- Email: [email protected]
Plasma edge model for tokamaks
PLASMA EDGE MODEL FOR TOKAMAKS*
M.R. GORDINIER Nuclear Engineeting
Department.
University of Wisconsin. Madison. Wisconsin, USA
and R.W. CONN School of Engineering
and Applied Science, University of California, Los Angeles. Los Angeles. California MQ4.
USA
The discharge characteristics in fusion experiments, particularly tokamaks, are often determined by the couplingbetween the core plasma and the region between the limiter and the vacuum wall. A zero dimensional scrape-off model is developed which incorporates the essential atomic physics and plasma/wall interactions. This model is coupled to a one dimensional radial transport code for the analysis of tokamak discharges. It is demonstrated that the discharge characteristics predicted
can be substantially altered by the inclusion of such a model.
1. introduction
Plasma simulation work to date has emphasized the development of models for the central region of the plasma discharge. Codes now exist which incorporate most of the essential physical processes present in the core region. By contrast, serious modelling of the “edge region” is only now receiving detailed attention. The scrape-off zone is recognized as being critical to the modelling of tokamak discharges since this region is the source of impurities which degrade plasma performance. Complex plasmawall processes have a strong effect on the overall density and energy balance by serving as sources and sinks of ions due to diffusion, trapping, and reflection phenomena. So strong is the coupling between core and edge that in many current devices agreement between simulation and experiment is only achieved by judiciously varying the phenomenological parameters present in crude recycling models. In assessing the possible impact of plasma-wall interactions on the burn dynamics of future fusion reactors, extrapolabased upon such device-dependent tions parameters is unwise. We have developed a genera1 mode1 of. the scrape-off zone of a tokamak which incorporates * Partially supported by US Department of Energy Contract no. DE-ASO3-76STtXKW
the relevant atomic physics and plasma-wall interactions with a minimum of adjustable parameters. Using this mode1 in conjunction with our one-dimensional transport code we have addressed the following questions. (1) Which edge effects have a substantial impact on the main-body plasma burn dynamics? (2) Can these processes be tailored to yield discharge improvements? (3) Can a better understanding of the physics of the scrape-off zone point the way to new operating scenarios and/or novel impurity and ash control schemes? 2. Boundary model Our model is based upon the previous work of Hinnov, Fallon and Bishop [l], Hogan [2] and Fielding [3]. The mode1 consists of a series of coupled ordinary differential equations for the density and temperature of the atomic and molecular species present in the scrape-off zone. The species are deuterium (D$, Dt, Do, D+) and tritium (T!, T;, T)‘, T+), alpha ash and, as an impurity, one of several wall materials (C, Ti, Va, MO, Fe). Coupling between species and isotopes occurs due to ionization, dissociation, and chargeexchange events. In addition, sources from the plasma core and from the diffusion of wall trapped gas into the scrape-off zone are included.
Jaumal of Nuclear Materials 93 & 94 (1980) 420-430 @ North-Holland Publishing Company
420
M.R. Gordinier, R.W. Conn I Plasma edge model for tokamaks
rethermalization, Radiation, charged-particle surface recombination, sputtering, sheath effects, and charged-particle parallel transport have also been incorporated. The model is self-consistent in that the edge density and temperature as well as the fuel and impurity influx to the core are evaluated by solving the coupled density and temperature equations subject to the constraint that the plasma outfluxes predicted by the one-dimensional radial code [4] are matched. The resulting influxes and edge values determine the boundary conditions for the radial model. This in turn determines the discharge characteristics of the core and the iterative coupling between core and edge.
(9)
00)
where S cure S*care Lll
3. Equations The zero-dimensional equations solved in the scrape-off zone for the spatially averaged densities and temperatures are:
L:, nlrp rtre
(2)
S Will S$“, S gas-Puff
(3) (4) (5)
= particle source from core, = power source from core, = particle loss in parallel direction, = power loss in parallel direction, = neutral particle flux to core, = neutral power fixu to core, rethermalization between specie i and j, = radiative loss of power, = loss/source of particles due to recombination at wall, = source of gas diffusing from wall, = sputtering source of neutral impurities, = neutral source due to gas puffing. =
Qi.j
L rad L recomb,
(1)
421
kmnb
For compactness, the above equations have been written in terms of hydrogen densities and temperatures. However, since the model treats deuterium and tritium as separate species, the above system represents 18 coupled ordinary differential equations plus two equations for alpha particles and impurities.
(6) 4. Modelling of plasma-wall physics 4.1. Impurities (7)
(8)
The impurity density and radiative losses are calculated using a modified version of the timedependent noncoronal equilibrium model developed at PPPL [5]. It solves for the number density of each charge state of the impurity taking into account sputtering sources, charged-
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
422
particle parallel losses and such atomic processes as impurity ionization, dielectronic recombination, three body recombination, etc. 4.2. Atomic physics After an extensive literature search [6-111, nineteen reactions were included in the model and represent the most probable reaction channels at characteristic scrape-off temperatures (table 1). Treating deuterium and tritium as separate species and allowing for cross-terms, the number of reactions increases from 19 to 55. Knowing the raw experimental cross-section data or analytic fits to such data, the 55 reaction rates were obtained by numerically integrating the expression for the reaction rate assuming two Maxwellian species.
