax [a°differential + a1(B/Br)]x. Similarly, A°,, B~, A1, and and first-order pieces a°and a
[4—
A
1x~1+ B1X1 (1
—
is analogous to ax,
~f+l/2)AfXf_l +
+
&B~x~ is analogous to a°x,
B)x~_1)
is analogous to ~
(A1.1) (A.1.2) (A.L3)
[4—eq. (5.2.6d)] is called BOUND in the code. Given the elements of a, A and B may be derived from eqs. (A.1.1—3).
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/ One-dimensionalplasma transport code
In the definitions below, the subscripts of aK/ refer to the different matrix elements and must not be confused with the radial zone index j. aaa= [t’a_
b=1,2~
b=~,hBr
—
~
D~n,,+ ~
b=1,2
[
a~~=
D~~anb(Z2)bj_~a=1,2,
+
—
+ [Daa +
D~”n~ +
~
(A.1.4)
b=e.h
b=1,2
b=1,2
a=
+
~
DL~nb(Z2)bJ~-,
a=
~‘,h,
(A.1.5)
b-e,h
1,2,t’,h; /3=
1,2,é~,h,e,i; a #/3,
(A.1.6)
aea= [C365t’~are7~]+ [—x~7~]~--, a= 1,2,
(A.1.7)
aea
(A.1.8)
a=
= [Xe7(Z)a]~,
te,h,
aee= [(F~1)Xe]~,
(A.1.9)
(A.1.10)
aej=0,
a1~= =
0,
/3 =
{
~_
[Daa
~
—
(F
(A.1.11)
i)x~+ ~
a=1,2
~
+
1,2,t’,h,e, (D~a(Za)b_
D~0(Z)b)nbI B’~a
b~,h
[Daa_F—1X1+
~
D~n~+
~
DKz2)bnhj.~_~.
b~-1,2
~ [—
+
a=e~,h +
~
D~n,,— ~
b=1,2
D~(Z)bnbI-I_[na(Z)Q]
b=e,h
~D~a((Z2)b/2+(Z)b)nbj~~i
a=1,2[
2)b + a ~th +
b
12
D~((Z)~/2 + 1)nb +
-i--,
[(F— 1)~1]
b
~
(Z)Q(Z)b)nbl n
-~
BT
-~
(h D:~((z
(A.1.12)
modified if C 110> 0. The following symbols are defined in section 2. D00 [4— eq. (2.6a)], v0 [4— eq. (2.7e)], D~ [4— eq. (2.4.ld’)], D~0 [e— eq. (2.4.lc)], D~j’[4— eq. (2.4.lc)], D~ [~eq. (2.4.ld)], D01 [4— eq. (2.4.11)], t’~jare [f— eq. (2.71)], x1 [4— eq. (2.6d’)], Xe [4eq. (2.6d)] and x1 [4—eq. (2.6c)], F, the ratio of specific heats, is set to 5/3. Note that additional “impurity drag” terms not documented here are included in subroutine CONVRT when C217 > 0.]
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2.C~ The radial zone index SREC +
Caa
=
a
j is omitted here for clarity.
ss
na
a
° ,
=
1,2,
(A.2.1)
5S
C00=—.~, a=t”,h, C~~=
+
Q!E
n~(1~— T1)
(A.2.2)
—
E~
(F— i)[
~
a~’,h
B(WEIRS)0]
(A.2.3)
Here, B(WEIRS)0/B7~is computed from analytic derivatives of the radiation tables from subroutine OFIT. ç~IE
C~1= ~‘ n~(1~T1) ~
Cie+
(A.2.4)
,
IE
~
(A.2.5)
,
fle(7~Ti)
C
~
—
QIE
n1(1~—T1)
+
E~
+
F—i B(WICHXS) n1 BT1
(A 26)
All other elements of C1 are zero. The particle and energy sources are defined in section 2.9 and 5.2.5: 5REC [4— eq. (2.9.12a)], Q~EC [4— eq. (2.9.12b)], ~2’ [4—eq. (2.9.3a)], Q~[~- eq. (2.9.5b); j = e,i], QIE [4—eq. (2.9.9a); j = e,i] and WEIRS and WICHXS (cf. section 5.2.5). 3.D~ Again, the radial zone index Da=SaN+S~+S:, Da=S:+SaHE+S~MP,
De=
j is omitted here for clarity.
a=1,2,
(A.3.1)
a=e,h,
Q~+ Q~+ Q~+ QRAD~
(A.3.2)
Q+ QLH~ QEC+
TMP QOH+
~
+ QHE÷ QI
(A.3 .3) D 1
=
+ Q~+ Q~+ QLH + QEC + QREC
—
rB(WICHxS)
(A.3.4)
The sources are defined in sections 2.9 and 5.2.5: S~ [~ eq. (2.9.1—2)], S~ [4—eq. (2.9.2c)], Set’ [4—eq. (2.9.3a—b)], S~IE [4— eq. (2.9.lOa)], 5IMP [4— eq. (2.9.13c)], Q~ [4— eq. (2.9.lc)J, Q~ [4— eq. (2.9.2d)], Q~ [4— eq. (2.9.3c)], QRAD [4— eq. (2.9.4a)], QLH [~— eq. (2.9.6a)], QEC [4— eq. (2.9.7a); j = e,i], QOH [4— eq. (2.9.8a)], Q~IE [4— eq. (2.9.lOb)], Q~MP [4-.. eq. (2.9.13c)], Q~” [4— eq. (2.9.ld)}, Q~ [4— eq. (2.9.2e)], Q~ [4— eq. (2.9.3d)], QLH [4— eq. (2.9.6b)], Q~EC [4— eq. (2.9.12b)], and WEIRS and WICHXS (cf. section 5.2.5).
316
G.E.
Singer eta!.
/
One-dimensional plasma transport code
Appendix B. Formulae Notes: 1. In eqs. (B.1—B.5), (B.8—B.22) and (B.24—B.30), the first code variable listed is the coefficient and the second is the exponential in an expression of the form (coefficient) * (10. * * exponent). For example, eq. (B.1) is evaluated as CFEV*10.* *CXEV = 1.60210 x 10120. 2. The Cross-references colunm tells where to find definitions of symbols used in formulae. For example, the notation A~(*— eq. B.7) at eq. (B.42) means that the symbol A~,which is used in the formula for eq. (B.42), is defined in eq. (B.7).
G.E. Singer ci aL / One-dimensionalplasma transport code No.
Code
317
Formula
Units
Subroutine BLOCKDTA B.1 CFEV,CXEV 8.2 CFEVDK,CXEVDK
12° keV=1.60210X10 keV/deg = 1.16049 X 1040
(erg/eV)
B.3
FCAO,FXAO
a
B.4 B.5
FCAE,FXAE FCALFA,FXALFA
B.6
FCAN
B.7
FCAP
A~=1.00727663
(amu)
B.8
FCC,FXC
c = 2.997925 X 1010.0
(cm/s)
B.9
FCC1,FXC1
Gradi = 3.7405X105°
(erg cm2/s)
not used
B.10
FCC2,FXC2
Grad
8.11
FCE,FXE
e =1.60210x 10—20.0
B.12
FCES,FXES
e
B.13 B.14
FCF,FXF FCG,FXG
B.15 B.16
5.29167x iO.~0 4°
0
Comments
(K/eV)
not used
(cm) (amu)
not used
5.48597x10 a= 7.29720x 10-30 =
not used
1.0086654
(amu)
2 = 1.43879 X 10’~°
not used
(cm K)
not used
(emu)
not used
(statcoul) 2g1/2/mol) (cm” (dyne cm2/g2)
not used
Faraday G = 6.670constant x 10 8.0= 9.64870 x 1030
FCH,FXH
h = 6.6256X1027°
(erg s)
FCK,FXK
kB
(erg/K)
not used
B.17
FCLNAE,FXLNAE
ACompton = 2.42621 )< 10— 10.0
(cm)
not used
B.18
FLCNAP,FXLNAP
A
(cm)
not used
B.19
FCME,FXME
1, =1.32140X10 ma = 9.1091 x1028°
(g)
B.20
FCMN,FXMN
m~ 1.67482 x 10_240
(g)
B.21
FCMP,FXNUCL
m~=1.67252X1024°
(g)
B.22
FCNA,FXNA
NA
B.23
FCPI
‘IT
B.24
FCR,FXR
R~= 8.3143 X 1O~°
B.25
FCRE,FXRE
B.26
FCRINF,FXRINF
R,,~=1.0973731 X105°
B.27
FSCB,FXSB
a
11.28
FCSIGT,FXSIGT
~
B.29
FCVO,FXVO
normal value of perfect gas = 2.24136x104°
B.30
FCWIEN,FXWIEN
Wien displacement constant = 2.8978x101°
Subroutine UNITS B.30a FACU,FXAU
4.80298 x 10—10.0
1.38054x 10160
13°
= 6.02252 X 10~~°
ZADEF2(1) ZADEF2(2) ZADEF2(3) ZADEF2(4) ZADEF2(5)
not used
(mol ~‘)
not used
(erg/K mol)
not used
(cm)
not used
(cm1)
not used
(erg/cm2 s K4)
not used
(cm2)
not used
(cm3/mol)
not used
(cm K)
not used
3.1415926536 2.81777X1013°
=
5.6697
X
iO~°
= 6.6516 X 10250
m~/A~ = number of
grams
in 1 amu
Subroutine AUXVAL
B.31 B.32 B.33 B.34 B.35
not used
Aa =1.007825, a H A,, = 2.0140, a = D A,, = 3.01605, a = T A,, = 4.00260, a =4He A,, = 3.0, a =3He
See also WTGAS, WTIMP ( *— eqs. D.2328-2331)
318
No.
G.E. Singer ci at
Code
/
One-dimensionalplasma transport code
Formula
Units
Comments
Subroutine IMPRAD(1)
B.36
B.37
ZATCOF(6,4)
ZAHEO(9)
A~’J~”° =
n = 0,3 — 8.875780 13.06722 — 22.74558 4.332644 —23. 22560 0.07795893 — 23.22962 0.2169734
n =1,4 33.68544 2.780003 2.823806 1.994706 0.4390245 0.1040682 0.4768998 —0.1969521
A~4-~~e = —44.50917
24.42988
2.470931
Subroutine AUXVAL B.38 AIRED(4,4)
~
B.39
A’=
ZIIONM
0.5107924
—
6)/(A~+ A,,)
~
n,,/
a’-1,2,t,h
~
n,,A,
B.40
ZVS
v11 =[(T,
j=2 j=3 j=4
i = 0,7 i = 8,9
(amu)
reduced mass
(amu~’)
inverse
(cm/s)
mass parallel flow
mean ion
/a=1,2.~’,h
2
j =1
0.02505100
(7); 13.74394;
= (A,,A
not used
—10.25714
—0.3426362
0.007438675 —
n = 2,5 30.04414 0.2216913; 4.693808 0.3405031; 0.03994050 0.1536833; — 0.09974274 0.06531130;
+ 7~)/(A’)I”
velocity, cf.
A1 (4—eq. B.39)
No.
Code
Formula
B.41
ZJS
~
I,=~.= 0.25e(
n~)vii
~
Units
Comments
Cross-refs.
(statcoul/cm2)
Twice the
v 11
1”2e4
B.42
CTIONS
G.,
B.43
CTELES
G.E =
1”2G~ 1
1/2 B.44 B.45
CNUHYD CNUEL
GrH = (m1,/A1,)
Gr~= GrHA~/2
eq. B.40)
(i_
A 1,
0.75[m1,/(A1,’TT)]
C... (A~/2)
imtial current density in scrapeoff
(‘—
(4—
eq. B.42),
A~ B.4) A (~eq. 4-eq. B.7) 1,( ‘—eq. B.44),
CrH(
— B.46
CETA
C,, = m,,e2
B.47 B.48
CDNHS CDNHIS
G~= 8(2i~me)1”2(ec)2/3 CHI =
GHA,~1”2
eq. B.7)
GH(
eq. B.4)
‘—eq. B.47), eq. B.4)
‘—
B.49
CDNIIS
C 1 = C~ 112
GHI( GHI(
4‘—
eq. B.48) eq. B.48)
B.50 B.51
CDITIS CDETES
GHI/(2) CE = CH
CH( 4—eq. B.47)
B.52
CCNU
Cr = 3A~
A,( 4—eq. B.4)
11.53
CDBOHM
GB = 0.0625 c/c
G.E. Singer et at / One-di mensional plasma transport code No.
Code
319
Formula
Cross-refs.
Subroutine IMPRAD 8A~ ~ 1o~(A~1] (1,kav/0.1)”~1Z,2(01), ~kaV<0.1
B.54
(none)
L~,(1) = antilog
B.55
ZPLOSS
R”I’(7,)
A~’]~”~( ~—eq.11.36)
L
0.1< 0.2 < 2. < 20. <
22(7), Lz3(1~), L 54(1~),
Lz4(2O.), B.56
ZBETA
B.57
ZRECOM
B.58
ZVO
L~,~(l)( i—eq. B.54)
20.
~,keV < ~kcv
n~(~— eq. J.3.6e),
= [n~T~ + n,T1 + ~(n~+ nE,,)J/(B~/8’TT) (Note: nE,, term is not currently included) 13n,,([0.073529max(0.001,7k,V)]1”2
nE,,( 4—eqs.
=
[(2 x 103k,,vC
ZSIGV
2,
antilog
=
a = 3He,4He
76)/(A,,mp)]”
~,cV<2
~A~~~(log
antilog
A;+HC +
~,v)”
A~Mc (log ~
2.
<
~,eV
<10~
io~< ~—
B.60
1./ZIONIZ
AMa =
1./ZTOTAL
eq. B.37)
4-eq. B.58),
VHeo/(nc(tJVe)Hco)
VHeO(
Subroutine IMPRAD B.60a
B.233—4)
nc
B.59
Tekev <2. <0.2 7~kaV
-1
C~’=
Ha dr ne
f
Hr
dr/(G 7SAHe)I dv)
r1 = maxEo,r —(1— G73)r~,,,1]
rHe(
‘—C77)
(OVC)HCO( 4—eq.
B.59),
AHe( 4—eq. B.60) I
IMP
B.60b
1./ZTOTAL
C~”~ = [4~2Rfr.
SMPT
drl
5IMP( ‘—eq. 2.9.13c),
‘—CC)
r,~Ml’(
Subroutine INITIAL
B.61
RHOINS(2,55)
Subroutine GETCHI B.62 Q(55)
n1=
a
q=(rB~)/(RB9)
B.63
AJZS(2,55)
J = c(arB9/ar)/(4lTr)
B.64
XZEFF(2,55)
Zeif~{(ni+n2)I~”~a + ~
/( Subroutine TRCOEF B.65 GSPITZ(2,55) CALPH(55)
B.67
CLOGIS(2,55)
8.68
CLOGES(2,55)
D.994), t(4-eq.D.995)
a — 1,2/,h ~ n~
A~’= [Z~(2.67+
Zc
3.4O(l.l3+ 8r)j/[
B.66
~
n~
= [71cZeci/(thi
Z~
Zc Zct8( ‘—eq. B.64)
8~)]
Z~1,(‘—eq. ~ ‘—eq.
+ n2)] —1.
A~ = max[(1.0, ~
B.64) B.64)
5°~
A = (max(1.,ln[1.54x10b0(T3v/ne)1~~2/Zeff]}, T v< ~max(1.,ln[(1.O9x1O1h7~)(nh/2Zcr 8)] },
Zat4(
‘—
eq.
B.64)
320
G.E. Singer ci a!.
No.
Code
Formula
B.69
AHME.AN(2.55)
A~= ~
AIMASS(2,55)
A
1=
One-dimensional plasma transpori code Cross-refs.
A,,na/ ~
a —1,2
B.70
/
Ia =1,2
Aana/nj
~ a — 1,2,1,h
B.71
Z2
=
~
3”2AH)/(Z
a
B.72
TIONS(2,55)
= (CrjTj
2Aj)
G,~(‘—eq. B.42), AH( ‘—eq. B.69), Z2( ‘— eq. B.71), eq. B.67) C.,c(—‘—eq. B.43),
3~2)/(flcAc)
X~(‘—eq. B.68)
B.73
TELECS(2,55)
; = (C~7
B.74
XNUHYD(2,55)
v~= [(GrHBT)/(Tj
IB
3AH)/(rT~)]1~2
cH( ‘—eq. B.44), AH( ‘—eq. B.69)
0 I)]E(R B.75
XNUEL(2,55)
IB
~ = L(CreBT)/(;
3/(r7)]1’~2
Cr~(‘—eq. B.45), ;( ‘—eq. B.73)
9 I)][R B.76
ZD,ASPINV(55)
B.77
ZFT(55)
B.78
ZXI
B.79
ZC
B.80
ZO
= r/R
= 0.58 +O.2Z~,,
C~=[.56(3.—
Zcit( ‘—eq. B.64)
Z~,,(—
Z~,,)J
Zaii)]/[Zcti(3.+
fTi = fT/(1 + EI~c*)
eq. B.64)
fT( ‘—eq. B.77),
eq. B.78), — eq. B.75)
‘—
8.81
FTRAP(2,55)
B.82
ZYO
CR( ‘—eq. B.79)
fT~( ‘—eq.
= (1—[min(C
B.80)
1/~’)1/2 57,C58)/C58]
Subroutine XSCALE B.83 SLTES B.84 SLNES B.85 SLBPS
rr,= 11/(81/8r)l i, = n,/(~ln,,/8r)I r l(8/arXB
These gradient scale lengths are numerically
9/r) B, rTI—IT~/(aT,/ar)I
smoothed except for r3-c
B.86
SLTIS
Subroutine TRCOEF B.87 ZGYROI B.88 ZESV
~ = c(2AHm1,)” v~,=n1J/e
B.89
ZVTHE
Va = (1~/m~)’/2
B.90
ZVSDI
V,
B.91
ThE 1 ‘ EXBETA
2/e
—
2 1 = [1/A~m1,f/ (t) = rIp
(P. C
145 B.92
ZRIPL1(55)
AH( — ‘—eq. B.69)
~(r, 0,
AH(
variable ripple -
ZMSO,ZMSOO
i) = ~0(i)+[&,or(i)~
<~2)o~
$~~(t)(‘—eq. D.2437—D.2O46)
~o(i)j(r/r,crf’47
~o(i)(
lo, for N,~,,11,4= ‘ff(2$)’/2 .IT/2[1 2. —O.278443269.IT(2$) (0.8862269255/I IT(2$)1”2]}
for B.94
XX
eq. B.90)
fixed ripple
1121 8.93
‘—
1T(2~)’/2> 2.
‘—eqs. D.1451—1460),
‘~,~,(‘X where ‘—eqs.
D.1472—1481)
— eq. 2.3h), 8v,,ñ,,blc( ‘—eq. 2.3i), $( ‘—eq. B.91) 8=
8fixcd(
a = Be/[N,,,, 11,Bz6(r, 0, z)]
6(r, 0, t)( ‘—eq. B.92)
G.E. Singer ci at / One-dimensionalplasma transport No.
Code
Formula
a*
8.96
G = [(3X211’2)/(161T)Jf dO{sin
(c sin ~
=
here, G(a*) [10] B.97
ZG
=
for 0 8 ir 28(1 + t cos 8)[8( r, O)/8(r, 0)]3”2C( a*) [(32X21”2)/(3’IT)](cos 1a~))(1 — (a* )2)1/2 — a*cos la* ]3/2) a*( ‘—eq. B.95)
G = antilog 2 — $1~~2(1. —0.243a — 0.13a2)] x antilog 10[11.641a +0.234a 10[ — (2.346 + o.462pu/2 )][(1 + 0.64379( a ~2)10.317 for variable ripple or for fixed ripple when C149 0.
B.98
ZO
Th, = m~A,/A~ 1”2
8.99 B.100
Z1 Z2
LY.
ZO
i~i,(‘-eq. B.98)
= (T/Th.) = min[r,cr,(c/e)8iji~j/Be]
DR = 3(ir/2)
ñ1 1( 4-eq. B.98), ii,( ‘-eq. B.99) i~(‘— eq. B.99), p~~(‘-eq. B.100)
1~~/R
2mc)/(41Te2) B.102 B.103
ZKADO ZYRFE
Wee =
B.104
ZGYROE
c~[c(2me)112]/e
B.105
ZGYRFI
W~
GK=(c (Bze)/(mcc) =
A~( ‘—eq. B.69) Cpe( ‘—eq. B.104)
1= (BZe)/(AHmPc)
2/Bz
B.106 B.107
ZRHOET ZRHOIT
PZc = PZI =
cc~” G~
2/B~
G~
1T~” B.108 B.109
ZK1 ZK.2
a( ‘—eq. B.94), $( ‘—eq. B.91) A1( ‘-eq. B.70)
1”2~ B.101
321 Cross-refs.
Subroutine AVERGE B.95 ASTAR1 GALPHA(55)
code
32Zcii)]
K1 = (0.53+ Zcff)/[Zeff(l + l. K 2 = 0.82(3— Zcif)/[Zcff(2.57+
$~=
B.110
Z1BETA
B.111
Z1ZETA
8.112
Z2BETA
P2
B.113
Z2ZETA
~2 = 0.25 +O.l 1T/[’. +(/3jVc*)I/2 + ~
B.114
ZK(11), ZK(12)
K1~
B.115
ZL13
L
=
Zcff)]
O.4+O.llZ~
1(‘4-eq. -eq. B.87) Zcir( B.64) Zcii(
*—eq. B.64)
Zcit(
4-eq. B.64)
0.55 +0.255Zcff
Z~,,(‘—eq. B.64)
0.05 +0.O345Z~ 43Zcff
Z~,,( Z,, ‘- eq. B.64) eq.75), B.64) 1,e*( 1(~ ‘-eq.
