Standard test cases for the BALDUR transport code

Computer Physics Communications 49 (1988) 399—407 North-Holland, Amsterdam

399

STANDARD TEST CASES FOR THE BALDUR TRANSPORT CODE Martha H. REDI Plasma Physics Laboratory, Princeton University, Princeton, NJ 08544, USA

Fourteen standard test cases developed for the BALDUR code are described. Selected output is presented.

1. Introduction BALDUR [1] is a multifluid transport code which solves time-dependent one-dimensional radial diffusion equations for the evolution of plasma density, energy and poloidal magnetic field. We have developed fourteen test cases to benchmark the BALDUR code. This report documents these test cases and results of simulations run with BALDUR version BALDP47M (March 1986). Version BALDP47M is of particular interest since it is the version of BALDUR documented in PPPL-2073 and published in Computer Physics Communications [1] (this issue, p. 275). BALDP47M comprises a number of new features not present in the previous standard version of BALDUR, BALDP17M (June 1984). BALDUR now includes a discrete Kadomtsev helical flux function based sawtooth model, a new pellet model which treats pellet ionization more realistically, an improved neutral impurity influx model and a more general formulation for specification as well as printout of the plasma coefficients. Five of the test cases are based on analytical solutions of the diffusion equations. The analytical test cases [2] check the accuracy of the numerical simulation of particle diffusion as well as energy convection and conduction. Additional benchmark cases were developed to exercise the four impurity capabilities of BALDUR, especially the neutral impurity influx model and the semiempincal formulation of transport coefficients, and to document BALDUR’s simulation of specific tokamaks. Most of the cases were verified by

comparison with the test cases run with version BALDP17M which were documented in refs. [2,3]. Sections 2 through 4 document the test cases run with BALDP47 (March 1986).

2. Analytical test cases The first set of test runs are based on analytic solutions to diffusion equations in cylindrical geometry for the evolution of the current density .~

—

a

C

F ~ arB0 1

3

4.1~~ and the particle densities —

~

i

—

(2.1)

c-~-,(-uiJ~),

~ + Sa, 3t r a = (hydrogenic and impurity densities),

~

=

—

-~-

(2.2)

as well as the electron and ion energy densities

a ~

a =

.~

=

—

[~.FIJTJ}

8r rn~1—~ ~T’.T.}+ 1a[ 3T. T

—

-“

Q1 (j

=

i, e),

~

(2.3) where B9 is the poloidal magnetic field, ~ is the plasma resistivity and J~is the current density due to fast beam ions. Particle densities are computed for hydrogen, deuterium, oxygen, iron and other impurities. The convective particle flux I~= ~D~~~+flaVa,

OO1O-4655/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(2.4)

M.H. Redi / Standard test cases for the BALDUR transport code

400

includes both diffusive and advective (pinch) term effects, S0 is the source of particles arising from charge exchange of thermal and beam neutrals, is the convective heat flux; Q1 is the source rate of plasma heating. Four types of analytic test cases have been used to verify the numerical solution of these equations with initial and boundary conditions specified for the plasma temperature and density, with specific sources of plasma density and energy, and for particular transport coefficients. BALDUR’s neutral beam, neutral transport, recombination, bremsstrahlung, cyclotron radiation, impurity radiation and ohmic heating sources as well as the scrapeoff and divertor models have turned off in these tests. The test cases concern a TFTR-size deuterium plasma and include up to four impurity species. Results follow for the five test cases with BALDP47M based on diffusion analyticalequations solution for of the one-dimensional radial the evolution of plasma density, energy and poloidal magnetic field,

though the equations remain coupled through the effects of the particle density on the electron energy density. Eq. (2.1) is decoupled from eqs. (2.2) and (2.3) in all of these studies by setting the resistivity to zero, i.e., no ohmic heating. The error in the simulation time constant T compared to Tanalytic arises from several sources. BALDUR solves the diffusion equations with the plasma boundary set at the midpoint of the zone just outside the last physical zone whose outer boundary is at the minor radius a. For this reason a* a(1 + 1/2N) where N the number of (equally spaced) physical zones. The analytic case corresponds to N cc. Then we expect the exact numerical solution to have time constant =

—*

Ta~aIytica1 =

1 1 1+—+—— Dx~~ N 4N2)’