Table 1 Hydrogenic atomic reactions e+H+H++2e e+Hz+Ht+2e e+Hr+H+H++2e e+Hr+2H+e e+Hs+H+H++e e+Ht+2H++2e e+Ht-+2H H++H+H++H H++Hr+Ht+H H;+Hr+Ht+Hz H++Hr+H++Hs+e Hi+Hz+Wf+e H++H+2HC+e H;+H+H++Hr Ht+H+H++H$+e H;+Hr+H++H+Hr Ht+Hr+2H++Hr+e H$+H+H++2H Hf+H+2H++H+e
4.3. Reflection-wall concentration model The reflection and/or trapping or ions and neutrals at the wall and limiter of a tokamak is modelled using the particle and energy reflection coefficients of Oen and Robinson [12]. For par-
titles not reflected from the wall, a calculation based on monoenergetic range theory [13] is performed to determine the spatial distribution of implanted atoms (see fig. 1). However, since
IMPLANTATION PROFILE a -. d
I R PLASMA TEMP.
DEPTH
-
100.00
[EVI
~flNGSTROMS1
Fig. 1. Depth profile for the implantation of deuterium into stainless steel. The incident deuterium energy distribution is a Maxwellian with a temperature of 100 eV.
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
423
MOLECULAR RELEASE RATE
IVPLRNT FLUX -
:: :: d 0.0
2.5I
5.0#
7.5I
TIME Fig. 2. Release rate of molecular
the incident flux is representative of a Maxwellian distribution of particles, the range calculation has been modified to reflect this. Given the net implantation profile for the spectrum of incident energies, application of a modified version of the time-dependent wall concentration model of Baskes [14] provides the release rate of implanted gas to the shadow region in response to concentration gradients (see fig. 2). 4.4. Sputtering Sputtering of the walls and limiter is assumed to occur by D, T, (Y,and impurity impact. These fluxes arise both from the core and shadow the yield regions. For light ion sputtering expression developed at Garching [15] has been integrated over a Maxwellian energy distribution of ions to obtain an average yield coefficient. For impurity self-sputtering, the yield expression developed by Smith [16] is used. In addition, the probability of impurity redeposition after sputtering is included as a phenomenological parameter. 4.5. Sheaths The presence of an ion accelerating sheath [17]
10.0I
ISECONDSI hydrogen
t.00n1015 [w'Clw+t2-SECI
9 12.5
15.0
I 17.5
2
0
from stainless steel versus time.
at the limiter-plasma interface has been included in modelling limiter sputtering and in evaluating the magnitude and energy of the hot recycling flux from the limiter surface. To illustrate the consequences of such sheaths, consider an edge electron temperature of 50 eV. Calculations show that at this temperature oxygen is six times ionized. Neglecting the effect of secondary electron emission or magnetic field, the sheath potential is approximately four times the electron temperature. Thus the oxygen ion would impact a poloidal limiter with an energy in excess of 1200 eV! Since sputtering and reflection coefficients are strong functions of energy, the sheath effect warrants inclusion in any realistic model of the edge. In addition, hot deuterium and tritium neutrals recycling from the limiter can penetrate more deeply into the discharge than neutrals characterized by the Franck-Condon process. 4.6. Parallel transport Charged particle parallel transport to the limiter is modelled by assuming free-streaming ions at a sound speed characterized by the local ion temperature. Both poloidal and toroidal
424
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
limiter geometries are considered. Given a specific limiter design, such as in ref. [18], with a prescribed pumping efficiency f, we allow (1 - f) of the incident parallel flux of D, T, a, and impurities to recycle into the plasma core at energies consistent with the presence of sheaths and finite energy reflection coefficients. The parameterization of finite fuel, ash, and impurity recycle is a powerful and necessary tool with which to investigate the possibility of divertorless reactor operation [19]. 4.7. Assumptions Since a reliable data base for the more complex plasma-wall interactions is incomplete at the present time, we have not included several processes or effects which may play a significant role in recycling. These include the presence of oxide layers, the effect of wall erosion and radiation damage on the wall concentration model, the presence of water and hydrocarbons on the torus walls, chemical sputtering, the effect of sample history on the mechanical and trapping properties of the wall, and the synergistic effects of simultaneous hydrogen and helium implantation. Including such effects must await the creation of a set of reliable prescriptions for their behavior.