=
j = 1,2
fT(4- eq. B.77), $~(4-eqs. B.110, B.112), ~ ‘-eqs. B.111, B.113)
13 = (1 + K1)K11[1 B.116
ZL23
L23 = (2.5
B.117
ZNV
~‘cZ =
—
+ K2)K12[1
(K1K11)/(1 + K1)] —
(K2K12)/(2.5 + K2)]
(1 + Zcrr)/; 12GNc~,7,cvT,ev(AH)1”3
B.118
ZNCRIT
n~1=
2.2X10
B.119
ZCRITO
CNCfl,
=
B.120
ZNCRIT
~
6.Ox 10’3CNcnt1~v
=
E3/2[qRA~(1+ Zeff)]~’
K1( ‘-eq. 0.108), K11( ‘-eq. B.114) K2( 4- eq. B.109), K12( ‘- eq. B.114) Zcff(
‘-eq. B.64),
B.73)B.119), CNC~I(‘—eq. AH( ‘-eq. B.64) i~(‘-eq.
Ac( 4- eq. B.68), Zdf( — eq. B.64) GN,,,,I( ‘—eq. B.119)
322 No.
G.E. Singer cia!.
/ One-dimensional plasma transport code
Code
Formula
Cross-refs.
Subroutine HRCAL B.121a HRB(6,11,26)
H,,(~i,r)
neutral beam deposition function for nth energy component
B.121b TAUMAJ(6) B.122
TAUMIN(6)
see last entries in appendix F.3.8 r
6”
see last entries in appendix F.3.8
Subroutine HRSET B.123
HENER(30)
energy of nth energy component of the neutral beams (see F.3.8a)
E,,NB
Subroutine DPOSIT B.124 HNSRCS(10)
j~NB=
B.125
S,,NBI
HDEPS(10,10)
j for which ~ = W,,/J. Number of particles per second which are injected in the n th energy component of the neutral beam (see F.3.8a)
1,,°f~,,
summed over those b and
local rate at which neutral beams deposit particles as a function of ~t = V1/V for each energy component [see eq. (2.9.2.2a)]. This array is reset in each BALDUR zone by a call to DPOSIT which interpolates between elements of H,(~s,r)3 (see B.121a above)
B.126
SHCHXS(2,55)
ñ~,(r)
sink rate of thermal hydrogen due to charge exchange between neutral beam atoms and plasma ion [see eq. (2.9.2.2h)]
B.127
YFRACT(6,3)
~
current fraction in the nth energy component of the bth neutral beam ‘—eqs. D.1234_1243), W~,(‘—eqs. D.1159—1168) injected power in each energy component (see F.3.8a) J,,0(
5
B.128 B.129
ZPOWR(4,10) YR(26)
p~’ ,~
B.130
ZHR(10)
H,,(r)
B.131
YRLOSE(6)
L~~B
(this is R2 in HRCAL, HRSET and IJECT, it is set by DPOSIT) =
f1 H~(it,r) d~s
H(r) for the nth energy component (see B.121a) this is set by DPOSIT from HRLOSE which is calculated
in HRCAL (see F.3.8b)
(
B.132
ZHRT
H( r)
B.133
ZMU
~jNB(r)
B.134
ZDEV
aN11(r)={~ENB!J~f1H(~ r)[~s2
B.135
ZDN
ñ~~B(r)
(see F.3.8c)
B.136
ZCEX
ñNBacx(r)
(see
B.137
ZLOSS
LNB
(see F.3.8d)
B.138
ZMU
<~NB) =(2/a2)f,L~ih(r)rdr
B.139
ZDEV
(aj~> = (2/a2)f o,~B(r)rdr
B.140
ZZDN
B.141
ZZCEX
8.142
ZDN
p~NB= (2i~)2Rof
B.143
ZCEX
Nacx = (2i~)2Rof ñ~,(r)r dr
= >.E,~BNJ~JBH,,
=~E~
f’H,,(IL r)jz d~/~E!~’~H,,(r)
ñ~(r)r dr
(2/a2)f ñ~,(r)rdr nNB(r)r dr
~NB(r)2]d~/~EN8P~NBH(r)}
B.126)
G.E. Singer ciat No.
Code
/
323
One-dimensional plasma transport code
Formula
Cross-refs.
Subroutine BEAMS B.144 LHEMAX
Ea,, = HEI(LHEMAX)
B.145
LHEMIN(55)
E~,,= HEI[LHEMIN(J2)]
B.146
HEI(10)
E~the upper boundary of each fast ion energy bin
B.147
HFI(10,10,55)
fB(~
B.148
HNPLAS(55)
EJ’~S
B.149
HEPLAS(55)
]V,P’~’
(see B.213)
B.150
SHBEMS(2,55)
h~,
(see eq. 2.9.2.2fl
B.151
WEBEMS(55)
Q~
(see eq. 2.9.2.2k)
B.152
WIBEMS(55)
Q~
(see eq. 2.9.2.2e)
B.153
ZSIGCX
(see eq. 2.9.2.2a)
E, r)
(see 8.211)
where the charge-exchange cross section is: 2 [1—0.0673 log(E/Ab)] a 15 B.154
ZLOGE
B.155
ZLOGI
8.156
ZBARLG
B.157
ZEFFLG
cm2
[1 + 1.112x 10_15(E/cv)33] 55(E)= 6.937x10 log A~,Coulomb logarithm for ion—electron collisions log A~= log{(2’trm~/nc)”2[k1~,/(he)])—1/2 log A,, Coulomb logarithm for ion—ion collisions if E
(n~/n
~
0)Z~logA1 allions 0 total neutral hydrogen density
B.158
ZNO
n
B.159
HSCATS(10,55)
!,~=
ire~Z,,2~(n
(see eq. 2.9.lj)
log2A,
1/n~)Z, 1 2
(see eq. 2.9.2.2a)
(2AbmbE3) / 8.160
ZNUE
4(2gm )“2e~Z,,~nlogA a 3 2 a
~-~e=
3A,,rnp(kTa) B.161
12A
ZGAMMA
\1/2
‘ITe4( —k J
=
(see eq. 2.9.2.2a)
/
Z~~n
B.162
HSLOWS(10,55)
1(Z,2/A,) log A~ \ 1,/ j H53v,~/log(y,,+ v,~E])/(’y,i+ v,0E~.1))
B.163
HCHEXS(10,55)
e,,,
B.164
ZSIGMA
2 1n°a~,[2E/(A,,m~)]” aDT(E) = (4.09 X i05 + 5.02 X 10~/[1+ (1.368 X 105E
(see eq. 2.9.2.2a)
8(r)( ‘- F.3.7c) is calculated ;from HSLOWS (see eq. 2.9.2.2a)
=H
—
1.076)12)
/(E[exp(1453./V~)— 1])x 10— 24 cm2 where E is in eV
B.165
HEINJS(55)
EJ”J
(see B.211)
8.166
HNINJS(55) ZNUDT
N~ 1~ 7 = NDorTa~(E)[2E/(Abmp)1/2
(see B.213)
B.167 B.168
HALFAS(55)
ñ~T(r)
(see eq. 2.9.2.2g)
B.169
SHBLOS(2,55)
n~
8.170
HELOSS(55)
EJ!055
5
(see eq. 2.9.2.2a)
(see eq. 2.9.2.20 (see B.211)
324
G.E. Singer ci at
/ One-dimensional plasma transport code
No.
Code
Formula
Cross-refs.
B.171
HNLOSS(55)
NI~55
(see 8.213)
B.172
ZDDSIG(10)
a
B.173
HDDS
Ni~
2,where °DD( 00(EX2E/A,,m~)~’ E) = (482./([1 + (3.08 x 104E — 1.177)2] x E[exp(47.88/E1/2) —1])) x 10—24 cm2 where E is in keV 0n= (211)2Rof nDDfl(r)r dr
(see eqs. 2.9.2.2n and B.320)
J Nrnn dt
B.174
HDDTOT
N~0~ =
(see F.3.4t)
B.175
RHOBIS(2,55)
n~
B.176
RHOBES(2,55)
Z,,nB
B.177
HEBEMS(55)
EB(r)
(see F.3.7b)
B.178
ZBJ
~~aan,
(see eq. 2.9.2.2c)
(see eq. 2.9.2.2j)
B.179
ZEPSLN
B.180 B.181
ZRTANG ZMUB
Rtang = ~
(R~~ +~
cos
B.182
ZM1
1—rn
1— 2/[(1
+ e)js~]
B.183
ZKM
K’(rn)
(see eq. 2.9.2.2d)
B.184
AJBS(55)
J,,~,,,/CJ
(see eq. 2.9.2.2e)
8.185
HECMPS
EJ°mP
(see B.211)
8.186
HNCMPS
Ncornp
(see B.213)
Subroutine HPRINT B.187 ZZNB
NB
(see F.3.7m)
8.188
ZZEZ
U5
(see
B.189
ZZEP
UI’
(see F.3.7u)
B.190
ZZEZP
(see F.3.7k)
B.191
ZZMUB
(see F.3.7o)
B.192
ZZETP
~
(see F.3.7e)
B.193
ZZET
UB
(see F.3.7n,w)
B.194
ZTAUS
rB(r)
(see F.3.7c)
B.195
ZDEN
nB(r)
(see F.3.7d)
B.196
ZFUSE
h~T(r)
(see F.3.7i)
B.197
ZCHEX
f~~(r)
(see F.3.7h)
B.198
ZSOURC
ñ~,,,(r)
(see eq. 2.9.2.20
B.199
ZIFRAC
f.B(r)
(see F.3.7j)
8.200
ZEPAR
Ei?(r)
(see F.3.7a)
B.201
ZETOT
EB(r)
(see F.3.7b)
83b
=
~
sin (see eq. 2.9.2.2d)
F.3.7v)
B.202
ZEDEN
n~(r)
(see F.3.7e)
B.203
ZMUBAR
~B(r)
(see F.3.7f)
B.204
ZMUDEV
OB(r)
(see F.3.7g)
B.205
ZZMUDV
(see F.3.7p)
8.206
ZZCXFR
(see F.3.7q)
G.E. Singer ci a!.
/
325
One-dimensional plasma transport code
No.
Code
Formula
Cross-refs.
B.207
ZZSORC
N,B
(see F.3.7r)
B.208
ZZFUSE
NDT
(see F.3.7s)
11.209
ZZFRAC
F?
(see F.3.7t)
B.210
ZZCHEX
(see F.3.7x) m~)
Eyias + E~’°
B.211
Zi
B.212
ZZECHK
B.213
ZO
N’~”~ =
B.214
ZZNCHK
F~= 1— N~”/N5
(see F.3.7A)
B.215
ZZLOSS
0.
(not used)
B.216
ZZHEAT
pB
(see F.3.7y)
B.217
ZZALFA
NOT
(see F.3.7s)
~
= =
~ (E”’J = E!oss — all zones j
1— Ek/UB ~
(see F.3.7z)
(Jy~flJ—
lOSS
—
~pIas + NCOrnP)
all zones j
Subroutine ALPHAS B.218 EFUSEI
E,, = 3.5 MeV DT = exp[ — 22.712/T,°275 — 23.836 — 9.393 x 10 2~ +7.994x104T,2 —3.144x106T 3] (T, in keV), see ref. [37] 1
B.219
ZSIGV
B.220
ZSIGV
B.221
AFUSES(55)
ñ~T(r)
(see eq. 2.9.3.lb)
B.222
SHFUS(2,55)
5,,”, a = D,T
(see eq. 2.9.3.lb)
B.223
ZDDFUS
n~,,,(r)
(see eq. 2.9.3.lc)
B.224
ADDS
NDDn
(see F.3.4q)
B.225
ADDTOT
NI~’Dfl
(see F.3.4s)
B.226
AOLOSS(55)
L,,(r) local fraction of alphas born on unconfined orbits
8.227
ZCOEFE(75)
Gec
B.228
ZCOEFX(75)
l/Vthc
B.229
ZCOEFI(75)
G,,
ZECRIT
(see eq. 2.9.3.ld)
=1/V21/m~
(see eq. 2.9.3.ld)
4log A,/mp)~n,Z,2/Aj
1 B.230
=16V’A,,Z~e~n~ log Ac/(3me)
=
(4.sA,,Z~e
E~= AaTa[~I1Trn 1,rnc
(see eq. 2.9.3.ld)
2/3
~ ~ ~‘
(see eq. 2.9.3.le)
B.231
ZBIFRC(55)
h1(X,,) when NTYPE = 1
(see eq. 2.9.3.le)
B.232
ASLOWS(55)
1/;,,
(see eq. 2.9.3.1!)
B.233
ALPHAI(55)
n,,(r)
(see eq. 2.9.3.10
B.234
EALFAI(55)
(see eq. 2.9.3.lj)
B.235 B.236
SALFS(55) WEALFS(55)
nE,,/n,,(r) 1’, a =4He S,, Q~’
B.237
WIALFS(55)
Q~’
(see eq. 2.9.3.li)
Qe
(see F.3.9a)
Subroutine APRINT 0.238 ZQE
(see eq. 2.9.3.lh)
B.241
ZTFUSE
nDT(r)
(see F.3.9b)
8.242
ZFALFI
f~,,(r)
(see F.3.9c)
326
G.E. Singer ct at
/ One-dimensional plasma transport code
No.
Code
Formula
Cross-refs.
8.243
ZNA
(see F.3.9d)
B.244
ZEA
(see F.3.9e)
B.245
ZTHERM
<~DT)
(see F.3.9f)
B.246
ZFUSN
<~DT>
(see F.3.9g)
B.247
ZOLOSS
L,,,,
(see F.3.9h)
B.248
ZTFUSE
(see F.3.9i)
8.249
ZTOTFI
F,,,
(see F.3.9j)
B.250
ZNA
N,,
(see F.3.9k)
B.251
ZEA
U,,
(see F.3.9()
B.252
ZTHERM
NOT
(see F.3.9m)
B.253
ZFUSN
NOT
(see F.3.9n)
B.254
ZLOSS
‘~DT,Ioss
(see F.3.9o)
B.258
ZTFUSE
NOT
(see F.3.9p)
8.259
ZPIALF
(see F.3.9q)
Subroutine HE3
B.260
ZENE(6)
8.261
ZAMU(6)
B.262
ZRHO2S
8.263
ZRHO3S
B.264
ZRHO4S
B.265
Z11
13.266
Z12
nOD,, 1D~He
B.267 B.268
Z13 SHFUS(2,55)
‘
8.269
SALFS(55)
4He
(see eq. 2.9.3.2j)
B.270
ALOSS(55)
5F, a a =3He = S,,”,
(see eq. 2.9.3.2k)
B.271
ZLAMDA
h(x,)
(see eqs. 2.9.3.2e,m)
B.272
WEALFS(55)
Q~’
(see eq. 2.9.3.2~’)
E 01 A,
initial energy and mass number
of the 6 charged fusion reaction products in eqs. (2.9.3.2a—d)
3Hc = exp[ — 27.76/1’,°~~ —31.02 + (cm3 s )2.79 X 10~2~-• (oc) —5.53x1O4T,2 D +3.03x106T,3 —2.52X109T4] (cm3 s1) for T, <80 keY (T, in key) = exp[—52.1 +5.84 log T 25(log 7..)2] for T, > 80 keV 1 —0.5 but = 0 for T 1> 240 keV 3T 095] exp(— 19.3/T3)/1~6(cm3 s~)
+ 5.4 x 103T,°92]exp( — 19.8/73)/T 6 for T, <80 keV (T, in keY) 1 = exp[—46.51+2.561og T 6(log T)2l for T, <80 keV 1 —0.1 but = 0 for T 1> 240 keV
(see eq. 2.9.3.2e) (see (see eq. eq. 2.9.3.21) 2.9.3.2g)
S,~’, a = D, P
(see eqs. 2.9.3.2h,i)
B.273
WIALFS(55)
Q,l~
(see eq. 2.9.3.2m)
B.274
ZX
x~T= Eo,/(A,7[3/4(~ITmp/rnc)1/2~(njZj/Aj)]2/3)
(see eq. 2.9.3.2e)
CE. Singer ciat No.
Code
/
327
One-dimensional plasma transport code
Formula
Cross-refs.
Subroutine PDX B.309
ZIIONM
Th~=—~ ~
n,,/ ~
Pa—1,2,~,h
B.310
ZTNEU
~ noE,,°/ ~
Tj~=
a—1,2
B.311
ZMNEU
ZVO
B.313
ZYI
E,,°(4-eq. 2.9.ld),
fa—1.2
ffi~=rn~~
~A,,/AP I
a—1,2
8.312
A~(’-eq. B.7)
n,,A,,
a—1,2,~’,l,
(ñ~(‘— eq. 2.9.le) (n~(’-eq.2.9.le)
~ a=1,2
A~(‘—eq. B.7) ~~5( 4- eq. 11.310), 4-eq. B.311)
2
= (3T
0s/Th~)l~’
i5~= [2APTI/(AHmP)11”2
A~,(4-eq. B.7),
AH( ‘-eq. B.69) B.314
ZE
v~[(13~)2+(i3~)2I
E5i
A~,(4-eq.B.7), ~o(‘—eq. B.312),
2A~
4-eq. B.313) (0.6937x 10~~)(1 —0.155 log B.315
8.316
ZSIG
=
10E 1 . ~) 1+0.1112X10_14(E~iev)3~3
[32] —
E~icv( ‘-eq. B.314)
(
ZFREQ
2
5
~ a—1,2
)~a:j~i~2 +(ii~)]i~~2
ñ~(‘-eq. 2.9.le), i3~(‘-eq. B.312), ii~(‘-eq. B.313), ‘-eq. 11.315)
Subroutine HEAT B.317 (none)
Wpe(4lTflce2/rna)”2
B.318
w
ZWPIHZ
2/mi)1/2 1,,
B.319
ZWRAT
Subroutine MPRINT B.320 HDDS
=
rn
(4ITn~e
1( ‘-eq. D.2252)
ec[41Tm~n~/(B~ + By)]
W~/Wce
nD(r)r dr 1”2 (E)[2E/(Am)]
NDDn = 8,,o(21r)2f
x fEm=o
dEJ1 f,~Qi,E, r) d~s Emax(
‘--
eq. B.144),
E,,~,(‘-eq. B.145), °D[),,( ‘-eq. B.172), A,,( ‘-eqs. D.1069—1078, D.1401—1410), ‘— eq. B.147) all beams
B.321
ZPOWR
=
~
f 15IW,,/j
b—I
fjb( ‘-
eq. B.127)
j—1,3 eqs. D.1234—1243) ‘-eqs. D.1159—1168)
4-
B.322
ZFUSN
(P,/l7.6
MeV) = NDT + NOT
N~T(4-eq. B.208) N~’T(4-eq. B.252)
328
G.E. Singer eta!.
/
One-dimensional plasma iranspori code
Appendix C. COMMON blocks; listing of common blocks by affiliation with descriptions ALPHAS (alpha particle package) COMBIG COMBNC BALDUR (main calculation) COMALF — alpha particle slowing down variables. COMBAS — basic system parameters. COMBEM — standard neutral beam variables. COMCNS — conservation checks. All COMCNS quantities beginning in A- and F- are computed in subroutine SOLVE and are CHI conservation checks, as in TOTAL!. All quantities beginning in E- and W- and B-poloidal energy check variables and are computed in subroutine SOLVEB, except EBIN!. BEGIN! and EBINI are computed in subroutine START. COMCNV — conversion factors (see subroutine UNITS). COMCOE — transport equation and boundary condition coefficients. COMDDP — diagnostic and development parameters. COMDF2 — plasma quantities computed by TRCOEF or used by TRCOEF. IH is a hydrogen species index, !I and 112 are impurity species indices, J is boundary/zone index, IZ selects boundary-centered value if !Z = 1, zone centered if IZ = 2. COMDIF — diffusion coefficients. All diffusion coefficients are in standard units. In this common block, J is a zone index, !H and 1H2 are hydrogen species indices (density of species IH is parameter IH — LHYD1 + 1 in CHI), II and 112 are impurity species indices (density of impurities species II is parameter II — LIMP1 = 1 in CHI). COMDIM — dimension of variables and important indices (see subroutine AUXVAL). COMEXT — extrapolation variables (see subroutine RESOLV). COMFLG — code flags and variables. COMFUN — fundamental constants. Each fundamental constant is stored as two real numbers, which are the coefficient and exponent of the base-lO floating point representation of the constant. They are stored separately because some constants are close to the largest or smallest number that can be represented on some machines. COMIN — inputted variables (see subroutine AUXVAL). All variables in this common block represent values in external units. COMMSC — miscellaneous constants. COMMSH - mesh variables. COMNEU — standard neutral gas variables. COMOUT — output parameters. COMOVF - overflow variables. COMSTA — arrays describing state of plasma. COMTIM — time dependent conditions. COMTOK — tokamak and plasma parameters. CORONA (coronal radiation package) COMCOR — coronal equilibrium fit coefficients. FOKKER (neutral beam injection) COMFOK — Fokker—Planck neutral beam. FREYA (neutral beam deposition package) COMLOC - input variables for FREYA. COMMCP — local Monte Carlo particle parameters.
G.E. Singer ci a!.
/
One-dimensional plasma transport code
329
neutral atom parameters. COMNUP neutral beam injector parameters. HE3 (D—D fusions) COMTST MONTE (Monte Carlo neutral gas package) COMUSR user interface. COMPRT particle variables. COMSPL — splitting variables. COMXS cross-section variables. PELLET (pellet injection package) COMPEL MISCELLANEOUS NEOION quantities having to do with nearly-exact neoclassical transport model. APOLAR — quantities having to do with reading of toroidal ripple input file [17]. TEMPRY — miscellaneous calculations. COMNUB
— —
— —
—
—
Appendix D. Input variables
Notes: 1. * * or * preceding a variable name means that the variable is in NAMELIST/NURUN1/, $$ or $ means it is in NURUN2 and * * or $$ means it is in RESET. 2. A variable is referred to (“USED IN EQS. *“) a particular equation whenever possible. If it is not used in any particular equation, it is referred to the section to which it applies. 3. Appendix numbers in the cross references refer to the extended user’s manual [17]. 4. Index of NAMELIST input variables. Variables are listed in the format D.xxxx?? NAME, where D.xxxx is the equation number associated with the variable, and ?? is one of the symbols * *, *, $$ or $. * * or * indicates that the variable is in NAMELIST/NURUN1/, $$ or $ that it is in NURUN2 and * * or $$ that it is in RESET. D.2527 D.2534 D.1441—1460 D.1461 D.2351 D.494—993 D.1068 D.2529 D.2530 D.1462-1481 D.1018 D.1482-1483 D.1484—1485 D.1486-1595 D.1596—1597 D.1598—1599 D.1600—1709
$ $$ $$ $$ $$ * *
$$ $ $ $$ * *
$ $ $ $ $ $
AFSLOW APRESR BPO!D(IT) BZ BZSTAR CFUTZ(500) CJBEAM CPVELC CPVION CURENT(IT) DELMAX
DENGAO(IH) DENGA1(IH) DENGAS(IH,J) DENIMO(II) DENIM1(II) DENIMP(!I,J)
D.1710-1729 D.1730—1784 D.1785 D.1786 D.2697-2776 D.2777—2856 D.2617—2696 D.2537-2616 D.2881—2904 D.1787 D.1788 D.1789 D.1790 D.1791 D.1792 D.1793—1794 D.1795—1796
$$ $ $ $ * * * * * * * * * *
$ $$ $$ $ $ $ $ $
DENMON(IT) DENS(J) DENSO DENS1 DFUTZE(N,K) DFUTZI(N,K) DFUTZD(N,K) DFUTZV(N,K) DXSEMI(N,IX) DTINIT DTMAX
DTMIN EBFIT EEBFIT EEFIT EEHFIT(!H) EEIFIT(I!)