~

(

(2.1.3)

in addition to terms arising from differencing er2. rorsAnwhich be proportional 1/Norder Bessel initialwill density profile of atozero function with D 20000 cm2/s and a 85 cm should decay with time constant =

50 Tanalytic

2.]. Plasma evolution at constant temperature

=

=

=

63.7 ms,

(2.1.4)

for 50 zones and T~a1ytic 64.9 ms for 25 zones. Note that BALDUR sets the density edge value at =

The electron density evolution equation is derived from eq. (2.2) with tie ~2ahba, Z 0 is the charge state of the a th ion species. With no sources, no pinch and constant diffusion coefficient D, this becomes 2~a D 8n (211) ~3r2 8t r 3r =

‘

‘

the center of the dummy zone just outside the wall radius, so a’ 85 + 85/100. The edge and initial central temperatures were 2.0 keV. =

case electron DEN47aand with zones, decay the average of In thetest central ion50density times were Tne = 64.0 ms, Tm 63.9 ms. For 25 zones, Tne 65.0 ms. Decreasing the number of zones =

=

If we assume that n 0 =f(r)g(t), this becomes the zeroth order Bessel’s equation with exact analytic solutions of the form

increases the time constant compared to that of the analytic solution T~aIytjc 62.4 ms (fig. 1). Fig. 1 shows that the simulation time constants for

ne(r,

cases with 50 and 25 zones are very similar to the analytic temperature solution timeofconstants. density central DEN47aThe evolve as and ex-

=J0(kr) exp(—t/T), (2.1.2) 2. If the boundary condition is where aT 1/Dk t) n 0, a constant,2/(Dx2) then ka x,,, a root and Tanajyije a *is constant in space It of canJ0(x), be shown that if T~ and time, and if n 0(r, t) is a solution of eq. (2.1.1), then E~ ~T~n0 is a solution of eq. (2.3). Thus, this test case can be used to evaluate the accuracy of eqs. (2.2) and (2.3) separately, alt)

=

= ~‘,

=

=

*

=

=

=

pected. The slight drop in I~(0)and the oscillation in the edge temperature during the simulation is probably due to “upstream differencing” which prevents an instability which can occur when energy transport is primarily convective, and the flux computed might otherwise transport out more

M.H. Redi

/ Standard test cases for the BALDUR

particles then reside in a zone. Since the density at the edge of the plasma is small and fixed, this oscillation does not affect the code results in practice. This test case also includes the four impurities carbon, nitrogen, oxygen and iron. The impurities were specified as 1% of the deuterium density, identical initial profile puritywith density lifetimes were foundshapes. to be The equalimto the electron density lifetime as expected since the impurity diffusion coefficient (C22) was also 20 000 2/s. cm 2.2. Plasma evolution at constant density For a plasma with density constant in time, eq. (2.3) can be written 3T181 8T. 3n—I rn-X 1-~—~ + ~D~T-~—~+ ~ni7’ivi] t r3r~ + w~. (2.2.1) The analytic solution of this equation with D 1 to v~ 0 has been tested in TEM47a. Heat flow due energy conduction with constant diffusivity x =

=

1

and constant density n1 causes T1 to exhibit the same behavior as the density in test case DEN47a. If there are no sources of heat, W~ 0, we find 2T (2.2.2) 3 8T’~ -~x~ 1 aT + xi—~. 88r =

-~ -~-

=

The ion temperature evolution is governed by a zeroth order Bessel’s equation just as the electron density was [eq. (2.1.1)]. As in section 2.1, this case tests BALDUR’s solution of eqs. (2.2) and (2.3). At t 0, we let the initial temperature distribution be T(r) = (i~ T0)J0(kr) + (2.2.3)

transport code

IC Den 47x Test Cases S

90 60

~.