5. Sohrtion scheme The coupled ordinary differential equations describing the time evolution of the spatially averaged edge densities and temperatures are solved using the DRODE [20] difIerential equation package developed at Sandia, Lawrence Livermore, and Los Alamos laboratories. In response to the source and loss terms from the core and wall, the system of equations is carried to a quasiequilibrium. Typically, this occurs in the range 10-4-10-3s. Having obtained the spatially averaged parameter values, and assuming that the charged particle densities and temperatures in the scrape-off layer can be approximated by an expression of the form [21]
the boundary conditions for the core calculation,
&(a), can be determined. The e-folding length, ajp for variable j can be evaluated using different formulations. For example, Sj corresponding to the density is often given as
4 = [(D*$j]"2, where DL is the perpendicular diffusion coefficient and ~1 is the mean time for particle flow along the field line. As an economy measure the core calculation is advanced in time with these boundary conditions fixed until the core outflux has changed sufficiently to warrant another quasiequilibrium edge calculation.
6. Resdtsand discus&m Two scenarios are considered: a fixed boundary (case A) and an evaluated boundary (case B). The tokamaks parameters are: a major radius of 513 cm, minor radius 144cm, plasma current equal to 5 MA, and a magnetic field on axis of 6T. The scrape-off zone is 1Ocm wide. Transport scaling is neoclassical and empirical (l/n,). External heating is supplied by 100 MW of RF for 0.5 s. 6.1. Case A The edge temperatures are fixed at 5 eV, while the edge density is fixed at 1 x 10” cmm3. It is assumed that ash and impurity outflux is pumped while all fuel out&r crudely recycles to the plasma. Those charge exchange neutrals not reflecting from the wall are lost from the calculation. Figs. 3-6 show various profiles at 0.6 s into a plasma burn modelled with &red boundary conditions. Several figures of merit are also tabulated in table 2. Several observations can be made: (1) the temperatures near the edge are large (-8OOeV), (2) the gradients near the edge are artificially steep, (3) the edge particle confinement time is large (-0.5 s), (4) the crude model underestimates the erosion of both wall and limiter, and (5) the power loading to the limiter is excessive. Since sputtering and impurity transport are to edge temperature and edge sensitive gradients, it is readily seen that tixing the boundary can lead to unrealistic values.
M.R. Gordinier, R. W. Conn 1 Plasma edge model
for tokad~~
ELECTRON DENSITY
Fig. 3. Electron density versus radius at 0.6 s into plasma bum for case ‘A.
ELECTRON TEMPERFlTURE
i
s
O_ d-
Fig. 4. Electron temperatures versus radius at 0.6s into plasma bum for case A.
425
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
ION TEMPERRTURE
Fig. 5. Ion temperature versus radius at 0.6 s into plasma burn for case A.
S. S. a316
DENSITY
‘O-
fiYG.
UENSITY
TItiE (MSECI -
ICI+31 -
5.1D*lO'
599.85
Fig. 6. Ion impurity density versus radius at 0.6 s into plasma burn for case B.
M.R. Gordinier, R. W. Conn / Plasma edge model for t~h~ks
6.2. Case B In this simulation the edge model produces a set of self-consistent boundary values and influxes. Alpha ash and impurity outthrx are allowed to recycle freely to the plasma. Figs. 7-10 show various profiles at 0.6s into the burn with boundary conditions determined with the edge model. For a comparison with Case A, refer to table 2. In contrast to the fixed boundary calculation: (1) a cool (-lOOeV), thick (-9 cm) plasma blanket was created, (2) temperature profiles are observed to flatten near the edge, (3) the edge particle confinement time is substantially reduced due to rapid recycling (-0.07 s), (4) the more sophisticated model predicts enhanced sputtering, resulting in larger limiter-wall erosion rates, and (5) the limiter power loading is reduced by a factor of 3. These preliminary results underscore the need for accurate modelling of the scrape-off zone.
427
Table 2 Comparison of models. Case A has fixed boundary conditions. Case B utilizes the edge model described in the paper to compute edge conditions. The fusion power output is the same for each case Parameter
Unit
Case A
Case B
ri; n, nz T, T,
cm3 cm3 cm3 keV keV
nita)
cm3
6.7 x 10” 1.8 x 10” 3.1 x 10’ 5.7 6.6 1 x 10” 1 x 10’ 5x106 5 5 4.2 x lOI5 720 70 3 2.2 1 0.6
1.2 x 1.9 x 5.2 x 5.3 5.8 3.1 x 8.5 x 1.4 x 35 45 3.6 x 126 71 4.2 3.1
n&) h&) T.(a) Tife) Fcx Tat P, P rad PCX PI,
,“: eV eV cm-2. s-l eV Mw h4w
Mw hfw
7p b-k4
s
ELECTRON DENSITY RVD. DENSITY WI-31 -
1.19~1011
THE OtlSECI - 599.85
k A
0.0
0 18.0
t 36.0
l 54.0
7i.O
PLASflfl RRDIUS
I DO.0
ICIil
1 108.0
126.0
Fig. 7. Electron density versus radius at 0.6 s into plasma burn for case B.