330
G.E. Singer ci at
/ One-dimensional plasma transport code
D.1797
$
EETEFT
D.1285—1339
$$
HR(J)
D.1798 D.1799 D.1800-1801 D.1802—1803 D.1804 D.2532
$ $ $ $ $$ $$
EETIFT EFIT EHFIT(H) EIF!T(II) EIONIZ ELECDO
D.1340—1349 D.1350—1359 D.1360 D.1361—1370 D.1371—1380 D.1381—1390
$$ $$ $$ $$ $$ $$
HRMAJ(B) HRMIN(IB) HTCHEK HTOFF(IB) HTON(IB) HWIDTH(IB)
D.2533 D.237 D.1019 D.1805 D.1807
$$
ELECTE
*
ELLIPT ERRMAX
D.13-32 D.1023 D.1043
$
ETEFIT
D.2332
$
ETIFIT
D.1024 D.33
D.994 D.995 D.1808—1847 D.1848—1887 D.1888—1889 D.1890—1891 D.1892—1911 D.1912—1951 D.1952—1971 D.996 D.997 D.998 D.2420 D.999—999a D.1000 D.1001 D.1002—1005 D.1006-1009 D.1069—1078 D.1079—1118 D.1119—1128 D.1129—1138 D.1139—1148 D.1149—1158 D.1159—1168 D.1169-1178 D.1170-1188 D.1189-1198 D.1199—1228 D.1229—1233e D.1234—1243 D.1244-1253 D.1254 D.1255—1264 D.1265—1274 D.1275—1284
* *
* * * *
$$ $$ $ $ $$ $$ $$ * * * * * * * * * * * * * * * * * *
$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$
EXTF EXTZEF FLGAS(IH,IT) FLIMP(!!,IT) FRACTH(IH) FRACTI(I!) GFLMAX(!T) GFRACT(IT,!H) GFTIME(IT) GNCHEK GRECYC GTCHEK GWMIN GXDOME(2) GXMAXE GXMINE GXPH!(4) GXTHET(4) HABEAM(IB) HANGLE(4,IB) HAPER(IB) HAPERV(IB) HD!V(!B) HDIVV(IB) HEBEAM(IB) HEIGHT(IB) HFOCL(B) HFOCLV(IB) HFRACT(IE,IB) HFUTZ(1O) HIBEAM((IB) HLENTH(IB) HNCHEK HPBEAM(IB) HPROF(IB) HPROFV(IB)
D.9 D.2251 D.2418 D.2252—2253 D.1O1O DlOlOa D.1O11 D.1012 D.1013 D.1391-1400 D.1401-1410 D.1411 D.1412 D.1413—1422 D.1423a-j D.1424—1433 D.2417 D.1434 D.2254—2255 D.1O D.1435 D.1436 D.1025 D.1437 D.76 D.1028 D.1 D.1029 D.1014 D.1015 D.2528 D.1016 D.1017 D.77—85 D.86—135
* *
NADUMP(20) NATOMC NBDUMP NBFIT NBOUND NCLASS NDIARY NEDIT NFUSN NGAS(IH) NGPART NGPROF NGSPLT NGXENE NGZONE NHAPER(IB) NHBEAM(IB) NHE NHMU NHPRFV(IB) NHPROF(IB) $$NHSHAP(IB) NHSKIP NHSRC NIMP(II) NIN NIPART NIPROF NITMAX NIZONE NLCHED NLDIFF NLEDGE NLEXTR NLGLIM NLGMON NLGPIN NLGREF NLGSPT
* *
NLHEAD(9)
* *
NLOMT1(50)
* * * * * *
$ * * * * * *
$$ *
$ * * * * * * * * * *
$$ $ $ $ $$ $$ $$ $ $ * *
$$ $$ * *
$$ * * * * * * * * * * * * * * * *
G.E. Singer ci at / One-dimensional plasma transpori code
D.136—185 D.186—235 D.1048—1067 D.1030 D.236 D.2 D.1031 D.1032 D.2367—2416 D.2333 D.2334—2335
D.2336—2337 D.3 D.1026 D.4 D.34—53 D.2338 D.2339 D.2340-2359 D.2419 D.1044 D.54 D.5 D.2360 D.11 D.6 D.7 D.8 D.2361 D.1045 D.489 D.490 D.491 D.492 D.493 D.12 D.1046 D.2362 D.55
D.2363 D.1047 D.2364 D.1027 D.2365 D.1440 D.56—75 D.2366 D.2913
* * * * * * * * * * * * * * * *
$$ $ $ $ * * * * * * * * $$ $$ $$
$$ * * * * * *
$$ * * * * * * * *
$ * * * * * * * * * * *
$$ * *
$ * *
$ * *
$$ $ * *
$ * *
NLOMT2(50) NLOMT3(50) NLPOMT(20) NLRCOM NLREPT NLRES NLSORC NLSORD NLXXX(50) NNFIT NNHFIT(IH) NNIFIT(II) NONLIN NOUNIT NOUT NPDUMP(20) NPEL NPEL2 NPELGA(IT) NPELOU NPLOT NPOINT NPRINT NPUFF NPUNCH NREAD NREC NRESUM NRFIT NRGRAF NRLDMP NRLEEX NRLIEX NLRMOD NRLPAR NRUN NSEDIT NSKIP NSUB NTEFIT NTGRAF NTIFIT NTRANS NTTY NTYPE NVDUMP(20) NZONES QMHD1
D.2914 D.2536 D.1972—2026 D.2535 D.1438 D.1439 D.238—257 D.258—277 D.278—297 D.298—317 D.318 D.319 D.320 D.321 D.322 D.323 D.324-378 D.379 D.380 D.381 D.382 D.383-437 D.438 D.439—488 D.2421—2471 D.2027—2046 D.2047-2066 D.2067—2086 D.2087—2106 D.2915 D.2107 D.2108 D.2109—2128 D.2129 D.2130 D.2131 D.2151—2206 D.2207 D.2208 D.2209—2228 D.2229—2248 D.1020 D.1021 D.2472—2526 D.2249 D.2250 D.2256—2275 D.2276
331 * *
$$ $ $$ $ $ * * * * * * * * * * * * * * * * * * *
$$ $$ $$ $$ $ $$ $$ $$ $$ $$ $$ $ $ $ $$ $$ * * * *
$ $ $ $$ $
QMHD2 QSTAR RADIUS(J) RASTAR RCWALS RDWALS REHE(IT) REHI(IT) REOFF(IT) REON(IT) REX1 REX2 REY1 REY2 RELEO RLE1 RLEPRF(J) RLEPWR RLFREQ RLIO RLI1 RLIPRF(J) RLIPWR RLPARA(50) RLPOWR(50) RMAJOR(IT) RMINOR(IT) RPA(IT) RPELA(IT) RRSTAR SEDIT SPLOT TBPOID(IT) TCOLD TCOLDP TCOMP(IT) TE(J) TEO TEl TEDIT(IT) TGAS(IT) THETA THETAP TI(J) TIO TI1 TIMP(IT) TINIT
332
G.E. Singer ci a!.
D.2277 D.2278—2287 D.2288—2307 D.1022 D.2308—2327 D.2857—2880
$$ $$ $$ * *
$$ * *
TMAX TPELA(IT) TPLOT(IT) TSPARE VPELA(IT) VXSEMI(N,IX)
/
One-dimensional plasma transport code
D.2328—2329 D.2330—2331 D.2905—2908 D.1033—1042 D.2909—2912
$ $ * * * * * *
WTGAS(IH) WTIMP(II) XESEMI(N) XFUTZ(1O) XISEMI(N)
C.E. Singer ci al. No.
Symbol
Code (array size)
/
333
One-dimensional plasma transport code Default (logical value or
Used in eqs no.
Description
units)
OLYMPUS variables D.1
a * NLEDGE
0 (false)
Not used in present version of BALDUR (set by formatted READ input as well as NAMELIST)
D.2
*
*NLRE5
0 (false)
Not used in present version of BALDUR (set by formatted READ input as well as NAMELIST)
D.3
* *
NONLIN
5
Channel to write messages to teletype (do not change)
D.4
* *
NOUT
6
Not used for input (current output thannd; cf. ref. [7])
D.5
*
aNPRINT
6
Channel to
write main output (do not
change)
D.6
**
NREAD
1
D.7
* *
NREC
0
Channel to read main input file (do not change) Not used in present version of BALDUR (set by formatted READ input as well as by NAMELIST)
D.8
a aNRESUM
0
Not used in present version of BALDUR (set by formatted READ input as well as
D.9
* *
NDIARY
6
Channel to write short printout (do not change)
D.10
*
1
Not used for input (current input channel; cf. ref. [7])
D.11
NPUNCH
7
D.12
* *
NRUN
10
D.13 —32
* *
NADUMP(20)
0, 0, 100, 0, 1, 1, 100, 1, 0, 0 0
D.33
* *
NCLASS
0
Not
D.34 —53
* *
NPDUMP(20)
0
Not used
D.54
*
a NPOINT
1
Not used for input (most recent point reported, cf. ref. [7])
D.55
* *
NSUB
1
Not used for input (most recent subprogram reported; cf. ref. [7])
D.56 —75
*
0
Not used
D.76
a*NLCHED
D.77 —85
*
by NAMELIST)
a NIN
*NYDUMP(20)
*NLHEAD(9)
Channel for card output (not used) appendix F
Maximum number of time steps This OLYMPUS array has been modified to be used as an internal storage array and should not be used for input used for input (most recent subprogram class reported; cf. ref. [7])
0 0 (false)
Not used
334
G.E. Singer ci a!.
No.
Symbol
D.86 —135
* *
D.136 —185
* *
D.186
Code (array size)
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
NLOMT1(50)
0 (false)
appendix F
True if subroutine is to be skipped (cf. ref. [7]). If NLOMT1(3) = .T., subroutine PRESET is skipped 5 AUXVAL 6 INITAL 8 START 9 ERRCHK 10 UNITS
NLOMT2(50)
0 (false)
appendix F
True if subroutine is to be skipped (cf ref. [7]). If NLOMT2(1) = .T., subroutine(s) STEPON is skipped 2 COEF 3 NEUGAS (Monte package) 4 BEAMS (Fokker package) 5 HEAT,ICRF 6 IMPRAD 7 BOUNDS 8 SOLVEB 9 REDUCE 10 SOLVE 11 RESOLV 12 GETCHI 13 CMPRES 14 TRCOEF 15 CONVRT 16 CNVCOF 17 DPOSIT (Freya package) 18 XSCALE 19 ALPHAS, HE3 20 EMPIRC 21 PDRLVE (pellet package) 22 NCFLUX
0
appendix F
True if subroutine is to be skipped (cf. ref. [7]). If NLOMT3(1) = .T., subroutine(s) OUTPUT is skipped 2 MPRINT 3 SPRINT 4 GPRINT 5 HPRINT 8 IPRINT 9 APRINT,FPRINT
* *
NLOMT3(50)
—235
D.236
/
(false)
*
*NLREPT
0 (false)
Not used
Ellipticity for modifying beam deposition, numerical ripple input table and semiernpirical model D.237 K aELLIPT I 2.3f Ellipticity 3g 2. 2.le section 2.7 section 2.8
G.E. Singer cia!. No.
Symbol
Code (array size)
/
335
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
Total ERCH power to electrons is pFiC between times t~k and t(,ffk
EGRH heating parameters
D.238 —257
P,~
*
REHE(20)
(W)
2.9.7a
D.258 —277
p,~C
*
REHI(20)
(W)
2.9.7a
D.278 —297
t~j~
*
REOFF(20)
(s)
D.238 —277
Times to turn off ECRH
D.298 —317
t~k
*
REON(20)
(s)
D.238 —277
Times to turn on ECRH
D.318
*
REX1
(cm)
section 2.9.7
D.319 D.320 D.321
*
REX2 REY1 * REY2
(cm) (cm) (cm)
Coordinates of vertices of a rectangular window into which ECRH power is unifonnly distributed: the origin (0.,0.) for these coordinates corresponds to the magnetic axis. REY1 and REY2 are height with respect to toroidal midplane and REX1 and REX2 are distance of sides of window in midplane from magnetic axis
p,,EC =
Total ECRH power to ions is p,Ec between times t~k and i(,ffk
*
p~C
Lower hybrid and arbitrarily specified heating D.322 * RLEO
0
section 2.9.6
Relative heating of electrons at magnetic axis [for heating profiles specified by polynomial fit (‘-eq. D.492)]
D.323
*
RLE1
0
section 2.9.6
Relative heating of electron at r = rWall [for heating profiles specified by polynomial fit (‘-eq. D.492)]
D.324 —378
*
RLEPRF(55)
0
section 2.9.6
Relative heating of electrons at each zone center [for tabular input profiles (‘-eq. D.492)]
D.379
*
RLEPWR
106 (W)
section 2.9.6
Total power to electrons [for NRLMOD = 1 or 2 (i— eq. D.492)]
*
RLFREQ
5 X 10~ (Hz)
section 2.9.6
Frequency of lower hybrid heating source
D.381
*
RLIO
0
section 2.9.6
Relative heating of ions at magnetic axis [for heating profiles specified by polynomial fit]
D.382
*
RLI1
0
section
Relative heating of ions at r = ~ [for heating profiles specified by polynomial fit (e— eq. D.492)]
D.380
w
2.9.6 D.383 —437
*
RLIPRF(55)
D.438
*
RLIPWR
*
RLPARA(50)
D.439 —488
n
11
0
section 2.9.6
Relative heating of ions at each zone center (for tabular input profiles (‘-eq. D.492)]
106 (W)
2.9.6a—b
Total power to ions [for NRLMOD = 1 or 2 or to ions plus electrons [for NRLMOD = 3 (‘-eq. (D.492)]
0
2.9.6a—c, 2.9.6e—f
Index of refraction for mode i [1 NRLPAR ( ‘-eq. D.493)]
i
336 No.
G.E. Singer ci aL Symbol
Code (array
size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
D.489
*NRLDMP
3
section 2.9.6
Switch to choose lower hybrid heating model: 1 -. ion heating only — all energy deposited in radial zone containing lower hybrid resonance point, 2 —+ electron Landau damping only, 3 -. other model
D.490
*
NRLEEX
0
section 2.9.6
D.491
*NRLIEX
Exponent in radial variation of polynomials fit to electron and ion heating: P = RLj1 + (RLjO— RLj1) * [1—(r/r~,,ii)s * NRLJEX], j = E,I NRLEEX = electrons; NRLIEX = ions
D.492
*NRLMOD
3
section 2.9.6
Switch to select: 1-. tabular input, 2 —* polynomial heating profiles, 3 —. lower hybrid heating (cf. eqs. D.322—379, D.381—438) (cf. C~—C 63)
D.493
*
NRLPAR
0
section 2.9.6
Number of specified indices of refraction in lower hybrid wave spectrum (maximum is 50)
Plasma modeling parameters D.494 C~ —993
* *
CFUTZ(500)
Specified values multiplying various transport coefficients, and controlling transport, source models, and numerical methods. In what follows, “x” means “multiplies”: “A —* + B” means “B gets added into A”
C1
0
2.lm
x pseudoclassical contribution to DHH; H =1,2
C3
0
2.ln
x pseudoclassical contribution to D,1 I =
x,, x~
0 0
2.lo 2.lp
x pseudoclassical pseudoclassical contribution to
CC4 5
0
2.lm
x Bohm contribution to Dim; H1,2
C6
0
2.ln
x Bohm contribution to D11 I
C7
0
2.lo
x Bohm contribution to
x,,
C8
0
2.lp
x Bohm contribution to
x~
C9
1
2.5.1 a—d
x simplified neoclassical contribution to DHH, diagonal terms only; H = 1,2
C10
1
2.5.1 a—d
x simplified neoclassical contribution to D11, diagonal terms only; I =
C11
1
2.5.1 a—d
x simplified neoclassical contribution to
C12
1
2.5.1 a—d
x simplified neoclassical contribution to
C3
x~
=
G.E. Singer ci at / No.
Symbol
Code (array size)
One-dimensional plasma transport code Default (logical value or
Used in eqs no.
337
Description
units)
C
13
0
Not used (coefficient for nonimplemented model of trapped electron thermal diffusivity)
Drift wave contribution to
x,,
C14
0
2.lo
x~
C15
0
2.lh
D~+n(~5)[1+Cis(r/r~r)c40];
a = (~‘~) (modifiedby “soft Bohns limit”, cf. C51’) C16 C17 C18
0 0 1s~) (cm
2.1k
0
2.lp
x~ by —~ “soft” +(Cj7/n~)[q(r Bohm limit”,=cf. O)]Cis C
(modified
51) x
~T,
ion thermal gradient contribution to
xi 1
C20
0
C21
0 (cm2/s)
2.lc
DHH—s+C21 H=1,2
C22
0 (cm2/s)
2.ld
D,, —+ + C22, I
C23
0 (cm2/s)
2.le
x~—~+ C23
C24
0 (cm~/s)
2.lf
(225
(cm2/s) 0
2.lh
(226
0 2/s) (cm 0
2.lh
C 27
2.7e—f
x v~i~ (factor multiplying Ware pinch of
C19
hydrogen isotopes and electron energy) Not used in TRCOEF
—‘
+
C2~
1[1 + Gis(r/r,~)C40]; H = 1,2 (modified by “soft Bohm limit”, -~+ C25n cf. C 51) IDJ,~~9+GS6n~~[1+GlS(r/r,cr)C10]; = 1,2 (modified by “soft Bohns limit”, cf. C51). Not used in TRCOEF
2.lf
x. —~+
0 (cm~/s)
2.le
X~~+C29(r/r~,)~
(230
0
2.le
(modified by “soft Bohm limit”, cf. C~~)
C31
0
2.lm
x contribution for q ~ 1 to DHH; H
C32 C53
0 0
2.ln 2.lo
x contribution for q ~ 1 to D11 I = x contribution for q 1 to
C34
0
2.lp
x contribution for q ~ I to
(2~~
0 2/s) (cm 0
2.lc
DHH~+C3s(r/r,or)C36; ified by “soft Bohm limit”,H=1,2 cf. C (mod31)
C28 C 29
C~
0 1s (cm
1)
2.lc
C28/n~
= 1,2
x~ x~
338 No.
G.E. Singer ci a!. Symbol
Code (array
size)
C
37
C38 C39
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
0 1s1) (cm
2.lc
DRH + C37/ne; H =C 1,2 (modified by “soft —~ Bohin limit”, cf.
0 2/s) (cm 0
2.ld
D11 -. + Css(r/r~r)’39; by “soft Bohm limit”, cf.I = C !‘,h (modified 51)
0
51)
2.ld 2.lh
Do,, —~ + (~~)[l + a = (~‘~) (modified by “soft Bohm limit”, cf. C55)
C41
0
7~scales as (compression ratio)C~~ (if C41 * 0?)
C42
1s1) 0 (cm~
2.ld
“soft limit”, Icf. D11 -~Bohm + C42/ne; = C !‘,h (modified by 51)
C43
0 (cm2/s)
2.6a
~
0 (cm2/s) 0
2.6c—d
x.,’ maximum
2.1!’
If
C45
(keY/cm)
maximum allowed value; a
(245>
x~ C~ (247
C48 C49 C50 C51
C52
0 (cm2/s) 0 0 0 0 1010
=
1,2,t’,h
i,e
0., C49
DHH
D11
allowed value; j
=
—. +
C50 [C~ C48
+
C45n~17’-C~,
2.1!’ 2.1!’ 2.le 2.lc 2.ld 2.li
0
(modified by “soft Bohm limit”, cf. C351)
= max(C51,1.), DB is the “soft Bohm limit” applied to various transport coefficients (cf. C55, C17, C25, C26, C29, C30, C35_.~,C42, C57, C58) ~j~B
Reserved for use as maximum allowed value overriding C~
C53
0 (cm2/s)
2.5.2a
DHH=C53 for r>
C54
0 (cm~/s)
2.5.2a
D11=C54 for r> r~,if C54>0.; I=!’,h
C55
0 (cm2/s)
2.5.2a
x~=
2.5.2a—c
If C56> 0., the values of these diffusion coefficients are linearly interpolated in radius between Cs6r~rand r~r. Note that C53—C56 have 0.~ forCno effect if C120 = 0. x~—’~ x~ —. + C 1[158=0. ~(r/r~,)2]~”, (r/r,~,)<57n~ y 0 <1., x~—+Css/n~, (r/r~~)y0ory0>l. and is (y0 C the value of r/r,cr where the C5-, 58 terms are equal)
C56
0
C57
0 1s’) (cm
2.1k’
C58
0 1s1) (cm
2.1k’
c~r if
x~
C53>0.; H=1,2
C55 for r> r,,~r if C55> 0
G.E. Singer ci at No.
Symbol
C
59
Code (array size)
/
One-dimensional plasma transport code
339
Default (logical value or units)
Used in eqs no.