The initial central temperature is T0, the initial edge temperature is T0 and J0(ka*) = 0 as in section 2.1. The analytic time-dependent solution

is T(r,

t) =

(T0

—

T0) exp(—t/’r)J0(kr)

+

r (2.2.4)

TN~

20 62

66

70

74

78

80

r Imsec) Fig. 1. Plasma density (DEN47a) and temperature (TEM47a) decay times versus number of BALDUR zones.

analytic solution has 3 T

a”2

2

=

(2.2.5)

(2.405)2’

2/s and a = 85 cm so r5° In the test case 63.7 x1 = ms 3 X and iO’ T~alytic cm = 62.4 ms. The analytic 3/cm3, and the imhydrogen 2 x2 10’ purity ion density densitieswas were X 10’1/cm3. =

After 200 ms, a Bessel function heating source was turned on S(r) S0J0(kr), (2.2.6) with the same wave vector k as above. The analytic time-dependent solution is =

for

t > t’:

=

{

[~*

T(r, —

t)

(~*

—

i~(t’))

xexp((tt’)/T)J

—

T 0}J0(kr)+ T0. (2.2.7)

By analogy with the t < t’ solution, the t> t’ solution has the following expected properties. T is a Bessel function in space, and is separable in r and t. At t = t’, this becomes T(r,

t’) =

(T0(t’)

—

T~)J0(kr)+ T~.

(2.2.8)

The central temperature at t’ is T,~(t’),and the edge temperature at t’ is T0. At t cc, the plasma temperature is T(r, cc) = (T0* T0)J0(kr) + 1~. T0* is the central ion temperature for t>> t’; T0 remains the source edge ion temperature. The maximum effect of the heating is to determine the =

—

In test case TEM47a, an initial temperature profile of a zero order Bessel function in 5°= 63.3decays ms. The time with a time constant of T

-

Tern 47x Test Cases

z 40

=

—

401

402

M.H. Redi

/ Standard test cases for the BALDUR

central plasma temperature T0*

—

T0

=

In the test case PDD47a the electron density does not remain constant in time. An initial de(2.2.9)

~T.

(= 63.7 ms) remains the same as in the first part of the simulation as it depends only on the properties of the plasma.3.To reach = 5power keV, The totalT~~”’ input we need S0 = 0.339 W/cm is P 101 = f(S(r) dV= 5.28 MW, the ion heating power. Additional heating is required for the larger number of electrons due to impurity ionization. The heating profile S0J(kr) is input for the electrons and for the ions. TEM47a had time constant for the temperature rise T~5 = 63.0 ms. Fig. 1 shows the simulation time constant for the temperature evolution test case compared to the analytic curve. The differencing errors are less than those which arise from finite zone size. Ta~alytic

2.3. Analytic tests of pinch-diffusion balance Two test cases have been devised to test BALDUR’s numerical calculation of a balance between outward diffusion and inward convective pinch. Both cases specify the initial density profile as Gaussian ne(r, 0)

= tie0

exp(—Ar2),

(2.3.1)

and a constant edge temperature, equal to the initial central tempera’~u~c, 2.0 keV. No sources or sinks of plasma density or energy are specified and x = 0. C19 = 0 assures that there is no Ware pinch included but subroutine EMPIRC can be used to specify v arbitrarily. Eq. (2.2) becomes Bn —

Bt

=

I B F / Bn —I r~D—

—

rBrL

t,

Br

+

~1

nv)

~.

transport code

(2.3.2)

j

crease in the central plasma density is due to different calculations of B n/Br in subroutine XSCALE and implicitly in subroutine REDUCE. This causes a numerical instability in the temperature electron as eq. (2.3) no longer eq. (2.2.1). The density and thereduces electrontotemperature evolution for PDG47a are constant, The density profile remained exactly constant from 200 to 900 ms. These two test cases test subroutine EMPIRC as well as the differencing accuracy of BALDUR. 2.4. Analytic test of the poloidal magnetic field diffusion equation Test case PMF47a tests the numerical solution of eq. (2.1), the diffusion of poloidal magnetic field due to finite plasma resistance and the effect of neutral beam ions on poloidal field. Temperature and densities3/cm3). are constant space (1~ = 0.2 There in is no Ware pinch. keV, If ne ij, =the 3 xresistivity, 10’ is independent of r and = 0, the magnetic diffusion equation is a first order Bessel’s equation and leads to a diffusion equation for plasma current j(r, t) which is a zeroth order Bessel’s equation, similar to those we have solved for plasma density and temperature. A Bessel function program, FBES, was added to subroutine AUXVAL for the initial poloidal magnetic field B 0(r, 0). Input variable EEBFIT < 0 triggers the analytic magnetic field test. The magnitude of EEBFIT specifies the Bessel function root used. This has a current response j(r,

t)=J~+exp(—t/T)J0(k, r),

(2.4.1)

where 2,

(2.4.2)

Any t) D will and also v forsatisfy whichthe eq.energy (2.3.2)transport has a solution n(r, equa-

T =an 41T/ci~k to initial J

lion with Te

0(k, r) +Ja current distribution. The central density time constant was Tavg = 229.0 ms.