144.0
7
0.07
10’4 10” 10’
10” 10’ 10’
lOI
MR. Godinier, R. W. Chn
/ Plasma edge model for tohmaks
ELECTRON TEMPERATURE 3 IWO. m m3I- 6.25 1IHE mst;elSW.85 b_ ?4-
T 0” r(-
Fig. 8. Electron temperature versus radius at 0.6 s into plasma bum for case B.
IONTEMPERATURE -a
%, A
I
0.0
18,O
1
33.0
I
I
M.0 72.0 FLRSIIR RRDIUS
I
CCf?”
I
105.0
I 128.0
Fig. 9. Ion temperature versus radius at 0.6 s into plasma burn for case B.
114.0
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
DENSITY
S. S. x316 /
\A”G.
I
18.0
DENSITY ICH-51 -
‘hrIIE
/
b__ _ - 0.0
I
,
35.0
54.0
PLRSNR
429
I
WXOI
I
72.0
90.0
RADIUS
-
5.21*10’
599.85
,
IO&O
I
126.0
.O
ICE0
Fig. 10. Ion impurity density versus radius at 0.6 s into plasma burn for case B.
Future parameter studies will examine and quantify in greater detail which plasma-wall processes have a substantial impact upon tokamak discharges and follow the plasma as it evolves further towards the steady state bum condition. 7. Summary A zero-dimensional model of the scrape-off zone has been developed for studying the impact of plasma-wall interactions on the discharge behavior in tokamaks. It incorporates atomic and molecular species, sputtering, sheath effects, noncoronal equilibrium radiation, parallel transport, recombination/trapping, a wall concentration model, and finite recycling of fuel and ash. The model produces the self-consistent boundary conditions and influxes required by the one-dimensional radial transport code. Calculations were presented for both fixed and calculated boundary conditions and influxes. Fixing the plasma density and temperature at the edge results in large edge gradients, high nearedge temperatures, a large limiter loading, and subsequent large impurity generation. Using the
edge model, a cool near-edge region is obtained with smaller gradients. The particle confinement time is reduced by a factor of 8, resulting in lower global temperatures. More sophisticated sputtering models predict somewhat enhanced impurity levels but the limiter power loading is substantially reduced. Nevertheless, all indications are that the limiter erosion rates would be very high (on the order of millimeters per year). The results show the importance of experimentally determined actual limiter loadings and methods for resurfacing both limiters and, potentially, armor plating. Acknowledgements We are indebted to Dr. D. Post at the Princeton Plasma Physics Laboratory for making his noncoronal equilibrium radiation code available to us, as well as Dr. M. Baskes of Sandia Laboratory-Livermore who provided us with the wall concentration package. In addition, we thank H. Attaya for his help and advice in performing the range calculations. This research has been partially supported by the US Department of Energy.
430
M.R. Gordinier, R.W. Conn I Plasma edge model for tokamaks
References [l] E. Hinnov et al., Plasma Phys. 10 (1968) 291. [2] J.T. Hogan, in: Proc. Symp. on Plasma Wall Interactions, Jiilich, 1976 (Pergamon, Oxford) p. 155. [3] S.J. Fielding et al., J. Nucl. Mater. 76/77 (1978) 273. [4] W. Houlberg and R.W. Conn, Nucl. Sci. Eng. 64 (1977) 141. [5] D. Post, Princeton Plasma Physics Laboratory, private communication. [6] E.L. Freeman and E.M. Jones, Cuiham Report CLMR137 (1974). [7] E.M. Jones, Cuiham Report CLM-R 175 (1977). [8] C.A. Finan, Lawrence Livermore Report UCRL-51805 (1976). [9] B. Peart and K. Dolder, J. Phys. B (Atom. Mol. Phys.) 5 (1972) 1554. [lo] B. Peat-t and K. Dolder, J. Phys. B (Atom. Mol. Phys.) 7 (1974) 236.
WI A. Riviere and D. Sweetman, Proc. Phys. Sot. 78 (1961) 1215. WI SO. Oen and M.T. Robinson, Nucl. Instrum. Methods 132 (1976) 647. 1131 D.K. Brice, Sandia Laboratory Report SAND 75-0622 (July 1977). [I41 M. Baskes, Sandia Laboratory Report SAND 80-8201 (Jan. 1980). P51 J. Roth et al., Max-Planck-Institut fiir Plasmaphysik Report IPP 9/26 (1979). WI D. Smith, J. Nucl. Mater. 75 (1978) 20. 1171 R.W. Corm et al., A.P.S. Bulletin,Vol. 22 (1977) 8D3. WI R.W. Conn et al., Problems of Fusion Research (IEEE, San Francisco, 1979). 1191 B. Badger et al., University of Wisconsin, UWFDM-330. PO1 M. Baskes, private communication. Pll A. Mense, Ph.D. Thesis, University of Wisconsin (1977).