Description
0
2.1k’
(modified by “soft Bohm limit”, cf. C51) If 0. C~<1., then the rf lost power fraction, LRF, is G~ (LRF = 0, otherwise). This is used to calculate Q,,
0
on
C61
0 (s)
section 2.9.6
Time auxiliary heating is turned G~)
(cf.
C62
0
section
Time auxiliary heating is turned off (cf.
C63
(s) 0
2.9.6 section 2.9.6
G~) Int(ç3 + 0.1) = time step after which cxtensive printouts of lower hybrid heating [if activated, cf. C60, C61, C62and NRLMOD (‘-eq. D.492)] are made
CM
0
C65
0
CM
0
C67
0
CM
0
C69
0
C70
0
section 2.9.10
~ 0. — turns off He volume source mediated by C7j—C77, and its associated short printout. Note that even if C.,,~> 0., there is 4He willspecified be no He as volume an impurity sourcespecies unless (‘-eq. D.2254—2255)
71
0
2.9.lOa
(If C71 0., we set C71 = 0.9.) Fraction of helium ions crossing out at r = ~ which is recycled as a volume source (cf. C70)
C.,2
0
2.9.lOa
(If C.,2 0., we set C.,2 = 0.9.) Fraction of helium lost by parallel flow in scrapeoff which is recycled as a volume source (cf. C70)
C73
0
2.9.lOa
(If C73 0., we set C.,3 = 0.9.) Helium neutrals penetrating further than C.,3r~~ are distributed in volume source by normalization (cf. C70)
C74
0 (keY)
2.9.lOb
(If C74 0., we set C74 = 0.1). Electron energy lost per He° ionization (cf. C70)
C.,5
0
2.9.lOa
is set the C75 (unique) angle, (If C75 0He 0., we = 1.) C75 = cosfrom where to angle to circular flux surfaces normal at which He° atoms are launched (cf.
Reserved for use by ion cyclotron resonance heating package
Notused Not used
C
C~~) C 76
0 (keV)
section 2.9.10
(If C76 0., we set C76 = 0.04.) (Unique) energy with He°atoms are launched
340 No.
G.E. Singer ci a!. Symbol
(277
Code (array size)
/ One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
0
section
(If C
2.9.10
77 0., we set C77 = 1.) He° atoms are launched inward from the outer boundary of zone number N +2— C77 (cf. C0)
C78
Not used
C79
Not used
C80
Not used
C81
0
C82
0 (s)
C83
0
5.2.2.5d
Flow bias parameter: If C81 * 0., zone centers for some of the differencing formulas are offset by (v~ i)/( C81 ~ r), where v is the convective velocity Time at which BALDUR will re-read the input file with an abbreviated namelist (see subroutine STEPON)
2.5.2d—e
Sets diffusivities in scrapeoff to a multiple of Bohm value. (C83 0., r~,)~ x~
r>
1.5DB; a =1,2,!’,h; j and C120) =
C84
0
Xe
C85
0
Not used
C86
0
Not used
C87
0
Not used
C88
0
Not used
C89
0 (s)
=
i,e(cf. C51
enhancement factor inside a tearing mode island (not used in present version)
section 3.2 2.lp
Recycling coefficient for hydrogen isotopes is set to toC102 time6-regime C89 if C89 > 0 Contribution x, atfrom model is 1.5C~,JD6(cf. C 91)
0 C91
0
2.1
C91
C92
0
2.1B
Arbitrary multiplicative 6 (cf.factor C included in the definition of D” 91)
C93
0
2.IA
Arbitrary multiplicative factor included in the definition of DKP (cf. C91)
C94
0
2.1C
C95
0
2.1w
Arbitrary multiplicative factor included in the definition of ~ (cf. C91) Arbitrary multiplicative factor included in the definition of DTE (cf. C91)
C96
0
C97
0
2.lm
C98
0
2.ln
0. turns off effects of 6-regime model mediated by C~ — C96
Arbitrary multiplicative factor included in the definition of DTI (cf. C91) x Kadomtsev contribution to DHH; H = 1,2 x Kadomtsev contribution to D11 I =
G.E. Singer ci al. No.
341
code
Default (logical value or units)
Used in eqs no.
Description
0
2.lo
x Kadomtsev contribution to
0
2.9.4a
Fraction of local cyclotron emission included as an electron energy loss
0
section 2.1
2C101 — 1 = the number of averagings done in computing scale9,hea, heights(‘~,,eqs. r~7.1 and magnetic shear B.83—88)
102
0
section 3.2
Recycling coefficient for hydrogen isotopes (— eq. D.497) is reset to C102 at time C89
C103
0
Not used
C1~
0
Not used
C105
0
Not used
C1~
0
Not used
C107
0
Not used
C108
0
Not used
C1~
0
Not used
C110
0
2.4.2a
C110 > 0. activities “nearly exact” neoclassical particle diffusivities between each ionic species and all others, and replaces simplified off-diagonal neoclassical diffuSivities
C111
0
2.4.2a
Factor multiplying classical contribution to “nearly exact” neoclassical particle diffusivities
C112
0
2.4.2a
Factor multiplying banana-plateau contribution to “nearly exact” neoclassical particle diffusivities
C113
0
2.4.2a
Factor multiplying Pfirsch—Schluter contribution to “nearly exact” neoclassical particle diffusivities
C114
0
Not used
C115
0
Not used
C116
0
Not used
C117
0
Not used
C118
0
Not used
C119
0
C120
0
C121
0
Symbol
C
99
C101
Code (array size)
/ One-dimensional plasma transport
x~
C
Not used 2.lh,k’; 2.3h—i; 2.5a—e; 2.8e; 2.9.5e 2.9.5e
C120 > 0. activates scrapeoff modeling. r,~ is defined by the inner boundary of physical zone number (C120 —1). If C120 0., r,,~r is set equal to r~~i1 and C53—C56 and C129—C132 have no effect (If C121 = 0., we set C121 = 100.) “Second scrapeoff” with connection length C128 lies between r~~1 and the inner boundary of real spatial zone number (C121 —1)
342 No.
G.E. Singer ci a!. Symbol
C
Code (array size)
/
One-dimensional plasma transpori code Default (logical value or units)
Used in eqs no.
Description
122
0
C123
0
2.9Sf
Charge-exchange parallel frictionis turned on in scrapeoff if C123 = 1
C124
0
2.9.5a
(If C124 = 0., we set C124 = 1.) Hydrogen isotope loss rate due to parallel flow in scrapeoff is multiplied by C124
C125
0
2.9.5a
(If C125 = 0., we set C125 =1.) Impurity loss rates due to parallel flow in scrapeoff are multiplied by C125
C126
0
C127
0 (cm)
2.9.5e
Connection length for parallel flow in inner-most scrape-off layer
C128
0
2.9.5e
C129
0
Connection length for parallel flow in “second scrapeoff”, cf. C121 Minimum n~in scrapeoff (i.e., n~ C129 is forced by the code; if activated, this forcing nullifies conservation checks). Has no effect if C120 0
C1~
0
section 2.9.10
Minimum T~0~ in scrapeoff (i.e., 77,~,,, C130 is forced by the code; if activated this forcing nullifies E,, conservation check). Has no effect if C120 0
section 2.9.10
Minimum ~ in scrapeoff (i.e., T1~~> C131 is activated, this forcing nullifies E, conservation check). Has no effect if C120
(eV)
C131
0 (eV)
Not used
(If C126 = 0., we set C126 = N + 2). Between times specified by C161 and C162, the outermost radial zone subject to normal time-step zone controls is zone number (C126 —1), for C126 > 2
C132_138
0
Not used
C139
0
C139 = —1 sets simplified off-diagonal neoclassical transport coefficients to zero
C~
0
2.3a
Switch to control effects of toroidal field ripple on ion thermal conductivity. C1~ = 0. -. no effects, 1. 2.
3. 4.
+ X~T+X~’+X~, -.
+
-.
+ +
-
x~”~
5.~.+ X~+
C141
0
2.3c,d,h
xi”.
(If C141 = 0., we set C141 = 12.) Toroidal periodicity of fixed component of ripple
G.E. Singer ci at No.
Symbol
Code (array size)
/
One-dimensional plasma transport code
343
Default (logical value or units)
Used in eqs no.
Description
2.3h 2.3h 2.3h 2.3h
&r~d(r,
C145
0 0 0 0
CIM
0
2.3c,d,i
Toroidal periodicity of time varying component of ripple (cf. eqs. D.1451—1460, D.1472—1481 and D.2119—2128). C1~ 0, turns off time-varying ripple component
0
2.3i
(If C147 = 0., we set C14., = 2.) Exponent in radial variation of time-varying ripple; analogous to C1M above (cf. CIM)
0
section 2.3
C~> 0. —e use table for input of fixed ripple contours (cf. section 7.5 and appendix G of ref. [33])
C149
0
2.3b
C149 >1. —# use Goldston—Towner formula (‘-eq. B.96) [21] for coefficient G when computing X~T for fixed ripple (cf.
C150
0 (s)
t~151_160 C161
0 0 (s) 0 (s)
section 6
Not used in present version of BALDUR. (If C162 = 0., we set C162 =10~.) Certain radial zones, as specified by C126, C163 and CIM, are excluded from normal time-step controls at times C161 i C16~
C163 CIM
0
section 6
Between times specified by C161 and C162, radial zones C163 j C~ are excluded from normal timestep controls
Nc,,
If
0s
C142 C143 CIM
~
C147
C145
0., 9,
we set C145 = 0.2.) sin(C ~)[C142 +2)(C143 — C142) (r/?~,cr)C144)[exp(_ C1459 141~)] (ignored if GIM> 0.). [C142and C143 represent the fractional peak-average fixed ripple at r = 0 and r = r,cr, respectively; i.e,. unlike the input file whose reading is triggered by C148 > 0., C142 and C143 are not given in percent] =
Csse)
C162
(If C150 = 0., we 5~tC150 = 5(L~tmax) (4eq. D.1788). Time between successive computations of coefficient G for fixed ripple (‘-eqs. B.96—97; cf. C~49)
C16519.,
0
C198
0 (keY)
section 2.9.1
Not used in present version of BALDUR If C198 > 0., TCOLD (‘-eq. D.2129) is continually made the minimum of (1) the ion temperature at a radial point determined by C198 times the zone index of the outermost non-scrape-off zone, and (2) its originally prescribed value. Note that TCOLDP (‘-eq. D.2130) is not affected by C198
C1~
0
section 2.9.4
If C1~= 1.(2.) peaks in the impurity radiation losses are smoothed by a binomialweighted smoothing performed over 5(7) adjoining radial zones
344 No.
C.E. Singer ci al. Symbol
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
C
0.
2®
Used in eqs no.
Description
section 2.9.13
If C2® 0., prescribed impurity influxes (‘— eqs. D.1848—1887) are considered ionized. Setting C~®>0. activates the neutral impurity influx model, which is mediated by C201216 and C220_226 in conjunction with N~,IMP (‘- eqs. D.2254—2255), t] (‘-eqs. D.2256—2275) and F1~(~— eqs. D.1848—1887). However, only impurity species numbers 1 and 2 are allowed 0 and to NbemP). influxed If C as neutrals, (i.e., N1” 2’ 2® = 1., 1., F1~, prescribes neutral influxes of 2s1). impurity If C species q at times tJ; in (cm 2® = 2., F1,, prescribes the total number (integrated over the plasma volume) of first and second impurity species a ions desired at times 1]; the influx is adjusted (sometimes overstability) to attempt to meet these targets. The third and fourth impurity species are always ionized and treated as for C2® = 0 (cf. C225226). Note effect of C205 and C2®
C201
0 (keY)
section 2.9.13
(If C201 = 0., we set C201 = 0.01.) Prescribed incoming kinetic energy for neutrals of first impurity species species (cf. C2®)
C202
0 (keY)
section 2.9.13
(If C202 = 0., we set C202 = 0.01.) Same as C201, but for second impurity species (cf. C2®). Note: neutral impurity influx can only be prescribed for the first two impurity species
C203
0 (keY)
2.9.13d
(If C203 = 0., we set C203 = 0.0136.) Electron energy loss per initial ionization of a neutral atom of first impurity species (cf. C200)
C2®
0 (keY)
2.9.13d
(If C2® = 0., we set C2® = 0.0136). Same as C203, but for second impurity species (cf. C2®) [If C
C205
0 (keY)
2.9.l3g
0., we used to set C205 = 10 (in before BALDP27M).] Extra radiation loss due to nonequilibrium per atom of first impurity species (cf. C2®) 205
=
versions
C2®
0 (keY)
2.9.13g section
C20.,
0
section 2.9.13
versions as [If C2® before 0., we BALDP27M).] used to set C2®Same = 10 (in C 2®, but for second impurity species (cf. C2®) (If C207 = 0., we set C207 =1.) N +2— C2® = index of outermost radial zone allowing ionization of neutrals of first impurity neutrals of first impurity species (cf. C2®)
C.E. Singer et at / One-dimensional plasma transport code No.
Symbol
Code (array size)
345
Default (logical value or units)
Used in eqs no.
Description
208
0
section 2.9.13
(If C208 = 0., we set C208 = 1.) Same as C207, but for second impurity species (cf. C2®)
C2®
0
section 2.9.13
(If C2® = 0., we set C209 = 0.9.) Prescribed maximum penetration depth for neutrals of first impurity species, expressed as a fraction of r,cr (cf. C120, C2®). Neutral impurity density is set to zero beyond this depth
C210
0
section 2.9.13
(If C210 = 0, we set C210 = 0.9.) Same as C209, but for second impurity species (cf. C2®)
C211
0
section 2.9.13
(If C211 = 0., we set C211 = 1.) Prescribed cosine of the angle between the inwarddirected normal to the outer plasma
C
surface and the influx direction of neu-
trals of first impurity species, Used in a rough approximation of controlled gas feed. Limited so that 0.5 C211 1.0 (cf. C2®) C212
0
section 2.9.13
(If C212 = 0., we set C212 =1.) Same as C211, but for second impurity species (cf. C2®)
C213
0
section 2.9.13
Setting (2213 1. activates an adjustment of the ionization rates determined by an empirical formula so as to approximate within 15% values of Freeman and Jones [40] (cf. C2®)
C214
0 (s)
section 2.9.13
If G2®> 0., impurity ion source rates (before and after normalization with respect to the neutral influx) and nonequilibrium radiation enhancement factors are printed out for three time steps starting at time C214. If C217> 0., average hydrogenic flux velocity and any enhanced drag on impurities are sinularly printed out
C215
0
section 2.9.13
(If C215 = 0., we set C215 = 1.) Multiplies the ionization rates for neutrals of first impurity species (cf. C2®)
C216
0
section 2.9.13
(If C216 = 0., we set C216 = 1.) Same as (.2215, but for second impurity species (cf. C2®)
C217
0
section 2.9.13
Proportionality factor for enhanced drag on impurities attributable to hydrogenic. ion fluxes. The extra drag gives rise to an increment ~ in the flux of impurity species a ions: ~F,, = (C217 — n,,) (flux velocity selected by C214). The enhanced drag affects only the region selected by C219. If C217 0., this enhanced drag is not included
346 No.
G.E. Singer ci al. Symbol
C
218
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units) 0
Used in eqs no.
Description
section 2.9.13
If C218 = 0., enhanced drag on impurity ions is proportional to flux velocity difference between hydrogemc and impurity ions. If C218> 0., enhanced drag is proportional solely to the hydrogenic flux velocity (cf. C217) (If C219 = 0., we set C219 = 0.2) Enhanced drag is active only for radii > (1— C2i9)r,,,~ (cf. C120, C217)
C219
0
section 2.9.13
C220
0
section 2.9.13
(If C220 = 0., we set C220 = 1.) Adjustment factor for influx rate of first impurity species when using model involved by setting (.2209 = 2 (cf. C2®)
C221
0
section 2.9.13
(If C221 = 0., we set C221 =1.) Same as C220, but for second impurity species (cf. C2®)
C222
0 2s1) (cm
2.9.13 section
mum recycled neutral influx of first im(If C222 = 0., we set C222 = 1017.) Maxspurity species (cf. C 2®)
C223
0 (cm —2 s — 1)
section 2.9.13
(If C223 = 0., we set C223 = 1017.) Same as C222, but for second impurity species (cf. C2®)
C224
0
section 2.9.13
Not used (dummy switch available for future used by neutral impurity model; cf. C2®)
C225
0
2.9.13g
(If C225 = 0., we set C225 = 10.) Maximum radiation enhancement factor with respect to coronal equilibrium for first impurity species (cf. C2®)
C226
0
2.9.13g
(If C226 = 0., we set C226 = 10.) Same as C225, but for second impurity species (cf. C2®)
C227
i0~~ (eV)
Prescribed maximum ion temperature (cv) desired at the plasma edge (JSEPX1)
C228
0.90
Factor lowering the prescribed neutralimpurity influx if T,(JSEPX1) is too cold
C229
1.10
Factor raising the prescribed neutralimpurity influx if ~‘1 (JSEPX1) is too hot
C2se
0
section 2.9.11
(If C231 through C250 are all 0., we set C2~= 0.) C230 = 0. —‘ pellets are injected singly. C230 = 1. —. pellets are injected in multiple-pellets clusters. The time interval between the firings of successive pellets within a cluster is the same for all clusters (cf. C231). C2~= 2. —‘ pellets are injected in multiple-pellet clusters. The time interval between the firings of successive pellets within a cluster must be specified individually for each cluster (cf. C231250) (4- eqs. D.2278—2287)
C.E. Singer ci at No.
Symbol
Code (array size)
/
One-dimensional plasma transport code Default (logical value or
347
Used in eqs no.
Description
section 2.9.11
If C230 = 1., C231 is the time interval between the firings of successive pellets within a cluster for all clusters. If C230 = 1., C230.~1is the time interval between the firings of successive pellets within the jth
units)
C
231
0 (s) 0 (s)
cluster for j > 1 C251
0
section 2.9.11
(If C251 = 0., we set C251 = 50.) Maximum deposition enhancement factor employed when intervals between successive pellet firings within a cluster are less than the current time step (cf. (2230)
C252
0
section 2.9.11
(If C252 = 0., we set C252 = 100.) Maximum allowed number of consecutive firings per pellet cluster (cf. C230)
C253
0
section 2.9.11
C253 = 0., —, Send a message to the teletype every time a pellet cluster is injected unless NTTY = 0 (— eq. D.2365). Also do a short printout everyNPELOUth time a cluster is injected (~eq. D.2419). C253 > 0. suppress these outputs
C254
0
section 2.9.11
~‘255
0
section 2.9.11
The fraction of the pellet cloud which is ionized hydrogen. Values greater than 0.1 suppress ablation by fast ion and alphas The fraction of the calculated pellet partide source which is added to the plasma density (the rest vanishes)
C256
0
section 2.9.11
C257_230 C281
The fraction of the pellet cloud which is neutral hydrogen. C254 and C256 do not need to add to 1 Not used
0
Switch for
x~0
—‘
x~”~ 1-. XFH,
2 —e
(
C282
0
Fraction of minor radius that magnetic axis is shifted outward for computing xFH (= R~ of ref. [27]). Maximum value = 0.95
C283_289
0
Not used
C2®..292
0
2.lg’
x18=f,= 1, r/aC291, (r/a) 1+(C2® 1) 292
C291 C2®,
<
r/a
—
—
C292,
C292
C291 291
348 No.
CE. Singer ci at Symbol
C
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
293
0
2.lf
X15
IX1~=C293+(r/LTC)
C209
0
2.lf
X19’
X~I=C294+(r/LTC)
C295
0
2.ld
eA~l9 —e eA~l9+ v~./(C295 A = D, i, e, v for all n
C296
0
2.lh +
+
QMHDI q—q~, ~MHD2
~
+ C 296
C297326
0
Not used
C327_329
0
If C327 * 0, ignore coronal equilibrium calculation of impurity radiation. When C327 > 0., the total radiated power becomes C327 times total heating power. When C327> 0., the radiated power density is proportional to =
C328 +(1 — C32s)[r/(r~,5iC329)], r < rwajjC329, 2 0.25 025 [1-(r/rweii) J /[1—C~9] r r~~iC329 C330.
Not used
3’U
C~5
0
2.11’
If C345, C3~and C347> 0.,
2.11’
DHH D1, CX~ 349 — C350
(cm 1~—1)
C3M C347
0 (keY) 0
2.11’
1 C345n [~+ (~kev/C3~)~’I1/C~7
~348
C~8 CM9 C350 C351
0 0 0 1010
2.lc 2.ld 2.le
(modified by “soft Bohm limit”, cf. C351)
2.11”
~B = max(C351,1.) is the “soft Bohm limit” applied to the so-called PLT empirical transport models mediated by C4550 and C345350 Not used
C352_353
0
C354
0
2.4.ld’
Factor multiplying the anomalous hydrogen mixing coefficient which scales like neoclassical coefficients
0
2.11’
D,,,, -. + DPDX = C3SSr~6(BOT/ a =1,2,1,h Not used in present version of BALDUR
2.le
Model: time-averaging parameter for semi-empirical Xe,,, = C3MXe,e,(iJ_l)+(l —
C358363
0
C~
0
C.E. Singer ci at No.
Symbol
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
349
Used in eqs no.
Description
2.7f
C
2.1J 2.1K 2.1K
Not used QSTAR (alternate input form ~— D.2913) QMHD1 (alternate input form e— D.2914) QMHD2 (alternate input form 4- D.2915)
CM5
0
C309_376 C37., C378 C379
0
C380
0
2.9.5i
Effective neutral path length from divertor
C381
0.1
2.9.5i
“Leakage” fraction of divertor reflux which does not contribute to divertor throat blocking (sometimes called ~
2.9.5m
Absolute value of energy loss per ionization in divertor chamber. C382 > 0 energy gain, so default value should not normally be used
C382
40(eV)
365 0 includes additional electron energy pinch [13] and omits the convective energy flow controlled by CPVELC (~— eq. D.2529)
C383
1
2.9.5m
Ion energy-per ion through divertor sheath
C384
1
2.9.5m
Electron energy-per ion through divertor sheath
2.9.5m
Neutral energy in divertor chamber
C385
10(eV)
C386
0
2.9.5n
Heat flux limit factor for parallel electron heat conduction loss
C38.,
0
2.9.5n
C388
0
2.9.5p
Determines electron temperatures used in computing heat flux limit Determines temperature used in computing s~
C3®
0
2.8e
C389,391..309
0
Changes scrape-off resistivity. (C3® = 1 gives problems near r = Not used
section 3.1
If C395of>the tance 0, plasma the effective is C external induc395
0
section 3.1
Switch to determine the type of boundary condition for the poloidal field diffusion equation 0—e total current or B0(a) specified, 4’eo~ specified 21 —e —e surface surface voltage voltagefrom calculated implicitly from specified ~ and ~
0
section 5.2.2
If C397 > 0., then the the B0e,~ diffusion min(C397,1)
C395 ~ C309
02/cm) (s
C 39.,
C398
0
°CN
eq.