Br

=

constant. In particular, we take

+nv=0,

(2.3.3)

by setting v = —D(Bn~/Br)(1/n~)in PDD47a, and, by setting v = 2DAr in PDG47a, we expect both test cases to show no change in n(r, t) or T(r, t).

3. Impurity influx test cases The impurity influx test cases provide benchmark data for the neutral and ionized impurity

M.H. Redi / Standard test cases for the BALDUR transport code Table 1 Impurity influx test cases C

200

NTI47a

2

N1147a

1

11147a

0

Table 3 Comparison of ionized and neutral impurity influx models

FLIMP

IMP 1,2

IMP 3,4

total no. of I & 2 impurities rate of influx of 1-2

neutral

—

neutral

rate of influx of 1-4 impurities

ionized

1

2

3

4

—

Target impurity 11147a

1.7817 1.822’~

1.78’~ 1.822’~

1.817 1.84617

1.8’~ 1.84617

ionized

NII47a

0.916916

0.916916

purity influx of capability of BALDUR BALDUR has for up to four species impurity. three models by which impurities are added to the transport simulations. Three BALDUR test cases, NTI47a, NII47a and 11147a, check that neutral and ionized impurities can be specified by total number or by influx rate. The scrape-off model was not used in these tests and the transport coefficients were held constant. In 11147a, oxygen, krypton, aluminum and carbon impurities were influxed with identical central and edge densities specified. N1147a and NTI47a test the influx of two species of neutral impurities only. In table 1 is presented the options investigated for the impurity influx option. In these tests, it was necessary to set NBOUND equal to 2 or 3, otherwise pedestal boundary conditions are specified for each ion. The input parameter C(200) is the switch by which the ionized or neutral state of the first and second impurities is set; and if neutral, whether their influx rate or a target total number is to be specified via input variable FLIMP. 11147a tests the ionized impurity influx model,

Results of NTI47a case

—

total number neutral impurity influx test

Impunty density evolution Time=0 200 332 Impurity 1 = oxygen target simulation Impurity 2 = krypton target simulation

403

=

It was identical to 11117k, as desired. Thewas constant 2/cm2s specified simurate offorinflux x 10’ within 4%. lated four 2.5 impurities NTI47a tests the neutral impurity influx model which specifies the total number of each impurity. Influx of only two neutral impurities is tested. The neutral impurity influx models should not be used for more than two impurity species as particle conservation is not maintained. The code only approximately meets the target numbers of neutral impurities (table 2). The oxygen overshoots the target value at 332 ms, but is within 1% of the target at 400 ms. The algorithm does not allow for loss of impurities so the krypton target values at 200 and 322 ms are not met, but the final target at 400 ms is reached within 1%. The particle conservative checks in NTI47a are good to better than i0~. N1147a tests the neutral impurity influx rate specification model. Table 3 shows a comparison of the final impurity totals for ionized and neutral impurity influx. Modification of the neutral impurity influx model algorithm has caused this test case to differ from N1117k for a constant flux rate of 2.5 x 1012/s. The plasma impurity densities should increase by 1.73 x lO~particles during the 0.25 ms simulation. The neutral impurity flux simulation is 1.23 X 1012/cm3s, about one-half the specified value of 2.5 x 1012/cm3s. This can be avoided by specifying neutral impurity influx values at more than two points in time. ,

400ms

4.7

2.016 2.0

3.3~ 4.5

~ 9.7

4.7~ 4.7

2.0 4.7

3.3~ 4.7

9.8~ 9.7

4,715

Impurity

4. Machine specific test cases The machine specific test cases exercise special features of the code in parameter ranges appropriate for particular machines. Two test cases

404

/ Standard test cases for the BALDUR transport code

M.H. Redi

Table 4 BALDUR test cases Running time (mm) DEN47a density diffusion TEM47a temperature diffusion PDD47a pinch diffusion balance v ~ gradient of density PDG47a pinch diffusion balance