M.R. GORDINIER Nuclear Engineeting
Department.
University of Wisconsin. Madison. Wisconsin, USA
and R.W. CONN School of Engineering
and Applied Science, University of California, Los Angeles. Los Angeles. California MQ4.
USA
The discharge characteristics in fusion experiments, particularly tokamaks, are often determined by the couplingbetween the core plasma and the region between the limiter and the vacuum wall. A zero dimensional scrape-off model is developed which incorporates the essential atomic physics and plasma/wall interactions. This model is coupled to a one dimensional radial transport code for the analysis of tokamak discharges. It is demonstrated that the discharge characteristics predicted
can be substantially altered by the inclusion of such a model.
1. introduction
Plasma simulation work to date has emphasized the development of models for the central region of the plasma discharge. Codes now exist which incorporate most of the essential physical processes present in the core region. By contrast, serious modelling of the “edge region” is only now receiving detailed attention. The scrape-off zone is recognized as being critical to the modelling of tokamak discharges since this region is the source of impurities which degrade plasma performance. Complex plasmawall processes have a strong effect on the overall density and energy balance by serving as sources and sinks of ions due to diffusion, trapping, and reflection phenomena. So strong is the coupling between core and edge that in many current devices agreement between simulation and experiment is only achieved by judiciously varying the phenomenological parameters present in crude recycling models. In assessing the possible impact of plasma-wall interactions on the burn dynamics of future fusion reactors, extrapolabased upon such device-dependent tions parameters is unwise. We have developed a genera1 mode1 of. the scrape-off zone of a tokamak which incorporates * Partially supported by US Department of Energy Contract no. DE-ASO3-76STtXKW
the relevant atomic physics and plasma-wall interactions with a minimum of adjustable parameters. Using this mode1 in conjunction with our one-dimensional transport code we have addressed the following questions. (1) Which edge effects have a substantial impact on the main-body plasma burn dynamics? (2) Can these processes be tailored to yield discharge improvements? (3) Can a better understanding of the physics of the scrape-off zone point the way to new operating scenarios and/or novel impurity and ash control schemes? 2. Boundary model Our model is based upon the previous work of Hinnov, Fallon and Bishop [l], Hogan [2] and Fielding [3]. The mode1 consists of a series of coupled ordinary differential equations for the density and temperature of the atomic and molecular species present in the scrape-off zone. The species are deuterium (D$, Dt, Do, D+) and tritium (T!, T;, T)‘, T+), alpha ash and, as an impurity, one of several wall materials (C, Ti, Va, MO, Fe). Coupling between species and isotopes occurs due to ionization, dissociation, and chargeexchange events. In addition, sources from the plasma core and from the diffusion of wall trapped gas into the scrape-off zone are included.
Jaumal of Nuclear Materials 93 & 94 (1980) 420-430 @ North-Holland Publishing Company
420
M.R. Gordinier, R.W. Conn I Plasma edge model for tokamaks
rethermalization, Radiation, charged-particle surface recombination, sputtering, sheath effects, and charged-particle parallel transport have also been incorporated. The model is self-consistent in that the edge density and temperature as well as the fuel and impurity influx to the core are evaluated by solving the coupled density and temperature equations subject to the constraint that the plasma outfluxes predicted by the one-dimensional radial code [4] are matched. The resulting influxes and edge values determine the boundary conditions for the radial model. This in turn determines the discharge characteristics of the core and the iterative coupling between core and edge.
(9)
00)
where S cure S*care Lll
3. Equations The zero-dimensional equations solved in the scrape-off zone for the spatially averaged densities and temperatures are:
L:, nlrp rtre
(2)
S Will S$“, S gas-Puff
(3) (4) (5)
= particle source from core, = power source from core, = particle loss in parallel direction, = power loss in parallel direction, = neutral particle flux to core, = neutral power fixu to core, rethermalization between specie i and j, = radiative loss of power, = loss/source of particles due to recombination at wall, = source of gas diffusing from wall, = sputtering source of neutral impurities, = neutral source due to gas puffing. =
Qi.j
L rad L recomb,
(1)
421
kmnb
For compactness, the above equations have been written in terms of hydrogen densities and temperatures. However, since the model treats deuterium and tritium as separate species, the above system represents 18 coupled ordinary differential equations plus two equations for alpha particles and impurities.
(6) 4. Modelling of plasma-wall physics 4.1. Impurities (7)
(8)
The impurity density and radiative losses are calculated using a modified version of the timedependent noncoronal equilibrium model developed at PPPL [5]. It solves for the number density of each charge state of the impurity taking into account sputtering sources, charged-
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
422
particle parallel losses and such atomic processes as impurity ionization, dielectronic recombination, three body recombination, etc. 4.2. Atomic physics After an extensive literature search [6-111, nineteen reactions were included in the model and represent the most probable reaction channels at characteristic scrape-off temperatures (table 1). Treating deuterium and tritium as separate species and allowing for cross-terms, the number of reactions increases from 19 to 55. Knowing the raw experimental cross-section data or analytic fits to such data, the 55 reaction rates were obtained by numerically integrating the expression for the reaction rate assuming two Maxwellian species.