Not used in present version
used
to solve (5.2.2a) is
350 No.
C.E. Singer ci at Symbol
C
309
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
1
2.8c
Multiplies resistivity in current diffusion
5.2.2a
Tlb’ equation but not in ohmic heating; s~ = C309s18 the B~,,,[ ‘-eq. (2.8c)]. Note: since si *
1energy conservation may be poor C4®
0 0
Not used 2.8b
If C,,®> 0.,
1~his
C~ 1times Spitzer s~
if C~1 0., ~ is neoclassical s~ C~2— C~®= sawtooth model with magnetic insulation at q = 1 between sawtooth disruptions (C~2, C,,.~3, C~5);enhanced transport during disruptions (C404, C,1®- C,~®)
C,~2_4®
C~~2~13
DflH+C,~2DHH, XcCse3Xe (only on outermost computation zone which contains a q = 1 surface) when d In Pth/d r < C~5,where Pth n,,T~+ n1T1
C4®
Disruption trigger; sawtooth disruption on when din p,h/dr> C4®
C~5
Disruption off when d In Pth/d r < C~5
C4,®
DHH —e C4®DHH for q <1 when d ln p,8/dr> C4® (unless C4® = 0)
C,~7
DHH —e C~7D~11for q > 1 when d ln p,h/dr> C4® (unless C,~7= 0)
C~8 >
C4® >
C410
1
-. ~ for q <1 when d ln p,h/dr C4® (unless C~8= 0) —e ~ for q >1 when d ln p~h/dr C4® (unless G~9= 0)
Multiplier on electron—ion equilibration rate
C415450
Not used in present version eV/cm3))C452 of BALDUR 16
C451455
x~ C4Sl?i~a~~”(plh/(10 X = R~4”I~’, where Pth =
(n,,7,,v +
n ~ Not used C~
0
No sawteeth may occur before
t=
C 4®
(s)
C~1
0
No sawteeth may occur after
t
C~1
(s)
C~2
(s)
0
Sawtooth period for t C,~5_~7
C~ overrides
C~3
0 (s)
Sawtooth period for given by C462 if I C463, by C~ otherwise
C4®
0
Sawtooth period for i> C463 overrides
(s)
C465_
C.E. Singer ci at No.
Symbol
Code (array size)
C
465467
C468
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
.
351
Description
0
If C462 = 0 = C4®, the sawtooth period is given by C46STrXT Y where x = C4®, y = C467
0
Determines what quantities are mixed by the sawtooth model; if C468 0 all the following are mixed. If C468 is exactly divisible by 2 then electron energy is mixed, by 3 then ion energy, by 5 then hydrogen density, by 7 then impurity density, by 11 then fast ion density, by 13 then alpha particle density (if NTYPE = 1).
C469
Not used in present version of BALDUR
C470_474
0
Xe_ + XTFCD [C472+ (r/r,~,) C473] /C~2
C471
>
0,
+ [C471I/ne[C472 +(r/r,cr)C4h3] /C~2,
C471O,
where C4’~2 = C472 +O.5~473, 2] C47i(rm/I~)exp[($t/$~) 2] by 1 47e.0.5) (replace if C exp[(Pt/~m) 470 <0)
XTFCD /.t(.(C=
2)K”2r~~n
$~
/(2 Rq( r~~)), film = C470(1 + K
I~ = 2.5(1 + K2)amBT /[(R /a)Kq(r~~~)], rm= rear/100, am= r~~i/100, BT = Bz/104, K( ‘-eq. D.237) C 475,476,479_5®
Not used in present version of BALDUR
C47.,
0
C478
0
Plasma model, time step control and section 2.8 neutral gas parameters D.994 F,”~,, * *EXTF 1
D.995
z:~
D.996 D.997
B.64
1,,,+ ~ n~(Z>~] Zett = [(ni + n2)I~, a e,h /[(n 1 + n2)F~,,.+ ~b + ~ ne,
EXTZEF
1
~GNCHEK
0
Not used
GRECYC
1
Fraction of ions or neutral leaving outermost zone which are recycled as neutral gas (cf. eq. C®2)
* * *
GRECYC
2.8d
If C477> 0, then sawtooth mixing occurs when q(0) C477. Overrides sawtooth period specification C462, C4®..467 If C478> 0, the resistivity used in eq. (5.2.2a) is determined by eq. (2.8d). Since ~ * fl~,in this case the B~1energy conservation may be poor
* *
+ (Z~—1.)
G.E. Singer ci al.
352
No.
Symbol
/ One-dimensional plasma transport code
Code
Default
Used in
(array
(logical
eqs no.
size)
value or units) 0
Description
D.998
s * GTCHEK
Not used
D.999 —999a D.999 D.999a
* *
D.1000
*
*GXMAXE
io~ (eV)
Maximum energy for neutral outfiux spectrum ( ‘-eq. D.1067)
D.1001
*
*GXMINE
1 (eV)
Maximum energy for neutral outfiux spectrum (‘-eq. D.1067)
D.1002 —1003 D.1002 D.1003
*
*GXPHI(4)
(deg)
Defines azimuthal angle bins used in computation of neutral outfiux spectrum
GXDOME(2)
D.999—D.1009 control neutral outfiux spectrum calculation not implemented in in the present version of BALDUR. Ratio to ion temperature of minimum (‘-eq. D.999) and maximum ( ‘-eq. D.999a) energies used in least squares temperature calculation based on neutral outfiux spectrum ( ‘-eq. D.1067)
4 10
0 15
D.1004
30
D.1005
60
D.1006 —1009 D.1007 D.1008 D.1009
*
*GXTHET(4)
(deg) 0 15 30 60
D.1010
Defines bins for angle between neutral outfiux detector and normal to the wall, used in computation of neutral outfiux spectrum
* *
NGPART
500
section 2.9.1
Number of particles in Monte Carlo cornputation of each source of neutral gas
D.lOlOa
* *
NGPROF
10
section 2.9.1
Neutral profile every NGPROF time steps
D.1011
* *
NGSPLT
5
section
Number of splitting surfaces for neutral gas (cf. section 2.9.1); must be 10
2.9.1
D.1012
*
*NGXENE
31
D.1013
*
*NGZONE
20
Number of energy bins for neutral outflux spectrum printout (~eq. D.1067); must be 100. Not usable in present version of BALDUR section 2.9.1
A minimum for the number of spatial zones in neutral gas computation; NGZONE>
24
-~
we set NGZONE
=
24; [NOTE: NGZONE> NZONES -. NGZONE < min(NZONES,24) (‘-eq. D.2366)] D.1014
*
*NLGLIM
0 (False)
section 2.9.1
.TRUE. —e Include reflection off limiter; ignore TCOLD (‘- eq. D.2129) and NLGMON (‘-eq. D.1015) for recycling neutrals, and launch these neutrals with a Ma.xwellian energy distribution at the ion temperature of the outer dummy zone. Note that gas puffing neutrals are not affected by NLGLIM
G.E. Singer ci at No.
Symbol
Code (array
size)
D.1015
* *
NLGMON
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
—1 (True)
section 2.9.1
TRUE. -e Monoenergetic, (.FALSE. — Maxwellian) recycled neutrals with energy (temperature) TCOLD and gas puff neutrals with energy (temperature) TCOLDP (‘- eqs. D.2129 and D.2130); NLGLIM (‘-eq. D.1014)
D.1016
*
D.1017
*
*NLGREF
*NLG5PT
353
cf.
0 (False)
section
0 (False)
section 2.9.1
TRUE. —e Sputtering of iron by neutrals crossing out over r = r~~ 1 is computed. [Sputtered iron is added to impurities as an influx analogous to FLIMP (‘-eqs. D.1848—1887) if NIMP(1) = 26 or NIMP(2) = 26 (‘-eqs. D.2254—2255)]
2.9.1
.TRUE. —, Neutrals lose energy due to wall reflection; assumes a stainless steel wall
D.1018
8,,,,,
* *
DELMAX
0.1
6g
Maximum relative change allowed in any zone over one time step without reducing the time step (cf. eq. (6b)]
D.1019
~m~e,
* *
ERRMAX
0.01
6g
9CN
* * THETA
1
section 5.2.1
Maximum allowed extrapolated error in any zone [whenextrapolation is used; cf. eq. 6c and NLEXTR ( ‘- eq. D.1029)] Degree of implicitness in Crank—Nicholson differencing schemes. Fully implicit —e °CN 1
O~,
*
0
section 5.2.4
D.1020 D.1021
D.1022
*THETAP
* *
TSPARE
60
Degree of implicitness for iteration in predictor—corrector scheme. Fully corrected —. = 1 Not used in present version of BALDUR
(s)
D.1023
* *
D.1024
*
D.1025
NATOMC
2
2d’ 2.9.4a
*NBOUND
1
section 3
* *
NITMAX
25
section 6
D.1026
* *
NOUNIT
22
D.1027
*
*NTRANS
1
D.1028
* *
NLDIFF
—1 (True)
2 -~eCoronal equilibrium radiation [26,27], 1—e bremsstrahlung only, impurities assumed fully ionized, 0 —e neither bremsstrahlung nor line radiation. Type of boundary condition. Note that impurity influxes (~— C 2®, eqs. D.1848— 1887, D.2256—2275) require NBOUND = 2 or 3 Maximum number of iterations in one time step. If this maximum is exceeded, run is aborted Unit to read impurity radiation ionization (do not change)
section 2.1
If NTRANS <1 —e all diffusion coefficients are set to zero
appendix A
.FALSE. sets A = B = 0
354 No.
G.E. Singer ci at Symbol
D.1029
Code (array size)
*
D.1030
*
*NLEXTR
*NLRCOM
/ One-dimensionalplasma transport code Default (logical value or units)
Used in eqs no.
Description
TRUE. if extrapolation is to be done
—1
section
(True)
5.2.4 of ref. [33]
—1 (True)
sections 2.9.1—
TRUE. if recombination is to be ineluded
2.9.12
D.1031
* *
D.1032
NLSORC
* *
NLSORD
* *
XFUTZ(10)
—1
appendix
(True)
A
—1 (True)
appendix A
FALSE. sets C = 0 FALSE. sets D
0
D.1033 —1042
x~
D.1033
x
1
0.5
6g
Damper on rate of increase of time step; 0. —e no damper, 1. -+ no increase
D.1034
x2
1.5
6g
Maximum rate of increase of time step
D.1035
x3
2
section 6
Maximum on value of iteration or extrapolation error, which can be used to control time step ( ‘-eq. D.1018)
D.1036
.r4
2
section 6
Maximum on value of relative extrapolation error, which can be used to control time step. [~ eqs. (6.6), D.1018]
D.1037
x5
0.333
section 6
D.1038
x6
0.333
section 6
Time step reduction when negative values of dependent variables have been computed * 0. —. override relative extrapolation error method of decreasing time step, and decrease time step when necessary by x6
D.1039 —1042
x7 — x10
0
Not used
*NBDUMp
0
Not used
NPLOT
10
Every NPLOT time steps generate profile
3
plots. Units to write out information for
Output controls D.1043 D.1044 D.1045
*
* *
* * NRGRAF
Extrapolation and time step control factors
plots vs. radius and 3-D plots (do not change) D.1046
* *
NSEDIT
1
D.1047
* *
NTGRAF
4
D.1048 —1067
NLP
*
*NLPOMT(20)
Every NSEDIT time steps generate small edit appendix F
Unit to write out information for plots vs. time (do not change)
appendix F
Controls which pages are printed in long edits. Note that the neutral gas neutral beam, and alpha printouts are automatically omitted when these packages are not active, regardless of how the .N~LP are set
G.E. Singer et at No.
Symbol
D.1048
NILP
D.1049
N
D.1050
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
355
Description
0 (False) 0 (False)
.TRUE. —e omit page 1 of long edits [1(r), T,(r) profile output] TRUE. —~ omit page 2 of long edits [n,,(r), a =1,2,1,h, output]
Nt”
0 (False)
.TRUE. —e omit page 3 of long edits (energy balance output)
D.1051
N 1-” 4
0 (False)
.TRUE. —e omit page 4 of long edits (output of confinement times, profiles averages, etc.)
D.1052
N —1 (True)
.TRUE. —e omit printout of diffusion coefficients vs. radius
1-” 2
5LI’ D.1053
~
—1 (True)
.TRUE. —e omit printout of zone-centering parameters
D.1054
~
—1 (True)
TRUE. —e omit printout of ripple diffusivities
D.1055
N~-~
—1 (True)
1-”
D.1056
N9
0 (False)
.TRUE. —e omit printout of neutral beam distribution in energy and pitch angle in each zone, and fits to the distribution Not used in present version of BALDUR
D.1057
Nh”
—1 (True)
TRUE. —e omit printout of sawtooth quantities
D.1058
Nh”
0 (False)
.TRUE. —e omit first half of neutrals printout (temperatures, densities, chargeexchange losses)
D.1059
N
0 (False)
.TRUE. —e omit second half of neutrals printout (neutral particle source rates)
1’~”
D.1060
~
—1 (True)
.TRUE. — omit printout of neutral beam energy levels and densities at each energy level
D.1061
Nh”
0 (False)
.TRUE. —e omit profiles (average energy per particles, particle source rate, etc.) of main neutral beam printout
D.1062
Nh”
0 (False)
TRUE. —e omit summary at bottom main neutral beam printout
D.1063
~
—1 (True)
TRUE. —e omit printout
D.1064
N1~.,”
0 (False)
.TRUE. -. omit printout of H(r) for 1/1, 1/2, 1/3 energy components of neutral beams
D.1065
~
0 (False)
.TRUE. —e omit printout of total H(r), charge-exchange rates, etc. (neutral beams)
D.1066
Nh”
0 (False)
.TRUE. —e omit alpha particle or D-D fusion printouts
D.1067
Nh”
-1 (True)
.TRUE. -. omit neutral outfiux spectrum printout. Not usable in present version of BALDUR
of
beam driven current
No.
Symbol
Neutral beam parameters D.1068
Code (array size)
Default (logical value or units)
$$CJBEAM
0
Used in eqs no.
2.9.2.2e
Description
Factor multiplying beam-driven current in poloidal magnetic field equation
D.1069 —1078
Ab
D.1079 —1118
6~,,,
$HABEAM(10)
0
2.9.2.2c
(am)
81b
$$HANGLE(4,10)
0 (deg)
section 2.9.2.1
0
92b
(deg) 0
83b
= 0, we set Ab according to the default for the injected species ( ‘-eqs. B.31—35). Note: the present version of BALDUR requires the same value of A,, for all b Angles defining orientation of the line around which the distribution of beam neutrals has symmetry (called the “beamline” below). 8~,,is the poloidal angle of pivot point (‘-eqs. D.1340—1359); posi-
tive
(deg)
~1l,
is below the midplane (cf. fig. 4
of ref. [6].
0
84b
Mass of beam particles. If HABEAM(b)
82,,
is the angle between
beamline and the midplane of the torus; not °2 of fig. 4 of ref. [6].93b is what is usually called the “injection angle”. 03f, is
(deg)
the angle between the beamline (as projected onto the midplane of the torus) and the major radius vector at the pivot point; 0 is dead perpendicular injection (‘-eqs. D.1340—1359); ef. fig. 3 of ref. [6].84~,is
the angle between the vertical axis of the injector face and a line which lies in the plane of the injector face and is parallel to the torus centerline D.1119 —1128
wr°
D.1129 —1138
h~°”
D.1139 —1148
~
$$HAPER(10)
*
iO~
section
Width [diameter for Nt7~I =1
(cm)
2.9.2.1
D.1391—1400)1 of aperture normal to the beamline at pivot point (‘-eqs. D.1340—
(‘-eq.
IO~
section
(cm)
2.9.2.1
0
section
1359)
$$HAPERV(10)
$$HDIV(10)
*
*
(deg)
2.9.2.1
Height of aperture normal to the beamline at pivot point (‘- eq. D.1340—1351) ignored if N,f°’t= 1 ( ‘-eqs. D.1391— 1400) Beam divergence: probability of each partide 0H
leaving injector face at an angle
( 8v) in the Horizontal (Vertical) direc-
tion from the beamline is D.1149
~v
$$HDIVV(10)
W,,
$$HEBEAM(10)
0 (deg)
—1158
D.1159 —1168
*
0
(keV)
P(8H,8v)~exp([(OH/~H)2 +(9~/~~)2]/2+
section 2.9.2.1
Energy of full energy component of injected neutrals. If W’~,= 0., beam b is never turned on (‘-eqs. 3.1371—1380)
D.1169
h?~
$$HEIGHT(10)
*
—1178
D.1179 —1188 D.1189 —1198
*
$$HFOCL(10)
*
$$HFOCLV(10)
Present version of BALDUR requires b energy components, i.e., only 2 different
*
section
Height of injector face (ignored if ~
(cm)
0
2.9.2.1
1)
1010
section
Horizontal focal length of beam (cf. top
(cm)
2.9.2.1 section 2.9.2.1
view of fig. 2 of ref. [6]) Vertical focal length of beam (cf. side view in fig. 2 of ref. [6])
1010
(cm)
=
6. Present version of BALDUR also requires that there be no more than 6 distinct may be active at any time in the simulation.
Wb
No.
Symbol
Code (array size)
Default (logical value or units)
D.1199 —1228
HJ~R
$$HFRACT(3,10)
*
H~ H~,R
H~
Description
section 2.9.2.1
Current fractions of neutral atoms at full, half and one-third energy. Full energy fraction; Half energy fraction; One-third energy fraction
1 0 0
HII~R
D.1229
Used in eqs no.
$$HFUTZ(10)
1
“Approximation factors” for beam treatment
—1233e D.1229
H
1
1
2.9.2.2a
D.1230
H2
1
2.9.2.2e
D.1231
H3
0
2.9.2.2d
Factor multiplying correction for scattering of fast ions into banana-trapped orbits (for current drive)
D.1232
H4
0
2.9.2.2d
Factor multiplying averaging of parallel velocity Over fast ion orbit (for current drive). If H4 0., there is no averaging
D.1233
H5
1
section 2.9.2.2b
Factor multiplying beam slowing-down rate
D.1233a H6—H10 D.1234 —1243
of beam ions Factor multiplying electron collisionaiity in collisional toroidal electron current driven beams
Not used
$$HIBEAM(10)
**
0 (kA)
section 2.9.2.1
Total atomic current injected by a neutral beam (one atom counts as one electron). If I,,°= 0., beam b is never turned on (‘-eqs. D.1371—1380)
D.1244 —1253
$$HLENTH(10)
**
0 (cm)
section 2.9.2.1
Distance along the beamline from injector face to pivot point ( 4- eqs. D.1340—1359) (cf. figs. 3 and 4 of ref. [6]).
D.1254
$$HNCHEK
0.05
section 2.9.2.1
Maximum fractional change in n,, before recomputing beam deposition profile
D.1255 —1264
$$HPBEAM(10)
0
Not used
D.1265 —1274
$$HPROF(10)
0
Not used
D.1275 —1284
$$HPROFV(10)
0
Not used
D.1285 —1339
$$HR(55)
1
Not used in present version of BALDUR
D.1340 —1349 D.1350 —1359
I,,°
0
Multiplier on rate of charge-exchange loss
**
**
~
$$HRMAJ(10)
**
R~IN
$$HRMIN(10)
**
0 (cm) 0 (cm)
section 2.9.2.1
If HRMAJ(b) 0., we set R,,~= RMA.JOR(1) (‘-eqs. D.2027—2046). The major radius Rg.VO( and the vertical distance Z!VOI of the “pivot point” from the toroidal midplane (cf. side view in fig. 4 of ref. [6]) are defined as 1cos follows: ~1b, ~ = KRI~IN = sin R~’ 8lb +(e— R~°’ eqs. D.237, D.1079— 1118). If
R~IIN
0., we set R~~IN =
RMINOR(1), (‘-eqs. D.2047—2066) *
**
Present version of BALDUR requires b 6. Present version of BALDUR also requires that there be no more than 6 distinct energy components, i.e., only 2 different Wb may be active at any time in the simulation. Present version of BALDUR requires b less than or equal to 6.
358 No.
G.E. Singer eta!. Symbol
D.1360 D.1361
Code (array size)
Default (logical value or units)
$$HTCHEK
$$HTOFF(10)
1OFF
**
—1370 D.1371 —1380
z~
$$HTON(10)
/ One-dimensionalplasma transport code
* *
Used in eqs no.
Description
0.2
section 2.9.2.1
Maximum fractional change in 1’,, before recomputing deposition profile
0.
section 2.9.2.1
Time at which beam b is turned off
(s) 0 (s)
section 2.9.2.1
Time at which beam b is turned on. The species ( ‘-eqs. D.1401—1410), energy Wb (e— eqs. D.1159—1168) and current J,,~ ~—eqs. D.1234—1243) must all be specified in order for beam b to be included; if any of these are omitted, we set i,,°” = iOM so that the beam is never turned on
D.1381 —1390
~
D.1391 —1400
N,~’°’
1
$$HWIDTH(10)
**
$$NHAPER(10)
**
D.1401 —1410
$$NHBEAM(10)
D.1411
$NHE
**
0 (cm) 1 0
10
2.9.2.1 section
Width [diameter if ~ = 1 (‘- eqs. D.1424—1430)] of injector face 1 — circular port, 2 —e rectangular port (cf.