0.5

v cc gradient of Gaussian PMF47a poloidal magnetic • . — field diffusion . NTl47a neutral impurity — total number neutral and ionized impurity influx N1147a neutral and ionized impunty influx Ill47a ionized impurity !~Ux HMD47a H-mode simulation — — — . TCR47a TFTR compression, NBI test INT47a INTOR test cases TFT47aTFTR test cases ATC47aATC test cases PDX47a PDX test cases

1.5

1.5 1.5

tron and ion temperature maxima on the time evolution plots were identical to those of ATC17A. INT47a simulates a plasma heated to ignition with INTOR machine parameters but with final density pumpout. The code results are not identical to those of INT17a but do not differ much. In particular, the accumulation of helium ash in INT17a is 7.888 x 1020 at 10 s and in INT38A is 8.004 x 1020. The total number of fusion reactions in INT17a is 8.424 x 1020 and in INT38a is 8.539 x 10 20 The helium ash accumulation only ineludes thermalized alpha particles. TFT47a simulates TFTR. The density and ternperature time plots and three-dimensional plots are very similar to those in TFT17a. The safety factor and current profile plots are different because cfutz(478) was defaulted in BALDP17M to flatten the current profile if q falls below one at .

1

2.3 1.0 11

7.0 5.5 35 2.5 i

2

are included from recent tokamak simulation studies: HMD47a, a simulation of the PDX H-mode, which exercises BALDUR’s scrape-off model and empirical transport law formulation, and TCR47a, which simulates a neutral beam and compression heated TFTR upgrade experiment. The complete list of BALDUR test cases is given in table 4. This list also includes the four other machine specific test cases INT47a, TFT47a, ATC47a and PDX47a which have been used to verify past versions of BALDUR. (These test cases are used with run numbers, such as INT2Oa, “INT2O” signifying that BALDUR2OM is being run with the INTOR test case, and “a” signifying that this is the first such test.) The test run output appended, presents a typical confinement page summary from each of three machine specific test cases: HMD47a, INT47a and TFT47a. These cases illustrate BALDUR simulation of tokamaks in the H-mode, ignition and neutral-beam-heating operating regimes. ATC47a simulates a noncompression discharge with ATC machine parameters. The central elec-

the plasma center, and this is not the default in BALP47M which now includes a sawtooth model. HMD47a simulates the PDX H-mode and exercises BALDUWs~rape-offmodel and semiempirical transport law formulation. Differences compared to HMD17a arose from changes in the neutral impurity influx model and the new default of cfutz(478). Enhancement of transport in HMD47a inside q < 1 is more effective because of C

478 default does not flatten the current. Thus, q(0) = 0.7 in HMD47a but q(0) = 0.9 in HMD17a. PDX47a simulates the PDX experiment. It differs from PDX17a in the safety factor and current plots because of the new default value of cfutz(478) in BALDP47M. TCR47a simulates a neutral beam and cornpression heated TFTR upgrade experiment. This run was compared to TCR27b and TCR17a. The default of cfutz(478) and the treatment of the Ware pinch are different in these two versions of the code. TCRI7a had maximum electron temperature ~max = 11.2 keY. The previous default of C478 in BALDP17M flattened the current, broadened the electron temperature profile, and produced a lower central electron temperature. The maximum ion temperature in TCR47a (24.6 keY) and in TCR27b (23.5 keY) are in reasonable agreement. Backward compatibility with Ware pinch on energy transport was not possible before BALDP38M.

M.H. Redi

/ Standard test cases for the BALDUR transport code

5. Conclusion Continued development of large numerical computer codes such as BALDUR requires documented testing of new versions against standard test cases. This report documents analytic solution test cases as well as code exercise test cases. These test cases were developed and archived for use as diagnostics and benchmarks for future standard BALDUR versions against the performance of earlier code versions [3]. Version BALDP47M of the BALDUR transport code verifies the analytic test case results. It exhibits improvement and certain defects remaining in the impurity influx test cases and shows changes in the simulation of the machine specific experimental simulation test cases which correspond to developments in the code’s treatment of current diffusion and energy transport. New users of the BALDUR code will find these test cases a