Table 1 Hydrogenic atomic reactions e+H+H++2e e+Hz+Ht+2e e+Hr+H+H++2e e+Hr+2H+e e+Hs+H+H++e e+Ht+2H++2e e+Ht-+2H H++H+H++H H++Hr+Ht+H H;+Hr+Ht+Hz H++Hr+H++Hs+e Hi+Hz+Wf+e H++H+2HC+e H;+H+H++Hr Ht+H+H++H$+e H;+Hr+H++H+Hr Ht+Hr+2H++Hr+e H$+H+H++2H Hf+H+2H++H+e
4.3. Reflection-wall concentration model The reflection and/or trapping or ions and neutrals at the wall and limiter of a tokamak is modelled using the particle and energy reflection coefficients of Oen and Robinson [12]. For par-
titles not reflected from the wall, a calculation based on monoenergetic range theory [13] is performed to determine the spatial distribution of implanted atoms (see fig. 1). However, since
IMPLANTATION PROFILE a -. d
I R PLASMA TEMP.
DEPTH
-
100.00
[EVI
~flNGSTROMS1
Fig. 1. Depth profile for the implantation of deuterium into stainless steel. The incident deuterium energy distribution is a Maxwellian with a temperature of 100 eV.
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
423
MOLECULAR RELEASE RATE
IVPLRNT FLUX -
:: :: d 0.0
2.5I
5.0#
7.5I
TIME Fig. 2. Release rate of molecular
the incident flux is representative of a Maxwellian distribution of particles, the range calculation has been modified to reflect this. Given the net implantation profile for the spectrum of incident energies, application of a modified version of the time-dependent wall concentration model of Baskes [14] provides the release rate of implanted gas to the shadow region in response to concentration gradients (see fig. 2). 4.4. Sputtering Sputtering of the walls and limiter is assumed to occur by D, T, (Y,and impurity impact. These fluxes arise both from the core and shadow the yield regions. For light ion sputtering expression developed at Garching [15] has been integrated over a Maxwellian energy distribution of ions to obtain an average yield coefficient. For impurity self-sputtering, the yield expression developed by Smith [16] is used. In addition, the probability of impurity redeposition after sputtering is included as a phenomenological parameter. 4.5. Sheaths The presence of an ion accelerating sheath [17]
10.0I
ISECONDSI hydrogen
t.00n1015 [w'Clw+t2-SECI
9 12.5
15.0
I 17.5
2
0
from stainless steel versus time.
at the limiter-plasma interface has been included in modelling limiter sputtering and in evaluating the magnitude and energy of the hot recycling flux from the limiter surface. To illustrate the consequences of such sheaths, consider an edge electron temperature of 50 eV. Calculations show that at this temperature oxygen is six times ionized. Neglecting the effect of secondary electron emission or magnetic field, the sheath potential is approximately four times the electron temperature. Thus the oxygen ion would impact a poloidal limiter with an energy in excess of 1200 eV! Since sputtering and reflection coefficients are strong functions of energy, the sheath effect warrants inclusion in any realistic model of the edge. In addition, hot deuterium and tritium neutrals recycling from the limiter can penetrate more deeply into the discharge than neutrals characterized by the Franck-Condon process. 4.6. Parallel transport Charged particle parallel transport to the limiter is modelled by assuming free-streaming ions at a sound speed characterized by the local ion temperature. Both poloidal and toroidal
424
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
limiter geometries are considered. Given a specific limiter design, such as in ref. [18], with a prescribed pumping efficiency f, we allow (1 - f) of the incident parallel flux of D, T, a, and impurities to recycle into the plasma core at energies consistent with the presence of sheaths and finite energy reflection coefficients. The parameterization of finite fuel, ash, and impurity recycle is a powerful and necessary tool with which to investigate the possibility of divertorless reactor operation [19]. 4.7. Assumptions Since a reliable data base for the more complex plasma-wall interactions is incomplete at the present time, we have not included several processes or effects which may play a significant role in recycling. These include the presence of oxide layers, the effect of wall erosion and radiation damage on the wall concentration model, the presence of water and hydrocarbons on the torus walls, chemical sputtering, the effect of sample history on the mechanical and trapping properties of the wall, and the synergistic effects of simultaneous hydrogen and helium implantation. Including such effects must await the creation of a set of reliable prescriptions for their behavior.