2.9.2.1
eqs. D.1119—1128, D.1129—1138)
section 2.9.2.1
Sets the injected species: If HABEAM(b) = 0. ( ‘- eqs. D.1069—1078), NHBEAM(b) = 0 —e injector b not active (cf. eqs. D.1371—1380); 1—H; —2—’D; —3 —e T; other values not allowed in present version of BALDUR. If HABEAM> 0., NHBEAM = 1— H (0
section
section 2.9.2.2
fast ions. Present version of BALDUR
requires NHE D.1412
$NHMU
D.1413 —1422
D.1423 (a—j) D.1424 —1433 D.1434
**
~
10
10
section 2.9.2
$$NHPRFV(10)
1
section 2.9.2.1
$$NHPROF(10)
1
section 2.9.2
Not used
$$NHSHAP(10)
1
section
1 —e circular injector face, 2 —e rectangular injector face (cf. eqs. D.1169—1178, D.1381—1390)
2.9.2.1
$NHSRC
1
Present version of BALDUR requires b less than or equal to 6.
section 2.9.2
Number of it (cosine of the pitch angle) groups in numerical solution of Fokker—Planck equation for fast ions. Present version of BALDUR requires NHMU 10 NHPRFV(b) = 1 —. beam has a uniform distribution at the injector face. NHPRFV(b)=2-.if N~~~=1 (‘-eqs. D.1424—1433), beam has a triangular distribution of intensity across the injector; otherwise, not used
Not used for input in present version of BALDUR (reset internally by subroutine DPOSIT)
G.E. Singer et al. No.
Symbol
Code (array size)
/
One-dimensional plasma transport code Default (logical value or units)
Used in eqs no.
Description
359
D.1435
$$NIPART
iO~
section 2.9.2.1
Number of Monte Carlo particles used in each computation of beam deposition profile
D.1436
$$NIPROF
10
section 2.9.2.1
Maximum number of major time steps between computation of beam deposition profiles
D.1437
$$NIZONE
20
section 2.9.2.1
Number of zones used in beam deposition profile computation; NIZONE> 25 — we set NIZONE = 25;NIZONE> NZONES -e NIZONE = NZONES (e- eq. D.2366)
Alpha heating model D.1438
$RCWALS
0 (cm)
section 2.9.3.1
Major radius of the toroidal surface which limits a-particle orbits. If not set by input, is set in ALFINI to RMAJS
D.1439
$RDWALS
0 (cm)
section 2.9.3.1
Minor radius of the toroidal surface which limits a-particle orbits (cf. eq. D.1438). If not set by input, is set in ALFINI to RMINS
2
section 2.9.3
Type of a-particle model. 1 —e slow down where created but include first orbit losses; 2 —. slow down on first orbit; 3 — evolve orbit during slowdown; > 3 not allowed in present version of BALDUR. The wall position used in calculating orbit losses in all models is determined by eqs. D.1438—9
2.3i
Poloidal magnetic field at ~
D.14.40
N.~jp~
$NTYPE
Plasma model, timesiep control and neutral gas parameters D.1441 ~ t~) 0 —1460 8 0(t~) $$BPOID(20) (kG) D.1461
Bz
$$BZ
D.1462 —1481 D.1462
I(r~~,t~) ~
$$CURENT(20)
I(r~~,t~)
600
D.1463 —1481
~
(kA) 0 (kA)
D.1482 —1483 D.1484
np,,
$$DENGAO(2)
n~
$DENGA1(2)
—1485
t~) —e I(r~~,i~)
35 (kG)
(‘-eqs. D.2109—2128). If C1~> 0. then B9(r~~i, t~) — 8~(i~~) for f >10 table 1
2.3i
0 3) (cm 0 (cm3)
at time tB
Initial toroidal magnetic field at magnetic axis R(t~).Fort> t~,B~= 1/R (~eqs. D.2027—2046) 3 (‘-eqs. Total D.2104—2128), toroidal current ignoredatiftime BPOID(1) ij > 0. (‘— eq. D.1441). If C 1~> a, I(r~~i, :~)
4a
Initial a; ignored central if DENGAS density of(a, hydrogen 1) > 0 isotope
3b,4a
Density at all times in outer “dummy zone” for hydrogen isotope; ignored if DENGAS (a, 1) >0
G.E. Singer ci at
360 No.
Symbol
Code (array size)
D.1486 —1595
~
D.1596 —1597 D.1598
ni,,
$DENIMO(4)
nL
$DENIM1(4)
Default (logical value or units)
Used in eqs no.
Description
0 3) (cm
section 4
DENGAS(a, j — 1) = initial density of hydrogen isotope a in radial zone j = 2, 3 N + 2 (cf. fig. 1). If n~(r~~ 2, 0) is not specified, we set n~(r~~2, 0) = n~’(r~~1, 0). Boundary value for isotope a at any time is n~’(r,~~3, 0)
0 3) (cm 0 (cm3)
4a
a; Initial ignored central if DENIMP(a, density of impurity 1) > 0 species
4a
Density at all times in outermost radial zone for impurity species a; ignored if DENIMP(a,1)>0
$DENIMP(4,55)
0 (cm3)
section 4
$$DENMON(20)
0 (cm3)
section 3.2
DENIMP(a, j — 1) = initial density of impurity species a in radial zone ‘~, j 2,3 N+2 (cf. fig. 1). [Ifn~,(r~f 2,0) is not specified, we set n ~,( r~ 2’ 0)n (r~+ ~ 0).] Boundary value for species a at any time is ~ 0) Target density for feedback monitoring of n~at t~P(‘-eqs. D.1952—1971), j= 2, 3 20. DENMON(1) is used as a switch. If DENMON(1) 0., n~is held constant at n~’(t~) between times t~/.. 2and ~ j = 2, 3 10. If DENMON(1) = 1., n~ is linearly interpolated from n (~-1) to n(t~.2) between times ij~and t~.1, j = 2, 3 19. Do not set DENMON(1) to any other values
$DENS(55)
0 3) (cm
4section
Tabular input: used orwith FRACTH (‘eqs. D.1888—1889) FRACTI (‘-eqs. D.1890—1891) to specify initial densities which are not determined by eqs. D.1482—1709.
0,
$DENSO
5 x 3) 1013 (cm
4c
place DENS density (‘- eqs. Initial central for D.1730—1784); profiles to reignored if DENS(1) > 0
1,
$DENS1
iO’~3) (cm
4c
fig. 1) at value all times; ignored if DENS(1) Density in outer “dummy zone” >(cf.0 (‘-eq. D.1730)
10—6
section 6
Initial value of time step
0) $DENGAS(2,55)
—1599 D.1600 —1709
~
D.1710 —1729
n(t~
0)
9)
D.1730 —1784
D.1785
D.1786
/ One-dimensional plasma transport code
n
n
D.1787
$DTINIT
(s) D.1788
l~tma,,
$$DTMAX
2X103 (s)
6g
Maximum allowed time step
D.1789
i~:,,,,,,
$$DTMIN
iQ~ (s)
Minimum allowed time step
D.1790 D.1791
Yb
$EBFIT
1
Xb
$EEBFIT
0
section 6 4e 4e
Initial Jc1[1_(r/rwaiiyb]Yb. EBFIT is ignored if NBFIT *0 or if EEBFIT =0 (‘— eq. D.2332)
G.E. Singer et at
/
One-dimensional plasma transport code
361
No.
Symbol
Code (array size)
Default (logical value or units)
Used in eqs no.
Description
D.1792
x~
$EEFIT
3
‘Ic
Replace DENS (e— eqs. D.1730—1784) by (n
1, + (n01
—
n11)[1 _(r/r,,,~i)x]Y)
(~_
eqs. D.1785, 1786, 1799); ignored if DENS(l) >0 or if NNFIT *0 (‘-eq. D.2333) D.1793 —1794
x~
$EEFIT(2)
3
4a
Initial density of hydrogen isotope a is n~= n~ + (n~a— n~)[1 — (r/r~~,)~]’~ (e_ eqs. D.1482—1485 and D.1800—1801); ignored if DENGAS(a, 1) > 0 or if NNHFIT * 0 ( ‘-eqs. D.2334—2335)
D.1795 —1796
x~
$EEIFIT(2)
3
4a
Initial density of impurity species a is n’ = n~,,+(n~,,— n~)[1—(r/r~~i)~]~ (‘-eqs. D.1596—1599 and D.1802—1803); ignored if DENIMP(a, 1) > 0, or NNIFIT * 0 (‘-eqs. D.2336—2337)
D.1797 D.1798
XTa
xi.,
$EETEFT $EETIFT
2 2
4b 4b
Initial temperature profiles are fT11 j=i,e (e— eqs. D.1797—17987, D.1805—1807, D.2207—2208 and D.2249—2250); ignored if NTEFIT *0 or NTIFIT *0, respectively. See D.1792
D.1799
y~
$EFIT
1
‘Ic
see D.1792
D.1800 —1801
y~
$EHFIT(2)
1
4a
see D.1793
D.1802 —1803
y,’
$EIFIT(2)
1
4a
Outer exponents (‘-eqs. D.1795—1796)
D.1804
E10~,~
$$EIONIZ
0.04 (keV)
2.9.lb
Electron loss for each ionization of any neutral hydrogen isotope
~
D.1805
vTC
$ETEFIT
1
4b
Outer exponent (‘-eq. D.1797)
D.1807
y~
$ETIFIT
1
4b
Outer exponent (‘-eq. D.1798)
D.1808 —1847
I,(t~”)
0 2s1) (cm
3b
$$FLGAS(2,20)
Specifies hydrogen a gas puffing influx rate eqs. for neutral isotope a (e— D.2252—2253), linearly interpolated between times ij~ (‘-eqs. D.2229—2248). This is independent of density monitoring (‘-eqs. D.1710—1729) (cf. C 2®)
D.1848 —1887
Fa(tj)
0 2s1) (cm
3b, section
$$FLIMP(4,20)
Specified tral species theaflux across of impurity r = r~ ion or neu1,linearly interpolated between times tj (~— eqs. D.2256—2275) (unless C2®> 0)
2.9.13 D.1888 —1889 D.1888 D.1889 D.1890 —1891
f~ f~” f~ f,~
$FRACTH(2)
$FRACTI(4)
4c 1 0 0
4c
Used in conjunction with DENS, DENSO, DENS1 (‘- eqs. D.1730—1786) to get initial densities of hydrogen isotope im, purities that are not determined by eqs. D.1482—1709
362
G.E. Singer
ci at
/
One-dimensional plasma transport code
No.
Symbol
Code (array size)
Default (logical value or units)
Used in eqs no.
Description
D.1892 —1911
Fm~(tjG)
$$GFLMAX(20)
(cm
section 3.2
Maximum gas puffing rate which can be forced by density monitoring linearly interpolated between times ~ ( e— eqs. D.1952—1971)
D.1912 —1951
f,°(t~)
0
section 3.2
Fraction of hydrogen isotope i (‘-eqs. D.2252—2253), in gas puffing due to density monitoring, linearly interpolated between times tj( e— eqs. D.1952—1971). If we set f?(if)1—f~(t~)
D.1952 —1971
i~
$$GFTIME(20)
i0~ (s)
section 3.2
Times for density monitoring (‘-eqs. D.1710—1729, D.1892—1951, D.1912— 1951 and D.2360). Note that ~ is used as a switch; for t~°> 0, the initial density monitoring target is set equal to the initial line-average (NPUFF = 1) or volumeaverage (NPUFF = 2) electron density (— eq. D.2360)
D.1972 —2026
r]’
$RADIUS(55)
0 (cm)
section
Tabular input for real spatial zone locations (cf. fig. 1); zones determined instead by NRFIT ( ‘-eq. D.2361) if RADIUS(2)
D.2027 —2046 D.2027
R(t~)
R(t~)
130
D.2028 —2046
R(4)
(cm) 0 (cm)
D.2047 —2066 D.2047
r~~ii(t~)
D.2048 —2066
r~~i(i~) —r~~1i(t~)
D.2067 —2086
~
D.2087 —2106
r,,~,iat(t~’)
1016
2s
$$GFRACT(20,2)
$$RMAJOR(20)
$$RMINOR(20)
r~~,(t~)
D.2107
40 (cm) 0 (cm)
table 1
$$RPA(20)
0 (cm)
section 2.9.11
Minimum major radius along the flight path of pellet launched at time i)’ (e— eqs. D.2278—2287)
$$RPELA(20)
0 (cm)
section 2.9.11
Radius of pellet launched at time eqs. D.2278—2287)
appendix F
Generate major printouts every time interval SEDIT
0 (s)
D.2109 —2128
$$SPLOT t]~
Major radius of magnetic axis, linearly interpolated between times if( 4- eqs. D.2131—2150). If C 1~,> 0., R(t$) — f,11,(i~)for j >10 (‘-eqs. D.2119—2128) [33] Minor radius of wall, linearly interpolated between times t~ (e— eqs. D.2131 —2150)
$$SEDIT
D.2108
5.1.1
table 1 2.3i
$~~(i~B)
—R(t~)
1)
$$TBPOID(20)
table I
0 0. (s)
t1” (‘—
Generate plots every time interval SPLOT D.1441
Times for changing plasma current [or for changing variable ripple for j >10 if C146 > 0. (e— eqs. D.2037—2046)]. Note that if the combined total number of tj~plus t~ (‘— eqs. D.2131—2150) exceeds 19, the first 19 unique nonzero times will be used and the rest ignored. The if and t~ always are merged by the code into a single array of dimension 20 whose first location is set to zero
CE. Singer et at No.
Symbol
/
One-dimensional plasma transport code
363
Code (array size)
Default (logical value or units)
Used in eqs no.
Description
D.2129
$$TCOLD
0.003 (key)
section 2.9.1
Temperature of recycled neutral hydrogen [but cf. NLGLIM (‘-eq. D.1014)] and C 198
D.2130
$$TCOLDP
0 (keV)
section 3.2
[If TCOLDP 0, we set TCOLDP the originally prescribed value of TCOLD (‘-eq. D.2129) regardless of how NLGLIM (e— eq. D.1014) and C198 are set.] Energy of gas puffing neutrals
$$TCOMP(20)
0 (s)
D.2027 —2066
Times for changing ~ and R for adiabatic compression. See note at TBPOID (e— eqs. D.2119—2128)
$TE(55)
0 (keV)
section 4
TE(j —1) = initial electron temperature in radial zone ,~, j = 2, 3 N + 2 (cf. fig. 1). [If ~ 0) is not specified, we set 1~(r~~2) = 7~,(r~+1, 0)]
0.2 (keV) 0.01 (keV)
4b 4b
Initially, T,=Tie+(T~~Tie) [1_(r/r~~iyTh]YT~ (‘-eqs. D.1797 and D.1805), ignored if TE(1)> 0. (‘-eq. D.2151)
D.2131 —2150
t~
D.2151 —2206
~
D.2207
T,~
STEO
D.2208
T1,
$TE1
0)
D.2209 —2228
$$TEDIT(20)
0 (s)
appendix F
Major printouts at time steps immediately following times TEDIT
D.2229 —2248
$$TGAS(20)
0 (s)
D.1808 —1847
Times to change specified hydrogen gas influx
4b
Initially,
D.2249
7~
$TI0
0.2
(keV) D.2250
T0
D.2251 D.2252 —2253
N~AS
D.2252 D.2253
N~AS
D.2254
N,MP
7 = Th + (lb — T,1)[1 — (r/rw~i)~~T~])’ni (‘-eqs. D.1798 and D.1807); ignored if TI(1) > 0
$TI1
0.01 (keV)
4b
(4-eq. D.2472)
$$NEDIT
10
appendix F
Generate major printout every NEDIT time steps
section 2
Sets the hydrogen isotopes for species a. If WTGAS(a) = 0. (‘-eqs. D.2328—2329), NGAS = 1—H, NGAS=-2—D, NGAS=-3—T. If WTGAS>0, NGAS=1—H (0
$NGAS(2)
1 0
N2GAS
$NIMP(4)
0
section 2
5-’B, 2—’
10—. Ne,
6—C, 11—.
7-eN,
Na,
8-’O,
9-sF,
12 —‘Mg, 13 —‘Al,
14 —‘Si,
16—eS,
18-. A,
19—. K,
21 —. Sc, 28 —‘Ni,
22 —. Ti,
23 —. V,
24 —.
20
—‘Ca,
Cr, 26 —. Fe,
47—. Ag,
29 —‘Cu, 30—. Zn. 33 —‘As, 36—eKe, 40—.Zr, 41—Nb, 42—.Mo, 45—ellis, 50 —. Sn, 54 —. Xe, 55 —. Cs, 56 —. aa,
64
73 — Ta, 77—. Ir,
37—’Rb.
—‘ Gd,
83 — Bi,
79 -. Au, 80 -s Hg,
86 — Rn, 90 — Th, 92 — U,
others not allowed). Note: it is not necessary to include 4He to get heating from D : T fusion D.2256 —2275
tj
$$TIMP(20)
0 (s)
D.1848 —1887
Times to change specified impurity influx
364
G.E. Singer et at
No.
Symbol
D.2276
t
D.2277 D.2278 —2287
/ One-dimensional plasma transpori
code
Code (array size)
Default (logical value or units)
Used in eqs no.
Description
$TINIT
0 (s)
appendix F
Time before first time step
im~,,
$$TMAX
I (s)
appendix F
Computation is terminated if
iJ’
$$TPELA(20)
0 (s)
section 2.9.11
Times to launch pellets. For j >1, pellet is ignored if ~! 0
$$TPLOT(20)
0 (s) 0 (cm/s)
section 2.9.11
Velocity of pellet launched at time D.2278—2287)
Atomic weight of hydrogen isotope a. If WTGAS = 0, we set A,, according to the default for the species ( ‘-eqs. B.31_35) Atomic weight of impurity a. If WTIMP = 0, we set A,, according to the default for the species (cf. section 7.3 and appendix F of ref. [17])
0
D.2288 —2307
7)
Generate plots at times TPLOT
D.2308 —2327
Vi,eiiet(i
$$VPELA(20)
D.2328 —2329
A,,
$WTGAS(2)
0 (amu)
(throughout)
D.2330 —2331
A,,
$WTIMP(2)
0 (amu)
(throughout)
D.2332
Y,,
$NBFIT
0
t >
4e
x,
$NNFIT
0
‘Ic
If NNFIT
D.2334 —2335
x,,Ft
$NNHFIT(2)
0
4a
(‘-eqs. If NNHFIT(a) D.1793—1794) * 0, we set y,,~
D.2336 —2337
x~
$NNIFIT(2)
0
4a
If NNIFIT(a) * 0, we set y,,’ eqs. D.1795—1796)
D.2338
$$NPEL
1
D.2339
$$NPEL2
1
*
4-
eq. D.1790)
0, we set x1 = NNFIT1(=‘-eq. D.1792) NNHFIT(a)
D.2333
N~°~,(tJ’)
(‘-eqs.
If NBFIT * 0, we set Y 6 = NBFIT (
D.2340 —2359
t]’
NNIFIT(a) (‘-
Not used in present version of BALDUR section
Not used in present version of BALDUR
2.9.11
Hydrogen isotope (4- eqs. D.2252—2253) in solid pellet launched at time t]’ (— eqs. D.2278—2287). If NPELGA(J) does not follow the numbering conventions of NGAS (‘-eq. D.2252) the pellet does not contribute to the hydrogen density
$$NPELGA(20)
D.2360
$$NPUFF
0
section 3.2
Density averaging method used for density mombring. NPUFF = 1—s line average, NPUFF = 2 — volume average. cf. DENMON ~~eqs. D.1710—1729)
D.2361
$NRFIT
1
section 5.1.1
Plasma transport zones equally spaced in minor radius (NRFIT = 1) or volume (NRFIT = 2); ignored if RADIUS(2) 0. (‘-eqs. D.1972—2028)
D.2362
$$NSKIP
1
appendix F
Printout information on every NSKIPth zone in major printout =
NTEFIT (‘-eq.
YT =
NTIFIT (s— eq.
D.2363
Yr.
$NTEFIT
0
4b
If NTEFIT * 0, we set Yr. D.1805)
D.2364
Yr~
$NTIFIT
0
4b
If NTIFIT * 0, we set D.1807)
$$N’VFY
0
appendix F
Print short summary to teletype every N’TTY time steps, N’ITY = 0 —e suppress teletype output
fig. 1
Number of real spatial zones in plasma transport calculation; maximum = 50, section 5.1.1
D.2365 D.2366
N
$NZONES
20
G.E. Singer ci at No.
Symbol
Code (array size)
/
Default (logical value or units)
One-dimensional plasma transport code Used in eqs no.
365
Description
D.2367 —2416
$$NLXXX(50)
0
D.2417
$$NHSKIP
2
section 2.9.2; appendix F
Printout information on every NHSKIPth zone in beam printout
NFUSN
0
section 2.9.3
NFUSN = 1—s always include D : T fusion model. NFUSN * 1—’ include D : T fusion model unless D has been specified as a hydrogen isotope ( ‘-eqs. 3He as an impurity (‘-eqs. D.2252—2253) and D.2254—2255), in which case use D: D fusion model described in section 2.9.3.2
D.2419
$$NPELOU
1
section 2.9.11
Allow pellet printouts when every NPELOUth pdlet cluster is injected
D.2420
$$GWMIN
section 2.9.1
Minimum weight below which particles are no longer followed in Monte Carlo neutral algorithm
D.2421
*
section
Relative power to each lower hybrid mode; cf. n
2.9.6
(‘-eqs. D.439—488)
D.2418
*
RLPOWR(50)
1.E—6 0
Dummy input switches (not used)
11 —2470 D.2471 —2526
T~(i~, 0)
D.2527
f,’
D.2528 D.2529 D.2530
*
*TI(55)
$AFSLOW * *
C~’~ G~,,
NLGPIN
$CPVELC $CPVION
0 (key) 1 —1 (True)
Factor multiplying alpha particle slowdown D : T fusion model
section 2.9.1
.TRUE. -s include proton impact ionization in Monte Carlo neutral calculation If > 0 — multiply by T~,(for CPVELC) or l’~(for CPVION) to get energy per particle carried by convective terms in energy transport equation
section 2.1
D.2531
$$BZSTAR
D.2532
$$ELECDO
D.2533
$$ELECTE
1.0 (key)
D.2534
$$APRESR
8x1013 (keV/cm3)
D.2535
$$RASTAR
~
D.2536
$$QSTAR *
*DFUTZV
12 (kG) 13 (cm 4X103)
Normalization constants for semiempirical model B~
x40 (cm)
r*
2.2
q~ must be input through CFUTZ(377); before version BALDP22M
0
2.7b
0
2.1H
(6,25) * *DFUTZD (6,25)
rate in
section 2.9.3
1.5 1.5
D.2531 —2535
D.2537 —2616 D.2617 —2696
TI(j —1) = initial ion temperature in radial zone i~, j=2,3 N+2(cf. fig. 1). [If Tj(r,~~2, 0) is not specified, we set T,(r~~2, 0) = T1(r~~,÷1,0)]
eD,,,,
366 No.
C.E. Singer ci at Symbol
Code (array size)
Default (logical value or units)
/
One-dimensional plasma transport code Used in eqs no.