405

valuable tool in learning to run the code and for benchmarking performance against BALDP47M. ACIUIOW1~IgementS It is a pleasure to acknowledge many interesting and useful discussions with D.R. Mikkelsen, C.E. Singer and G. Bateman. This work was supported by the US Department of Energy Contract No. DE-ACO2-76-CHO3073. References [1] C. Singer, D.E. Post, D.R. Mikkelsen, M.H. Redi et al., Comput. Phys. Comnsun. 49 (1988) 275 (this issue). [21 M.H. Redi and D.R. Mikkelsen, Princeton Plasma Physics Laboratory, Applied Physics Division Report No. 32 (Janu~ 1985) 21 pp. [3] M.H. Redi, Princeton Plasma Physics Laboratory Report No. TM-377 (August 1986) 43 pp.

406

M.H. Redi

/ Standard test cases for the BALDUR

transport code

TEST RUN OUTPUT hmd47a/17nov83;baldp47m —6—

.ee time step

385

boldp47m/llmarB6 time

•.*

602.517

=

radius ( cm ): deuterium confinement time ( sec ): oxygen confinement time ( sec ): krypton confinement time ( sec ) ci. energy confinement time ( sec ): ion energy confinement time ( sec ): total energy confinement time( sec ): n—tau(energy)

10.00 5.478e+00 1.061e—01 1 .O7le—01 2.298e—02 3.068e—02 2.646e—02 9.447e+l1

experimental energy confinement times: thermal/ohmic 1.571e—01 total 4.630e—02 mean mean mean mean line

electron density . . • ion density electron temperature: ion temperature . . . avg. electron density.

loop voltage:

2.516e+13 2.307e+13 7.857e—01 8.431e—01 3.015e+13

dt

21.55 1.194e+00 1.069e—01 1 •878e—01 2.971e—02 4.532e—02 3.616e—02 1.200e+12

31.00 1.217e—01 9.639e—02 9 .668e—02 3.165e—02 4.798e—02 3.818e—02 1.144e+12

beam—plasma 1.955e+13 5.056e+12

internal f I ux= 0. 598

baldp47m —

4—

is.

time

5433.493

=

): ): ): ): ): ): )

experimental energy confinement times: thermal/ohmic 4.086e+03 total 1.634e+00 electron density . . . ion density electron temperature: ion temperature . . . avg. electron density.

loop voltage:

krypton

.

at zone

42

baldp47m/llmarB6

120 ...

radius ( cm deuterium confinement time ( sec tritium confinement time ( sec hel ium—4 confinement time ( sec el energy confinement time ( sec ion energy confinement time ( sec total energy confinement t ime( Sec n—tau(energy)

mean mean mean mean line

volt—sec

conservation: 5.643e—11 conservation: 1.149e—09 conservation: —6.006e—08 conservation: 1 .061e—10 conservation: 1.061e—10 conservation: —6.628e—04

INTOR CRAY TEST CASE time Step

beto—toroidal 2.754e—03 2.710e—03 2.854e—03 0. 8.319e—03

total 2.009e+13 5.137e+12

maximum change over one timestep was 16.230 7., for deuterium oxygen krypton el . energy ion energy b—poloidal energy

millisec

40.00 9.691e—03 1.974e—02 I .639e—02 3.244e—02 4.410e—02 3.733e—02 9.393e+11

beta—poloidol electron: 2.277e—01 ion: . • 2.240e—01 beam ion: 2.359e—01 alpha: . 0. total: . 6.876e—01

I ambda= 1 .064

thermonuclear 5.408e+11 8.105e+10

7.212122

=

sec sec

port/cu cm part/cu cm key key part/cu cm

7.111 e—01 volt

d+d reaction neutrons neutrons / sec neutrons (total)

millisec

5.795e+13 5.473e+13 2.493e+01 5.821e+01 8.940e+13

2.789e—02 volt

d+d reaction neutrons neutrons / sec neutrons (total)