5. Sohrtion scheme The coupled ordinary differential equations describing the time evolution of the spatially averaged edge densities and temperatures are solved using the DRODE [20] difIerential equation package developed at Sandia, Lawrence Livermore, and Los Alamos laboratories. In response to the source and loss terms from the core and wall, the system of equations is carried to a quasiequilibrium. Typically, this occurs in the range 10-4-10-3s. Having obtained the spatially averaged parameter values, and assuming that the charged particle densities and temperatures in the scrape-off layer can be approximated by an expression of the form [21]
the boundary conditions for the core calculation,
&(a), can be determined. The e-folding length, ajp for variable j can be evaluated using different formulations. For example, Sj corresponding to the density is often given as
4 = [(D*$j]"2, where DL is the perpendicular diffusion coefficient and ~1 is the mean time for particle flow along the field line. As an economy measure the core calculation is advanced in time with these boundary conditions fixed until the core outflux has changed sufficiently to warrant another quasiequilibrium edge calculation.
6. Resdtsand discus&m Two scenarios are considered: a fixed boundary (case A) and an evaluated boundary (case B). The tokamaks parameters are: a major radius of 513 cm, minor radius 144cm, plasma current equal to 5 MA, and a magnetic field on axis of 6T. The scrape-off zone is 1Ocm wide. Transport scaling is neoclassical and empirical (l/n,). External heating is supplied by 100 MW of RF for 0.5 s. 6.1. Case A The edge temperatures are fixed at 5 eV, while the edge density is fixed at 1 x 10” cmm3. It is assumed that ash and impurity outflux is pumped while all fuel out&r crudely recycles to the plasma. Those charge exchange neutrals not reflecting from the wall are lost from the calculation. Figs. 3-6 show various profiles at 0.6 s into a plasma burn modelled with &red boundary conditions. Several figures of merit are also tabulated in table 2. Several observations can be made: (1) the temperatures near the edge are large (-8OOeV), (2) the gradients near the edge are artificially steep, (3) the edge particle confinement time is large (-0.5 s), (4) the crude model underestimates the erosion of both wall and limiter, and (5) the power loading to the limiter is excessive. Since sputtering and impurity transport are to edge temperature and edge sensitive gradients, it is readily seen that tixing the boundary can lead to unrealistic values.
M.R. Gordinier, R. W. Conn 1 Plasma edge model
for tokad~~
ELECTRON DENSITY
Fig. 3. Electron density versus radius at 0.6 s into plasma bum for case ‘A.
ELECTRON TEMPERFlTURE
i
s
O_ d-
Fig. 4. Electron temperatures versus radius at 0.6s into plasma bum for case A.
425
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
ION TEMPERRTURE
Fig. 5. Ion temperature versus radius at 0.6 s into plasma burn for case A.
S. S. a316
DENSITY
‘O-
fiYG.
UENSITY
TItiE (MSECI -
ICI+31 -
5.1D*lO'
599.85
Fig. 6. Ion impurity density versus radius at 0.6 s into plasma burn for case B.
M.R. Gordinier, R. W. Conn / Plasma edge model for t~h~ks
6.2. Case B In this simulation the edge model produces a set of self-consistent boundary values and influxes. Alpha ash and impurity outthrx are allowed to recycle freely to the plasma. Figs. 7-10 show various profiles at 0.6s into the burn with boundary conditions determined with the edge model. For a comparison with Case A, refer to table 2. In contrast to the fixed boundary calculation: (1) a cool (-lOOeV), thick (-9 cm) plasma blanket was created, (2) temperature profiles are observed to flatten near the edge, (3) the edge particle confinement time is substantially reduced due to rapid recycling (-0.07 s), (4) the more sophisticated model predicts enhanced sputtering, resulting in larger limiter-wall erosion rates, and (5) the limiter power loading is reduced by a factor of 3. These preliminary results underscore the need for accurate modelling of the scrape-off zone.
427
Table 2 Comparison of models. Case A has fixed boundary conditions. Case B utilizes the edge model described in the paper to compute edge conditions. The fusion power output is the same for each case Parameter
Unit
Case A
Case B
ri; n, nz T, T,
cm3 cm3 cm3 keV keV
nita)
cm3
6.7 x 10” 1.8 x 10” 3.1 x 10’ 5.7 6.6 1 x 10” 1 x 10’ 5x106 5 5 4.2 x lOI5 720 70 3 2.2 1 0.6
1.2 x 1.9 x 5.2 x 5.3 5.8 3.1 x 8.5 x 1.4 x 35 45 3.6 x 126 71 4.2 3.1
n&) h&) T.(a) Tife) Fcx Tat P, P rad PCX PI,
,“: eV eV cm-2. s-l eV Mw h4w
Mw hfw
7p b-k4
s
ELECTRON DENSITY RVD. DENSITY WI-31 -
1.19~1011
THE OtlSECI - 599.85
k A
0.0
0 18.0
t 36.0
l 54.0
7i.O
PLASflfl RRDIUS
I DO.0
ICIil
1 108.0
126.0
Fig. 7. Electron density versus radius at 0.6 s into plasma burn for case B.