Description
D.2697 —2776 D.2777 —2856
* *DFUTZE (6,25) * *DFUTZI (6,25)
0
* *
0
2.11
e~,,k
D.2857 —2880
* *VXSEMI (6,6)
0
2.7b
v,~,a=1,2,t’,h,i,e
D.2881 —2904
* *DXSEMI (6,6)
0
2.1H
D,7,,, a =1,2,c”,h
D.2905 —2908
*
*XE5EMI(6)
0
2.11
x,~,
D.2909 —2912
*
*XI5EMI(6)
0
2.11
x~
*QMHD1
0
2.1J
MHD correction to 1X
QMHD2
0
2.1J
Singularity avoidance parameter for I; must also be input through CFUTZ (379) before version BALDP22M
*RRSTAR
140
2.1J
Normalization constant c~= r~/R~(‘-eq. D.2535). (Version BALDP21M will not read this variable in NAMELIST, so default values must be used)
D.2913
QMND1
*
D.2914
QMND2
* *
D.2915
R~
*
D.2916
* *
MEMPRT
0
2.1!
ee,k
91; must be input through CFUTZ (378) before version BALDP22M
Specifies term for which D~” and v~j,~” 5for first is printed term out. Default value gives v~”,D~”
Appendix E. Subroutines The main branches of the program are described. This list does not include OLYMPUS [7] and BALDUR utilities, dummy EXPERT calls, error sequences, nor the details of the neutral gas package, coronal radiation package, neutral beam package, alpha-particle package, pellet injection package or output routines. The order of events within routines is not always exact; e.g., the call to IMPRAD(1) by AUXVAL occurs somewhere in the middle of the routine, not at the end. Outer calling structure Program BALD UR 1. Set up I/O units (except for ripple input data file). 2. Call BLOCKDTA Store fundamental constants. 3. Call MASTER 1. Call BASIC Initialize OLYMPUS variables. Call MODIFY, which reads input deck header cord used for restart switches (not implemented in present revision of BALDUR). 2. Print date and time.
G.E. Singer et at
3. Call
/
One-dimensional plasma transport code
367
COTROL 1. Call LABR UN Store BALDUR version number. Read runs labels from input deck. Work revision number, labels, “PROGRAM BALDUR”. 2. Call CLEAR Clear common blocks. Includes calls to ACLEAR, GCLEAR, HCLEAR, ZCLEAR. 3. If run is not a restart Call PRESET Set default values for NAMELIST input variables (‘~—appendix D). 4. If run is a restart Call RESUME Restart capability is not implemented in present version of BALDUR. Do not use this branch. 5. Call DATA Read and write NAMELIST input. 6. Call AUXVAL 1. Call ERRCHK Check for obvious mistakes in input deck. 2. Call UNITS Computer units conversion factors.
3.
Set auxiliary values based on input data. This includes code flags, the radial grid, time dependent conditions, plasma conditions variables not defined in INITIAL or START, etc. (*— section 4). 4. If scrape-off model is on, output sheath limited current densities section 2.8). 5. Call IMPRAD(1) 1. Initialize neutral impurity influx parameters (*— section 2.9.13). 2. If scrape-off model is on Call PDX Initialize and compute scrape-off losses (~- section 2.9.5). 3. Initialize coronal radiation package. Read atomic data file FOR22 (‘c— section 7.3). 6. Call BEAMS(1) Initialize neutral beam package (+— section 7. If run is not a restart Call INITAL 1. Call IMPRAD(2) 1. If scrapeoff model is on Call PDX Initialize and compute scrapeoff losses (4— section (~-
2.9.2).
2.9.5).
Compute initial radiation terms, impurity charge states. Call coronal radiation package if selected (4— section 2.9.4). 2. Define physical initial conditions for x (4— section 5.2.3), B9. 2.
368
G.E. Singer ci at
/ One-dimensional plasma transport
code
Note that the electron density is redefined in START section 4). START 1. Set initial ~t, boundary conditions, volume integrals. Unlock neutral impurity influx flag. 2. Call GETCHI(2) 1. Call IMPRAD(2) Same as above, plus compute neutral impurity and He influxes. Includes call to PDX. 2. Compute boundary-centered densities and temperatures, zone-centered B 9 and J section 5.2.2). (~—
8.
Call
(~-
9.
10. 11.
12. 13.
14.
15.
Compute Z-effective, electron density. 3. Initialize packages for remaining source terms: call HEAT(1) (lower hybrid and input profile heating), ECRH(1) (electron cyclotron resonance heating), and either ALFINI(alphas) or HE3(1) (D—D fusion) as selected by input. Call OUTPUT(1) Initial alphanumeric output (call MPRINT, GPRINT, HPRINT, IPRINT, APRINT or FPRINT, RPRINT, SPRINT). Initial graphics output (call TGRAF(1), GRAFIX(1)). Call STEPON Do a time step of main computations. See ext. chart. Call OUTPUT(2) Control output flags. At times or time steps selected by input, do main alphanumeric output (call MPRINT, NCFPRT, GPRINT, HETPRT, HPRINT, IPRINT and either APRINT, FPRINT, as selected by input), short alphanumeric output (call SPRING, GPRINT), and/or graphics output (call TGRAF(2), GRAFIX(2)). CALL TSEND Test for completion of run. If run is not yet completed, do next time step: go to ~10. Call OUTPUT(3) Do main alphanumeric output for final time step if this has not been done already (same calls as in * 11). Do final graphics output and close graphics (call TGRAF(3), GRAFIX(3). Call ENDRUN Terminate run. Print final message, STOP.
Inner calling structure 10. Call
STEPON 1. Do short teletype output at appropriate time steps. Zero the iteration counter. CALL SAWMIX 2. Call RESOLV(1) Initialize flags and store values for predictor—corrector, extrapolation. 3. Unlock neutral impurity influx flag. 4. Call COEF
G.E. Singer ci at
/
One-dimensional plasma transport code
369
1. Call GETCHI(2) 1. Call IMPRAD(2) 1. If scrapeoff model is on, Call PDX Compute scrapeoff losses Call DIVER. 2. Compute radiation terms, impurity (Z> and
5. Call 6. Call
Compute sources due to neutral beam injection. This includes Fokker—Planck calculation of beam fast ion distribution, and every so often includes call to Monte Carlo computation of beam deposition profile. 6. Call HEAT(2) Compute sources due to lower hybrid heating or input profile heating. 7. Call ECRH(J) Compute sources due to electron cyclotron resonance heating. 8. If D—T fusion has been selected by input, Call ALPHAS(2) Compute sources due to D—T fusion. This includes calculation of alpha-particle distribution, and includes call to Monte Carlo computation of alpha-particle orbits. 9. If D—D fusion has been selected by input, Call HE3(2) Compute sources due to D—D fusion. 10. Call CONVRT 1 .If nearly exact neoclassical transport model has been selected, compute transport coefficients for this model. This includes call to NCFLUX. 2.Compute A, B, C, D (4— appendix A). 11. Call CNVCOF Convert A, B, C, D (f— appendix A) from standard to interval units. SOLVEB Solve B9 equation. Compute source due to ohmic heating. BOUNDS Compute boundary condition coefficients.
370
G.E. Singer ci at
/ One-dimensional plasma transport
code
7. Call
RED UCE Compute P, 0, R, S (4— eq. 5.2.3g). Reduce equations to first order. 8. Call SOLVE Solve transport equations for new x9. Call RESOLV Time-step control. Set next ~t. Control flags and store values for predictor—corrector. extrapolation, time step repetition. 10. Increment iteration counter. Check that maximum number of iterations has not been exceeded. 11. If time step is to be repeated, or the predictor portion of a predictor—corrector time step has just finished, or the first i~t/2minor time step of an extrapolation time step has just finished, go to #4. 12. If the t~tminor time step of an extrapolation time step has just finished, go to #5. 13. Call CMPRES 1. Compress the plasma: adjust x~ B9 radial grid. 2. Call other routines which may need to compress: BEAMS(3), HEAT(3) and either ALPHAS(3) or HE3(3) as selected by input. 14. Call
PDRIVE Compute sources due to pellet injection.
15. Call
GETCHI(1) Update quantities based on x: densities in standard units, ion and electron temperatures. Also update B9, B~ in standard units. Note that electron density (but not electron temperature) is recomputed in GETCHI(2). 16. Lock neutral impurity influx flag. 17. Call GETCHI(2) Almost same as COEF*1, but new x values were used, and there is no neutral impurity or He influx computation when IMPRAD(2) is called. 18. Advance the time. Time step is completed. The calling sequence and nesting level for the major plasma source routines are given below. Neutral gas fueling 5. NEUGAS 6. MONTE 7. GCLEAR 7. SETVAR 7. SPLITS 7. MCARLO 8. XSECT 8. LOCATE 9. VELOC 9. REFLEC 8. VSOURC 8. VELOC 8. FOLLOW 9. TIME! 9. VELOC
G.E. Singer ci at
9. ESCAPE 10. SPUTER 10. REFLEC 6. MATRX1 6. MATRX2 Neutral beams 5. BEAMS 6. DPOSIT 7. ICLEAR 7. HRSET 7. HRCAL 8. HRSET 8. CXSNEU 8. INJECT 9. SORSPT 9. ROTATE 8. ROTATE 6. ENDRUN 7. DAYTIM 8. DATE Alpha particles 5. ALPHAS
6. ALFEND 7. ENDRUN 8. DAYTIM
9. DATE 9. CLOCK 8. RUNTIM 9. SECOND 6. ALFSET 7. ALFEND 8. ENDRUN 9. DAYTIM 10. DATE 9. RUNTIM 10. SECOND 7. XINTER 8. ALFEND 9. ENDRUN 10. DAYTIM 11. DATE 11. CLOCK 10. RUNTIM 11. SECOND
/
One-dimensional plasma transport code
371
372
cE. Singer ci al.
/
One-dimensional plasma transport code
6. BOUNCE 6. ARLPUT 7. ALFEND 8. ENDRUN 9. DAYTIM 10. DATE 10. CLOCK 9. RUNTIM 10. SECOND 7. SQUEZE 8. ALFEND 9. ENDRUN 10. DAYTIM 11. DATE 11. CLOCK 10. RUNTIM 11. SECOND Pellet
4.
injection
[46]
PDRIVE 5. PELLET 6. ZINCON 6. INICON 6. PELRK4 7. PELRAT 8. PELABL 9. PELQE 9. PELQF
Appendix F. Sample nm Sample BALDUR run A number of test cases have been developed to benchmark the BALDUR code [36,52]. These include analytic test case comparisons for the energy, particle and poloidal magnetic field diffusion equations as well as machine specific test cases. The test cases simulate a variety of tokamak scenarios and are used to validate new versions of the code. They have been documented in refs. [36,52] and may be obtained from the MFECC by “FILEM read 4776 .sbvtst all”, see ref. [52] (this issue). Our sample BALDUR run is the neutral beam and compression heated TFTR test case from this series. The input deck for our sample run is shown in fig. 2. The top line of zeros begins in column 1. It is a holdover from old BALDUR versions which were restartable. The next four lines are comments used to label the run. The comments must begin in column 2 and may contain up to 48 characters per line. Below that are the two NAMELISTS, NURUN1 and NURUN2. NAMELIST input variables should be entered in the format shown. Column 1 should be blank. The end of each NAMELIST is indicated by a $END line. Appendix D described all the input variables for BALDUR. Here we call attention to several variables
C. E. Singer ci at
/
One-dimensional plasma transport code
Di
In
(0
N
373
— (0~
0(0-
V
F.
Di
W)r)
(0 II
I’)V ~I
—
C.C
0
(0
*-~—
.-4)
(‘5— (0.c—... N (0
N’
(0F..-
•—~.—* V .(0 (0
U)li(0
fl—fl .-VN
(0*..(0..-(0(0
•—.4.
I —.(0ON’-’(0
N
II — N (0(0(0 . ‘-.C—lI N -N
—
N
I —~
(0 N
Inlt)
In it) *
(0
3
(.)
o
‘.—‘~‘—‘
0
—
333 5- 5.. ~ .
(0
*0
0 A N-—
—0Vfl C 01 V V VCV
1*
II N
. .,.0).’..s (0ON—In(0 ii II ‘“I’) I’) —‘(0 N ‘...S-.’ N (0 — — N N ‘3-.-~.’ NNN’.-33 ...‘—Us-’.3 3 3 0 0 “-‘-“0 0 0 I’) (‘5
Di
N>SVC
‘4 0 V 0 (00.0 0 ‘- N — 01NV
II
(0 ‘-‘
(0
3—...CI-..C
—
3(0 5- *
0(0 * •.(0N —
N .-
1.—N .~‘5
(0
.-In* ut)*s.I 5-.’ N 5-.’ N — N-..’ 3 “3’.3’.-0
s-c) 0
(0 — *14) N (0 (0..
5- ‘I- 5- 5.- 5.-’.0000000000
5-5-
‘I- 5-
.4
-I’)
* * (0 It) InN “5* .4..
.
0
11
3
.
-
It) (0 N 11
** . (‘1
11 C.’.-
(0
I-’.-
0
~
-
.
0) (0
0)
In
*1’)
*In
.
O~VV LVCL(0
-
11(0
(0
C 0..’
. (‘5
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/ One-dimensional plasma transport
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383
not referenced elsewhere. NRUN (t-.- eq. D.12) and TMAX (4— eq. D.2277) are used to control the duration of the run. TINIT (4— eq. D.2276) sets the time at the beginning of the run. NLOMT1, NLOMT2 and NLOMT3 (i’- eqs. D.86—235) can be used to omit various packages and subroutines. The frequency of printed output is controlled by NEDIT, NSEDIT, SEDIT and TEDIT (t— eqs. D2251, D.1046, D.2107 and D.2209—2228, respectively). NLPOMT (i’- eq. D.1048—1067) is used to omit certain output pages. NSKIP (4— eq. D.2362) and NHSKIP (.t_ eq. D.2417) control the number of radial zones printed out. NPELOU (4— eq. D.2419), CFUTZ(63), CFUTZ(70), CFUTZ(214) and CFUTZ(253) are special output controls; consult appendix D for details. NTTY (t— eq. D.2365) controls output to the teletype. NPLOT, SPLOT and TPLOT (-*-- eqs. D.1044, D.2108 and D.2288—2307, respectively) control the frequency of graphics output produced by BALDUR’s companion plotting codes. A number of variables associated with restarting capabilities are documented to facilitate future recovery of this capability. The printed output from the run is extensive. We reproduce it for one time step in fig.2. Explanatory comments follow. The printout begins with some OLYMPUS [7] printout followed by run labeling comments and the values of all input variables, whether or not these were set explicitly in the input deck. The array CFUTZ is also printed out in ten-colunm format for easy reading. In runs which include a scrapeoff model, a printout of sheath-limited current densities will appear immediately below the second NAMELIST. F. 3.1. First page The next few pages describe some of the major models used in the run and are self-explanatory. The zeroth time step printout gives a detailed breakdown of the initial conditions. A major printout is always generated for the first and for the last time step, and for intervening time steps as specified by input. Notice that the run label from the input deck and the BALDUR version label are printed out at the top of each major printout page. The quantities printed out on the first page are as follows: zone: radius: te: reg: ti:
j 1, where j fig. 1) is the BALDUR zone index radius of zone center rZ electron temperature, 1~( ,~z) two-letter mnemonic for dominant term affecting electron energy transport ion temperature, T1 (,~) —
(+-
reg: ne:
two-letter mnemonic for dominant term affecting ion energy transport electron density, n e ( FZ) ni: thermal ion density, n1(~)[4— eq. (2e)] ez: toroidal electric field at jz: toroidal current density, J(rZ) safety factor, q(i~) total beta: local thermal toroidal beta ~ ( Ee + E1 )/( B~/8’n) The total at the bottom of the page are the total electron and ion thermal energy in the plasma, the total numbers of thermal electrons and ions in the plasma, and the total plasma current. These totals are computed as volume-weighted sums over all physical zone. F. 3.2. Second page The quantities printed out on the second page are as follows: zone: j 1, where j fig. 1) is the BALDUR zone index radius: radius of zone center nu-el: electron collision frequency (t— eq. B75) —
(—
384
G.E. Singer et at
/
One-dimensional plasma transport code
nu-h: ion densities:
hydrogen ion collision frequency (~ eq. n0(i~)for each hydrogen and impurity species
zeff:
Zeff(I~jZ) (#—
B.74)
a
eq. B.64)
F. 3.3. Third page zone: j 1, where j (.1— fig. 1) is the BALDUR zone index radius: radius of outer boundary of BALDUR zone j. ~ 4-if 2R ~~I.iXe~ b electron conduct: —
(F.3.3a)
—
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4R~2R~i{qe+ Xe~~ r r=rj*,b
—
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b
r
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(F.3.3d)
neutral losses:
—
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4~2RJ~*1(Q~)+ QHE+ Q1MP+
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4~2Rj’)+IQoHr
alpha heating:
4if2RfrJ+I(QF
+
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+
+
4)if2RfrJ*1(QN
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dr
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dr
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4-ii
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q1)
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e—i coupling:
+ QEC
4)if2RjrJ*IQIEr
—
QN
—
QREC
—
QN
—
QRAD
—
QHe
—
(F.3.3i)
Q~
dr.
The following symbols used above have been defined elsewhere: Xe(4- eq. 2.6d), q~(4—eq. 2h), 2.6c), q 1(~—eq. 2h), Q~(—eq. 2.9.lc), Q~~(’— eq. 2.9.12b), Q)(~~~ eq. 2.9.4a), QHE(4- eq. 2.9.lOb), QIMP(4- eq. 2.9.13d), Q~(—eq. 2.9Jb), QOH(4- eq. 29.8a), Q~(—eq. 29.3.lh), Q~(~ eq. 2.9.3.li), Q~(~— eq. 2.92.2k), Qr(~—eq. 29.2.2e’), QLH(+... eq. 29.6a), Q~’~’(+— eq. 2.9.6b), Q~(~ eq. 2.9.7a), QEc(4- eq. 2.9.7b), QIE(4- eq. 2.9.9.a).
x~(— eq.
F. 3.4. Fourth and fifth pages The fourth and fifth pages list the transport coefficients calculated at the outer boundary of each physical zone. The first two columns of each page list the zone numbers and the radii of the outer boundaries of the zones at which the transport coefficients are calculated. On page four: k-e tot!: total electron thermal diffusivity k-i totl: total ion thermal diffusivity
CE. Singer et at / One-dimensional plasma transport code
385
neoclassical Ware pinch on ions v~1e: neoclassical Ware pinch on electrons d-h tot!: tota! diffusion coefficients for hydrogen ions by species d-i tot!: total diffusion coefficients for impurity ions by species
v~iate:
On page five are printed the semiempirical transport coefficients specified through the DFUTZE, XESEMI, etc., variables Xe(1)... xe(5): 1st five terms in the semiempirical electron thermal diffusivity x1(1): 1st term in the semiempirical ion thermal diffusivity
v (0):
total semiempirical inward pinch total semiempirical diffusion coefficient
d(0):
Setting input variable MEMPRT = i, (i = 1,... ,6), causes the ith term in the semiempirical pinch and diffusion coefficients to be listed in the last two columns. Setting MEMPRT — 0 causes the total semiempirical pinch and diffusion coefficients to be listed. F. 3.5. Sixth page The confinement times printed at the top of the sixth page are computed for each species, for the electron and ion energy, and for the total energy: particles:
fonardr
a=1,2,e,h,
electron energy:
f~E0r’dr’ rqe+J~(Q~”+Q~+QIMP+QN)r~dr/
ion energy:
f~E~r’ dr’ , rq1+J~ç(Q~+ QREC)r~drF fr(E
total energy:
r(q~
n-tau is computed as:
+
-
q1)
+ Jr(QRAD
(F.3.5a) (F.3.5b)
,
(F.3.5c) +
E.)r’ dr’
+Q~+ QIMP + Q~+ Q~+ QREC),.~~dr’’
(total energy confinement time).
(F.3.5d) (F.3.5e)
In these expression, r is the minor radius at which the confinement time is computed. The four values of r are chosen so as to be roughly evenly spaced over the range 0
f(EJr)SrQoHr~ + E1)r’ dr’dr’
(F.3.50 J”s~r(E+E+nB)r’
total:
Jr~or(QOH+QF+
Q.~+Q~+Q~+
QLH~
dr’ QLH~
QEC+
The following symbols are defined elsewhere: Q~(4- eq. 2.9.8a), 2.9.3.li), eq. 2.9.2.2k), eq. 2.9.2.21), Q~(-—eq. 2.9.6.a), 2.9.7a), QEC(4- eq. 2.9.7b). The mean (volume-averaged) densities are: Q~(~
=
~f~’°’ti~~ dr,
Q~(~
j = e,i.
.
QEC)r~dr Q~(I—
Q~’(---
eq. 2.9.3.lh), eq. 2.9.6b),
(F.3.5g)
Q~(~ eq. Q~(+—
eq.
(F.3.5h)
386
G.E. Singer et al.
/
One-dimensional plasma transport code
The mean (volume-averaged) temperatures are: = SCSEIr dr/ Th 1r dr, j = e,i.
f
j
(F.3.5i)
The line-average electron density is:
~ —f 1
r~~ii
~
o
‘~edr.