thermonuclear 3.692e+18 5.946e+18

millisec

3822 2.246e+00 5.041e+00 2.080e+00 8.861e—01 2.408e+00 1 .545e+00 2.462e+14

dt

73.50 2.517e+00 5.796e+00 2.916e+00 9.089e—01 3.577e+0O 1 .858e+00 1.994e+14

=

100.000000

111.72 2.145e+08 3.956e+00 5.865e+00 8.566e—01 4.743e+00 1 .973e+0O 1.502e+14

millisec

147.00 3316e—01 0. 7.720e+12 8.220e—01 4.923e+00 1 .926e+00 1.116e+14

sec sec

part/cu cm part/cu cm key key part/cu cm lambda= 9.171 beam—plasma 8.849e+16 3.751e+17

beto—poloidal electron: 2.234e+00 ion: . . 4.928e+eO beam icr,: 1.693e—01 alpha: 1 .304e+00 total . 8 636e+00 internal flux=19,083 total 3.781e+18 6.321e+18

beta—toroidal 2.071e—02 4.567e—02 1.569e—03 1 .2e9e—02 8.003e—02 volt—sec

M.H. Redi / Standard test cases for the BALDUR transport code

pbstr— 5.28e+07 prfstr~0. poth~7.19e+07 qinst—

pbnow— 7.50e+07 pbabs— 7.28e+07 phbem— 6.69e+07 Rrfinj~0. phrf~0. lrf— 0.000 pabt— 4.07e+06 pholf~7.58e+07 eodot——3.10e+06

5.065

qss=

7.076

qcomp—

baldp47m —

4—.

s.c

qtech—

time =

149 ssc

1614.223

electron density . . . ion density electron temperature: ion temperature . . . avg. electron density. 1.696e—01 volt

d+d reaction neutrons neutrons / sec neutrons (total) pbstr— 2.98e+07 prfstr= 0. path= 3.50e+06 qinst=

4.888.4.13 4.529e+13 5.342e+00 1.540e+01 5.517e+13

deuter i urn tritium el. energy ion energy b—poloidol energy

at zone

dt =

42.50 2.873e—@1 5.392e+00 1.352e—01 7.127e—01 3.443e—01 2.151e+13

part/cu cm port/cu cm key key part/cu cm

50

qss=

0.879

conservation: conservation: conservation: conservation: conservation:

qcomp=

beam—plasma 7.836e+15 7.748e+15

0.883

—5. 638e—10 —5.272e—12 —7.408e—10 —7.4e8e—10 4.581e—05

64.60 2.876e—01 1.201e+00 1.250e—0I 9.140e—01 3.471e—01 1.972e+13

15.000000

millisec

85.00 3.e67e—02 2.663e—02 1.276e—01 7.940e—01 3.278e—01 1.602e+13

beta—poioidai electron: 3.038e—01 ion: . . 8.116e—01 beam ion: 2.159e—01 alpha: . 1.136e—01 total: . 1.445e+00 Internal

flux= 6.880

beta—toroidal 3.888e—03 1.039e—02 2.763e—03 1.453e—03 1.849e—02 volt—sec

total 2.316e+16 1.678e+16

pbnow= 3.20e+07 pbabs= 3.20e+07 phbem= 2.13e+07 prfinj= 0. phrf= 0. lrf= 0.000 pobt= I.87e+06 phalf= 4.31e+06 eadot= 4.60e+05

0.839

,

sec see

lambda= 2.017

thermonuclear 1.532e+16 9.031e+15

millisec

22.10 2.736e—01 3.967e+00 I.418e—01 5.561e—01 3.196e—01 2.124e+13

experimental energy confinement times: thermal/ohmic 2.761e+01 total 3.748e—01

loop voltage:

hel,um—4

baldp47m/llmor86

radius cm ): deuterium confinement time sec ): tritium confinement time sec ): ci. energy confinement time sec : ion energy confinement time sec : total energy confinement time( sec : n—tau(energy) :

mean mean mean mean line

4.889 7., for

7.985

conservation: 6.324e—12 conservation: 9.592e—13 conservation: —1.120e—09 conservation: —6.476e—11 conservation: —6.476e—11 conservation: 8.514e—07

TFTR CRAY TEST CASE time step

ebdot——7.28e+05 lbem— 0.118 ethdot— 2.45e+07 lath— 0.036 lobt 0.145

7.277

maximum change over one timestep was deuterium tritium helium—4 ci. energy ion energy b—poloidal energy

407

ebdot= 2.37e+03 lbem= 0.333 ethdot= 1.09e+06 lath= 0.089 lobt= 0.153 qtech=

0.884