144.0
7
0.07
10’4 10” 10’
10” 10’ 10’
lOI
MR. Godinier, R. W. Chn
/ Plasma edge model for tohmaks
ELECTRON TEMPERATURE 3 IWO. m m3I- 6.25 1IHE mst;elSW.85 b_ ?4-
T 0” r(-
Fig. 8. Electron temperature versus radius at 0.6 s into plasma bum for case B.
IONTEMPERATURE -a
%, A
I
0.0
18,O
1
33.0
I
I
M.0 72.0 FLRSIIR RRDIUS
I
CCf?”
I
105.0
I 128.0
Fig. 9. Ion temperature versus radius at 0.6 s into plasma burn for case B.
114.0
M.R. Gordinier, R. W. Conn I Plasma edge model for tokamaks
DENSITY
S. S. x316 /
\A”G.
I
18.0
DENSITY ICH-51 -
‘hrIIE
/
b__ _ - 0.0
I
,
35.0
54.0
PLRSNR
429
I
WXOI
I
72.0
90.0
RADIUS
-
5.21*10’
599.85
,
IO&O
I
126.0
.O
ICE0
Fig. 10. Ion impurity density versus radius at 0.6 s into plasma burn for case B.
Future parameter studies will examine and quantify in greater detail which plasma-wall processes have a substantial impact upon tokamak discharges and follow the plasma as it evolves further towards the steady state bum condition. 7. Summary A zero-dimensional model of the scrape-off zone has been developed for studying the impact of plasma-wall interactions on the discharge behavior in tokamaks. It incorporates atomic and molecular species, sputtering, sheath effects, noncoronal equilibrium radiation, parallel transport, recombination/trapping, a wall concentration model, and finite recycling of fuel and ash. The model produces the self-consistent boundary conditions and influxes required by the one-dimensional radial transport code. Calculations were presented for both fixed and calculated boundary conditions and influxes. Fixing the plasma density and temperature at the edge results in large edge gradients, high nearedge temperatures, a large limiter loading, and subsequent large impurity generation. Using the
edge model, a cool near-edge region is obtained with smaller gradients. The particle confinement time is reduced by a factor of 8, resulting in lower global temperatures. More sophisticated sputtering models predict somewhat enhanced impurity levels but the limiter power loading is substantially reduced. Nevertheless, all indications are that the limiter erosion rates would be very high (on the order of millimeters per year). The results show the importance of experimentally determined actual limiter loadings and methods for resurfacing both limiters and, potentially, armor plating. Acknowledgements We are indebted to Dr. D. Post at the Princeton Plasma Physics Laboratory for making his noncoronal equilibrium radiation code available to us, as well as Dr. M. Baskes of Sandia Laboratory-Livermore who provided us with the wall concentration package. In addition, we thank H. Attaya for his help and advice in performing the range calculations. This research has been partially supported by the US Department of Energy.
430
M.R. Gordinier, R.W. Conn I Plasma edge model for tokamaks
References [l] E. Hinnov et al., Plasma Phys. 10 (1968) 291. [2] J.T. Hogan, in: Proc. Symp. on Plasma Wall Interactions, Jiilich, 1976 (Pergamon, Oxford) p. 155. [3] S.J. Fielding et al., J. Nucl. Mater. 76/77 (1978) 273. [4] W. Houlberg and R.W. Conn, Nucl. Sci. Eng. 64 (1977) 141. [5] D. Post, Princeton Plasma Physics Laboratory, private communication. [6] E.L. Freeman and E.M. Jones, Cuiham Report CLMR137 (1974). [7] E.M. Jones, Cuiham Report CLM-R 175 (1977). [8] C.A. Finan, Lawrence Livermore Report UCRL-51805 (1976). [9] B. Peart and K. Dolder, J. Phys. B (Atom. Mol. Phys.) 5 (1972) 1554. [lo] B. Peat-t and K. Dolder, J. Phys. B (Atom. Mol. Phys.) 7 (1974) 236.
WI A. Riviere and D. Sweetman, Proc. Phys. Sot. 78 (1961) 1215. WI SO. Oen and M.T. Robinson, Nucl. Instrum. Methods 132 (1976) 647. 1131 D.K. Brice, Sandia Laboratory Report SAND 75-0622 (July 1977). [I41 M. Baskes, Sandia Laboratory Report SAND 80-8201 (Jan. 1980). P51 J. Roth et al., Max-Planck-Institut fiir Plasmaphysik Report IPP 9/26 (1979). WI D. Smith, J. Nucl. Mater. 75 (1978) 20. 1171 R.W. Corm et al., A.P.S. Bulletin,Vol. 22 (1977) 8D3. WI R.W. Conn et al., Problems of Fusion Research (IEEE, San Francisco, 1979). 1191 B. Badger et al., University of Wisconsin, UWFDM-330. PO1 M. Baskes, private communication. Pll A. Mense, Ph.D. Thesis, University of Wisconsin (1977).