(F.3.5j)
The betas are computed as follows: .~
electron:
/~e =
ion:
f3X =
(F.3.5k)
,
[B~/8-if]
beam ion: /3~=
alpha:
(2/3)[(2/rs~r)Jo”~Eerdr]
~
=
(2/3)[(2/ç~r)f~.’rEjrdr] [B~/8~] (2/3)[(2/r2
)JrscrnBr
scr
(2/3)[(2/
(F.3.51)
drj
(F.3.5m)
,
r)Jø~’.’flEa’
dr]
(F.3.5n)
,
[B~/8~I
total:
(F.3.5o)
~
Here x stands for 0 (poloidal) or z (toroidal). For poloidal-beta, B9 is evaluated at r expressions are defined elsewhere: n~ (~ eq. F.3.7e), nE0(— eq. F.3.9b). The loop voltage is:
=
~scr
The following
(F.3.5p)
21rRq~(J—J~~0)~) Ir=r.-,’
where ~h is the resistivity (+— eq. 2.8a), J is the toroidal current density (4— eq. B.63), and beam-driven current (— eq. 2.9.2.2e). The rates of neutron production by D—D reactions are as follows (neutrons/second): thermonuclear:
NDDn
beam-plasma:
NDDn,
total:
ND~ = NDDfl +
thermonuclear:
N~Dfl=
beam-plasma: total:
N~Dfl=
j N~, dt’, t NI~Dfldt’, j
NDDfl
N~Dfl+ N~Dfl.
=
=
~beam
is the
(F.3.5q)
~fl~(aV>DDfl,
(F.3.5r)
“~DDn’
(F.3.5s)
(F.3.5t) (F.3.5u)
D is the density of deuterium. ~ ) DDn (-f-- eq. B.220) is the thermonuclear D(D, n)3 He reactivity. N~Dfl is defined in appendix B (— eq. B.320). The upper limit of integration, t, is the time at which the printout occurs. The lower limit of integration, timt’ is the time at the start of the run (+— eq. D.2276). Qtech is defined as:
Qt~h=
~b
[i— (total gain)
I
r=r~/(ot~
heating)
I
,~,.-,]
(F.3.5v)
G.E. Singer et at
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One-dimensional plasma transport code
387
where Pf is the fusion power (+- eq. B.322), ~b is the input beam heating power (4-. eq. B.321), and the total gain (~ eq. F.3.3i) and other heating (~— eq. F.3.3h) are as defined for the third output page. Q~~h is defined and printed only when neutral beam heating is active. Note that for most cases of interest, “other heating” reduces to the absorbed beam power. The maximum change over one time step describes what is limiting the time step. It prints out ô~ (1— eq. 6b) in %, prints the physical quantity (density, energy density or B 9) which exhibited the maximum change in eq. 6b, and prints the number of the zone in which change was greatest. The conservation checks are (c.c. compression correction): for hydrogen species:
f~”[n~ — f,~,,(8n0/at’ S~)dt’] r dr + (c.c.)a
=
—
—
1,
a
=
1,2,
(F.3.5w)
,rwall
J
tla1~~1
0
for impurity species: (0na/8t’) dt’}rdr+(c.c.)a
t
jrw.u[fl a
CaCOflS =
J~
‘.‘
r
—
E1~~‘11~~a’ dr]
a
1,
=
(F.3.5x)
~
~
for electron and ion energy: =
c~cons
—
frW.’II
f
[(Ee + E~)—
~
aE
+
[(aEe/0t’
1/at’ [J~w5II(Ee+Ei)r
—
—
Q)
dt’] r dr + (c.c.)e
+
drj t~tmi)
(c.c.)~ —1,
(F.3 .5y) for B-poloidal: I
r
(ITR/2)f
=
“>
II
B~(r)rdr
0 ~ —
+ ~=InIt
f
c’rrRaEz(r = a)B9(r
=
1) dt
tm,
Q~Hrdr} dr + c.c.s}/[(~R/2)f0
rWII
2(r)r drj
—
1.
(F.3.5z)
B0
In these expressions, t is the time at which the printout occurs. The numerators represent values at the present time and integrals up to the present, while the denominators are taken at the start of the run, t,~ (+—eq. D.2276). The following expressions used here are defined elsewhere: (3fla/8t) (~eq.2a), S (~eq. 2.9.lla), (aE~/at),j= e,i (~eq.2b), Q~(~eq. 2.9.llb). If zones are added or removed during compression, then the compression corrections above account for the particles and energy which are added or removed. Compression heating and poloidal field energy changes are also included in the compression corrections.
The particle and energy conservation checks are generally of the order of 10-10 or less. The B-poloidal check is generally of the order of iO~or less. During neutral impurity injection (~— section 2.9.13), the conservation check value for the injected species may be somewhat higher. In cases where maximum values of density and/or temperature are set in the scrapeoff (+— C 129—C131), the particle and energy conservation checks are bypassed.
G.E. Singer et al. / One-dimensional plasma transport code
388
F 3.6. Scrapeoff summaty When a scrapeoff layer is included in the numerical model, page 7 of each large edit consists of a list of special quantities in the scrapeoff region: power (kW):
—
~
2j~f waIIQs
+
dr,
Q~)r
(F.3.6a)
the total power flowing to the divertor or limiter through the scrapeoff. ion fluxes into divertor (per second):
—
4if2Rf
maCSr dr,
(F.3.6b)
for each hydrogen and impurity ion. total ion flux into the divertor (per second): total of the ions in (F.3.6b). ion fluxes into the scrapeoff (per second): for each hydrogen and impurity ion. ion fluxes onto the wall (per second):
2RF~crFa(i~cr),
4TT
4¶R2rwall Fa ( rwaii),
(F.3.6c)
(F. 3.6d)
for each hydrogen and impurity ion. line den (ions/cm2):
f~’~’~e(1) dr.
separatrix radius:
r~
(F.3.6e)
(F.3.6f)
1.
zone:
j—
1,
r (cm):
rz
zone centered radius.
power (kW):
—
where j is the BALDUR zone index.
45rT2Rf‘~*1(QS +
ions/second:
—
4~2RJrJ+I~Ssr
the total ion flux for zone
(F.3.6h)
Q~)rdr,
the power following to the divertor or limiter in zone
(F.3.6g)
(F.3.6i)
j.
dr,
(F.3.6j)
j.
T~(eV):
7,(I~),
electron temperature.
(F.3.6k)
T~(eV):
Ti(rjz),
ion temperature.
(F.3.61)
ne:
n0(r~z), electron density,
(F.3.6m)
~‘~a(’~)’
a
total ion density.
(F.3.6n)
The remaining quantities are computed from the divertor model of section 2.9.5 and are evaluated at zone centers. ul (cm/s):
plasma flow velocity into the divertor.
(F.3.6o)
amachi:
the match number of plasma flow into the divertor.
(F.3.6p)
tau2:
divertor plasma opacity to returning neutrals.
(F.3.6q)
t2 (eV):
plasma temperature at the divertor plate.
(F.3.6r)
d2:
plasma density at the divertor plate.
(F.3.6s)
G.E. Singer et at
/ One-dimensional plasma transport code
389
F. 3.7. Neutral beam fast ion distribution summary The characteristics of the neutral-beam-injected fast-ion distribution are written by subroutine HPRINT. The definitions of the output quantities are given below. zone:
j
radius:
zone-centered radius,
—
1;
j is the BALDUR zone index.
where
,~.
average energy/particle:
parallel: local average of the parallel energy per particle Er(r)
=
fE~~ dEf
f~(ii,
E, r)~2d~i/n5,
(F.3.7a)
total: local average of the total energy per particle (see B.201) EB(r)
=
fE’~’,’~dEf
~
E,
r) d~/n8.
(F.3.7b)
taus: thermalization time; i.e., the time to slow from the local maximum fast-ion energy, Em~,to the local minimum fast in energy, E~ (see B.194) =
(3v~ 0)’log[(7~1+
1~)/(y~ + 1
v~0E1
v~eE~)j.
(F.3.7c)
beam dens: local density of fast ions (see B.195) 8(r) = dEff(~t,E, r) d~t, n
(F.3.7d)
j~m
ener dens: local energy density of fast ions (see B.202) n~(r)
(F.3.7e)
=nB(r)EB(r).
cos pitch ang: local average of ~ = v
mean:
1/v (see B.203) ,LB(r)
=
f~dEf
8, f(ti,
E,
(F.3.7f)
r)it d~/n
std.dev.: local standard deviation of the distribution in p~(see B.204) 1/2
I~E
a(r)=~f
2
f(ji,E,r)(~L—~) d~t/n~
mdEI
I~JEmi,
-
(F.3.7g)
~
cx power over heat + cx power: the fast-ion charge-exchange power loss divided by the sum of the beam heating power and the charge-exchange power (see B.197) f~~(r)
=
Q~(r)/(Q~(r)+ Qr(r)
+
Q~(r)).
(F.3.7h)
part sources: local rate at which fast ions slow down below E~ and enter the thermal ion population. ~a,th where a is the index for the beam injected species (see eq. (2.9.2.21) and B.150 and B.198). fusions: local rate at which beam—target D—T fusion reactions are occurring (see B.196) flDT(T) =n?J
cYDT(E)[2E/(AbmP)1
dEff(t&, E, r)~sdii,
(F.3.7i)
G.E. Singer etat
390
where n?
~ D for tritium beams, and n9 = 0 for hydrogen beams. ion heat fraction: fraction of local beam heating power which heat ions (see B.199) =
f.B(r)
fl~ffor
=
deuterium beams
/ One-dimensional plasma transport code
+
Q~(r)/(Q~(r)
~?
=
Q~(r)).
(F.3.7j)
The “totals” row entries are: avg. energy/particle: parallel: global average of the parallel energy per particle (see B.190) (Er)
U11B/NB,
=
(F.3.7k)
total: global average of the energy per particle (see B.192) (EB) =
UB/NB.
(F.3.7~)
beam dens: total number of fast ions (see B.187) NB = (2~rr)2RofOrdrf’E
dEf’f(r,
~s, E) d~L,
(F.3.7m)
ener dens: total energy in fast ions (see B.193)
~
=
cos pitch angle: mean: global average of
drfE~~.’~E dEf
,s
=
f~(ji,
E, r) dEL.
(F.3.7n)
v1/v (see B.191)
(,~)= (2~rr)~Rof°r drfEmxE dEJ’f~(~L,E, r)~dIL/NB, std. dev.: local standard deviation of the distribution in (aB)
=
(2~rr)2Rof°rdrf~’E dEf’f~(,i,
,i
(F.3.7o)
(see B.205)
E, r)(~i—
(~B)2
d~/NB.
(F.3.7p)
cx power over heat + cx power: volume integrated fast-ion charge-exchange power loss divided by the sum of the integrated beam heating power and the integrated charge-exchange power (see B.206) =
P~/(P~+ P~+ Pg).
(F.3.7q)
part source: volume integrated rate at which fast ions slow down below population (see B.207) =
(2~)2Rof °~B~h(r)r
~
and enter the thermal ion
dr.
(F.3.7r)
fusions: total rate of beam target D—T fusion reactions (see B.208) NDT
where n?
(2rr)2R0fñ~yr(r)
= liT
dr,
for deuterium beams,
(F.3.7s) n?
=
~ I) for tritium beams, and
~?
=
0 for hydrogen beams.
ion heat fract: fraction of total beam-heating power which heats ions (see B.209) FB =
P~/(P~+ p,B).
(F.3.7t)
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One-dimensional plasma transport code
391
The totals at the bottom of page 6 are: total perp. energy: total perpendicular energy in the fast-ion distribution (see B.189) =
(2ir )2Rf r
drf5”E
dEf’J?(ti~ E, r)(1
—
~2)
d~,
(F.3.7u)
total parallel energy: total parallel energy in the fast-ion distribution (see B.188) 2dii,
(j~B=(2.~)2R 0frdrfEn~~xEdEJlfB(,j,
E,
(F.3.7v)
r)~
total beam energy: total energy in the fast-ion distribution (see B.193) UB
=
(F.3.7w)
U~B+ L~,
total fast-ion charge-exchange power loss (see B.210) =
(2rr)2Rof
n
drf ~~acx(E)[2E/(Abmp)1~/21
0(r)r
other energy loss rates: not presently used
(=
f’fB(,1
E,
r) d~,
(F.3.7x)
0.) (see B.215),
plasma heating rate: total beam heating power (see B.216)
2R
pB
+
=
=
(2’rr)
0f (Q~(r)+ Q~(r))rdr,
(F.3.7y)
energy check: fractional energy nonconservation (see B.212) F~= 1
—
(Ui~~ ~ —
—
~as +
mp)/~,
(F.3.7z)
particle check: fractional particle nonconservation (see B.214) F~=1 —(N~~ ~
(F.3.7A)
beam target rate: total rate of beam target D—T fusion reactions NDT (see F.3.7s above and B.217). F.3.8. Neutral beam profile printout The neutral beam deposition profiles are written by subroutine IPRINT. The first section characterizes the neutral beam injectors which are “on”. active injector no.: energy Wb:
beam index of each “active” beamline; beam voltage of the injector (see D.1159); power in energy compon.: injected power in each of the energy components (see B.128) pNB = (Wb/j)4°Hj~. The neutral beam deposition profile for each energy component is given in the next section.
zone: in-surface out: NB
,/
NB. ~‘)±i~
index of FREYA zone; minor radius of the inner and outer surface of the FREYA zone (see B.129)
(F.3.8a)
G.E. Singer et al.
392
/ One-dimensional plasma transport code
h (r) at energy levels: neutral beam deposition function H( r) for each energy component (see B. 121a)
H~(r)=
f’H~(~, r) d~
percent loss: fractional shinethrough for each energy component (see B.131) L~B= I
—
(F.3.8b)
(2/a2)j°Hn(r)r dr.
The total neutral beam deposition results, summed over all energy components, are presented in the bottom section. zone: avg. rad.: total h(r):
cos of pitch ang: mean: std. dev.: deposition:
index of FREYA zone; zone centered radius of the FREYA zone; total neutral beam deposition function, H(r), summed over all energy components (see B.132).
local average of ~ = v11/v over deposition function (see B.133) local standard deviation of ~i distribution (see B.134) aNB(r); local particle deposition rate (see B.124 and B.135) ñNB(r)
=
n~1
NB/~tasmHn(~
I2NB(r);
(F.3.8c)
sink rate of thermal hydrogen due to charge exchange between neutral beam atoms and plasma ions (see B.126) n~(r); % for total h (r): fractional shinethrough of total neutral beam injected power (see B.1 37) 2) jH(r)r dr. (F.3.8d) LNB = 1 — (2/a Average for: cos of pitch ang: mean: global average of s = v 11/v over deposition function (see B.138)
(2/a2)f
=
charge ex.:
ñ
(r)r dr;
(F.3.8e)
volume average of ñ~~(r)(see B.141) =
(2/a2)f°n~0)(r)rdr.
(F.3.8f)
Total particles/second for: deposition: integrated particle deposition rate (see B.142) ~rNB
charge ex.:
=
(2rr)2Rof
h~(r)r dr;
integrated source of neutrals (see B.143) 2Rofñ~X(r)rdr. ‘a = (2~r)
(F.3.8g)
(F.3.8h)
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393
Lastly, for the full energy component of each active beam the “opacity” of the plasma integrated along the beam axis is given. beam no.: index of each active beam. optical depth to minimum: minor rad: the opacity to the point along the beam axis which has the minimum minor radius (see B.122) ~ major rad.: the opacity to the point along the beam axis which has the minimum major radius and is inside the vacuum vessel (see B.121b) ~
F. 3.9. Alpha particle page Characteristics of the D—T fusion product alpha-particle distribution are written by subroutine APRINT. The definition for the quantities listed under the column headings are: zone: radius: hi-e alphas: energy; slowdown t.:
j 1; where j is the BALDUR zone index; zone-centered radius, i~ local particle density of suprathermal alpha particles (see B.233) n a( r); local density of suprathermal alpha particles (see B.234) nEO(r); thermalization time; i.e., the time to slow down from the alpha particle’s initial energy, E0, to thermal energy (see B.232) ;0(r);
therm-fusion:
local thermonuclear D—T fusion reaction rate [see eq. (2.9.3.lb) and B.221]
—
nDT(r)
rate-total:
=
local total D—T fusion reaction rate [see eq. (2.9.2.2g) and B.168] ñDT(r)
1st-orb loss: qe:
=
17.6 MeV
=
+
n~T(r);
f
nDT(r)
dy/f
(
+
Q~+ ~
dv;
(F.3.9a)
time-integrated D—T fusion reaction rate (see B.241) nDT(r)
ion fract:
nDT(r)
local prompt-loss fraction due to alphas born on unconfined orbits (see B.226) L0(r); The fusion Q for the volume inside the present radius (see B.238) Qe
total fusion:
nDnT
=
f
‘~DT(’)
dt;
(F.3.9b)
fraction of alpha heating which heats ions (see B.242) =
f10)(r)
Q~(r)/(Q~(r)+ Q~(r)).
(F.3.9c)
The column averages are: hi-e alphas: volume average of the particle density of suprathermal alpha particles (see B.243) =
energy:
(2/a2)j’~n0)(r)rdr;
(F.3.9d)
volume average of the energy density of suprathermal alpha particles (see B.244) =
(2/a2)f°nn5(r)r dr;
(F.3.9e)
394
G.E. Singer et at
therm-fusion:
/ One-dimensional plasma transport code
volume average thermonuclear D—T fusion reaction rate (see B.245) (n~~) = (2/a2)fñ~T(r)r dr;
rate-total:
(F.3.9f)
volume average total D—T fusion reaction rate (see B.246)
1st orb loss:
= (2/a2)f
ñDT(r)r dr;
(F.3.9g)
integrated prompt loss fraction due to alphas born on unconfined orbits (see B.247) L10~=
total fusion:
f~
dr/~DT;
(F.3.9h)
volume average of the time-integrated D—T fusion reaction rate (see B.248) (F.3.9i)
ion fract:
the fraction of total alpha heating which heats ions (see B.249) Fia
=
f
QE(r)r dr/{f (Q~(r)+ Qr(r))r
dr}.
(F.3.9j)
The column totals are: high-e alphas: total number of suprathermal alphas (see B.250) N0 energy:
=
(2~TT)2Rof
dr;
(F.3.9k)
total energy of suprathermal alphas (see B.251) 2R U,, = (2.rr) 0f nEa(r)r dr;
therm-fusion: volume integrated thermonuclear D—T fusion reaction rate (see B.252) 2Rof ñ~~r dr; volume integrated total D—T fusion reaction rate (see B.253) NDT
=
(21T)2R
(F.3.9n)
0JOañDT(r)r dr;
1st orb loss:
(F.3.9m)
(2~)
~
rate-total:
(F.3.9tf)
volume integrated alpha-particle loss rate due to alphas born on unconfined orbits (see B.254) NDTIoss =
(2~)2RofañDT(r)L,,(r)r
dr;
G.E. Singer et a!.
total fusion:
395
volume integral of the time integrated D—T fusion reaction rate; i.e., the total number of 3.5 MeV alphas and 14.6 MeV neutrons produced during the simulation (see B.258) NDT =
ion fract:
/ One-dimensional plasma transport code
(2.~r)2Rof’~nDT(r)r dr;
(F.3.9p)
total power for alpha heating of ions (see B.259) =
(2~)2RofQ~(r)r
dr.
(F.3.9q)
The lower hybrid heating, pellet injection and Hawryluk—Hirschman transport models each have associated printout pages which are activated by the models themselves and not by NLPOMTs. The He-influx model produces an additional line in the SPRINT short printout. The input-file ripple model produces a printout of values read from the file at time step 0; further ripple printout must be triggered by NLPOMT(7).
Appendix G. Units BALDUR uses three sets of units, called “external”, “internal” and “standard”. External units are the units used for input and printout. They may be changed without having to rewrite the whole program, but are fixed for a given version of the code. Standard units are cgs, and are used for formulae which one wants to be in the same form as “the literature”. For example, transport coefficients are computed in standard units. It is not expected that the choice of standard units will be changed during the life of BALDUR. Internal units are used when the transport equations are solved for a time step. They are supposed to be chosen so that quantities such as density and energy density are near 1. At present, they are fixed, but they may be changed from one version of the code to another. In all three sets of units, certain relations are assumed among the various units: Particle (and energy, etc.) densities are per cubic length unit; Current density (j) is in current units per square length unit; E-field is resistivity times current density; Power is energy per time; Flux is particles (energy, etc.) per square length unit per time unit; Area is square length, volume is cubic length, etc. In internal units, B, energy, length and time units may be chosen; the rest are determined by the previous relations and the following: energy = mv2/2, energy density = temperature X density. Maxwell’s equations in internal units are: ~‘x
B= 2ITJ+
(dE/dt),
2-iivx E= —dB/dt.
The last two equations determine E and J (and with the current density and resistivity relations, i~and I) from the four independent internal units. There are conversion factors to convert from one set of units to another. For external units, the names of the units are kept in variables. The conversion factors and names are set in subroutine UNITS.
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Unit
External
Standard
Internal
magnetic field (B) density energy (E) temperature (H) current (I) current densities (j) length (1) mass(m) power (P) resistivity (ii) time(t) electric field (E)
kG particles/cm3 J keV kA kA/cm2 cm g W m~2-cm s V/cm
G particles/cm3 erg erg stat-A stat-A/cm2 cm g erg/s s s stat-v/cm
10 G particles/p.m3 nerg (about 2/3 keV) 2/3 nerg 5 mA 5 mA/p.m2 p.m 10p.g 100 nerg/s (21T/S) x 10~’ m~l cm lOms 2~nV/cm
Conversion factors: To convert from standard units to internal, multiply: magnetic field density energy temperature current current density length mass power resistivity time electric field
by by by by by by by by by by by by
USIB USID USIE USIH USII USIJ USIL USIM USIP USIR USIT USIV
To convert from internal units to standard, multiply: magnetic field by UISB density, etc. by UISD, etc. To convert from standard units to external, multiply: magnetic field, etc. by USED, etc. To convert from external units to standard, multiply: magnetic field, etc. by UESB, etc. To convert from internal units to external, multiply: magnetic field, etc. by UIEB, etc. To convert from external units to internal, multiply: magnetic field, etc. by UEIB, etc.
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