PFC/JA-%-1
MIT Contributions
to the 5th Plasma Edge Theory Conference*
Asilomar, CA
December 4-6, 1995
D. J. Sigmar, et al.
December 1995
MIT Plasma Fusion Center
Cambridge, Massachusetts 02139 USA
This work was supported by the US Department of Energy under contract DE-FG02-91ER-54109.
Reproduction, translation, publication, use, and disposal, in whole or in part, by or for the US
Government is permitted.
*These papers will appear in Contr. Plas. Phys. 36, Berlin (1996).
Preface
This report is a collection of 14 papers (1 invited, orals, posters) which were
presented at the 5th Plasma Edge Theory Conference in Fusion Devices in Asilomar,
California, December 4-6, 1995 (conference abbreviation PET-5).
The principal
authors are permanent or visiting researchers at MIT's Plasma Fusion Center, in
strong collaboration with US and international co-authors as named below.
The topics cover a rather wide range of current results in scrape-off layer and
divertor plasma physics which can be assigned to four groups:
1. An overview of the status and understanding of plasma detachment.
2. Analytical papers on tokamak edge transport theory, self-similar variable
solutions of neutral particle transport, effect of molecular recombination on
detachment, impurity radiation driven instabilities of the edge plasmas.
3. Semi-analytical/semi-numerical investigations of scaling laws for divertor
plasma checked against 2D divertor modelling, effect of including perpendicular
transport on impurity radiation driven losses, and 1D fluid theory of plasma-neutral
interaction effects on detachment.
4. Full scale numerical simulations of SOL divertor plasmas using (i) 2D fluid codes
(simulation of detached Alcator C-Mod divertor plasmas and of ITER divertor
plasmas); (ii) using the kinetic (particle following) codes W1 and W2 (kinetic models
of ELM bursts and detached and ELMy SOL plasmas) and a newly developed (finite
element) Fokker-Planck code ALLA (simulation of ELM bursts and of electron
transport) .
1
Divertor Plasma Detachment: Present Status of Understanding
S. I. KRASHENINNIKOV (ab)
(a) Massachusetts Inst. of Tech., Plasma Fusion Ctr., Cambridge, MA, U.S.A.
(b) Kurchatov Institute of Atomic Energy, Moscow, Russia
Abstract
We consider the main features of plasma detachment and try to find the most natural way to
explain them. In general, we focus more on the qualitative analysis of the problem,
illustrating the physics of the processes, rather than discussions of particular solutions of
the plasma transport equations.
1. Introduction
Experiments on most diverted tokamaks have demonstrated detached divertor regimes (see
Refs. 1-4) which are characterized by: 1) high energy radiation losses from the scrape off
layer (SOL) region; 2) low plasma temperature near the divertor plates; 3) relatively high
neutral gas density in the divertor volume; 4) strong decrease of the~plasma particle and
energy fluxes onto the plates; and 5) strong plasma pressure drop along magnetic field lines
in the divertor volume. In current tokamaks these regimes either occur at relatively high
plasma density (forced by hydrogen puffing) or are stimulated by impurity puffing. Due to
the very low heat loads on the divertor plates observed in these regimes, they look attractive
from the International Thermonuclear Experimental Reactor (ITER) [5] divertor design point
of view. Experiments on linear machines [5-91 also show strong decreases in the plasma
energy and particle fluxes onto the target, and strong plasma pressure drop along magnetic
field lines as the neutral gas pressure is increased. These are the main experimental
observations of plasma detachment (on both tokamaks and linear devices) which theory has
to explain. In this review we will focus more on the qualitative estimates illustrating the
physics of the detachment in general rather than on particular solutions of the SOL plasma
transport equations.
2
2. Qualitative Analysis of Energy and Particle Balance in SOL Plasma.
Before proceeding with the details we look at what useful insights we can get based on the
most general energy and particle conservation laws. Recall that divertor detachment regimes
in tokamak experiments were found for so called high recycling conditions. For these
conditions the plasma flux from the bulk to the SOL plasma, r, is much smaller than the
plasma flux onto the target, rd, (see Fig. 1) which is determined by the ionization and
recombination processes (neutral and/or plasma recycling) taking place in the SOL region
with open magnetic field lines. The inequality r
r,<<d implies some restrictions on
plasma/neutral parameters. The neutral ionization mean path, X ion, should be smaller
than characteristic plasma width in the divertor, A,, so that neutrals (impurity neutrals in
particular) will be ionized before they reach hot bulk plasma. Notice, that for a deep slot
divertor this requirement can be replaced by XKim<<Lslot, where L.i 0 t is the slot depth.
Rather simple access to high recycling conditions is a very important advantage of diverted
tokamaks in comparison with limiter machines. For rs/rd <<l, in the zero order
approximation one can neglect the influence of the particle flux r, and consider the SOL
region a "closed box" with a fixed number of the particles in it. Only an energy flux Qs
comes into the SOL from the bulk plasma in this extreme limit.
One intuitively expects that with decreasing Q, the mixture of plasma and
neutrals in this "closed box" becomes cooler, so that the number of ionization events drops
resulting in a decrease of plasma flux rd. Indeed, since ionization of the neutrals "costs"
energy Eien, the energy (Qs) and the particle (rd) fluxes must satisfy inequality
rd < Qs/Ej 0 0 , confirming our intuition. Here Eion is the energy "ionization cost" of a
neutral particle; for hydrogen atoms at an electron temperature larger than 3-4 eV, Eon -30
eV. Moreover, if some part of the energy flux Qs can be dispersed in the SOL due to
impurity radiation and/or energy transport to the walls by the neutrals, then less energy is
left to ionize neutrals and we expect that the plasma flux will decrease. Indeed, an estimate
for rd based on an energy balance in the SOL plasma gives
rd s(Qs -Qd -QN)(Eion +ypTd)< Q/Eion
(1)
where Qr and QN are the energy losses from the SOL due to impurity radiation and
neutral flux onto the sidewalls and target, Td is the plasma temperature near the target, and
y, -5-8 is the plasma energy transmission coefficient. The term y , Td in Eq. (1) accounts
for the energy removed from the "gas box" by the plasma flux onto the target. Generally
speaking, the temperature Td is unknown and must be determined self consistently. But, in
the limit (Q, - Qr - QN) - 0 one can make the reasonable and natural assumption that
3
Td decreases monotonically to zero. Then, from Eq. (1) we conclude that plasma flux onto
the target decreases with (Qs - Qrad - QN) -- 0 Next we consider whether the decrease in the plasma flux Td is compatible with
the neutral influx into the high temperature region where neutrals are immediately ionized.
With increasing energy loss one might expect that the neutral gas density near the walls
remains the same or increases (see for example Ref. 10), resulting in a constant (or even
increasing) influx of neutrals into the plasma. But how can the decrease in rd be balanced
by what appears to be an increase in the neutral influx? There are two natural explanations
of this "paradox". The hot temperature region of the SOL plasma is screened from the
neutrals by a relatively dense cold (no neutral ionization!) plasma layer (see Fig. 2) with
characteristic width
Ac>%Ni,
(2)
where %Ni VThi/vNi =VTi/KiNnc, nc is the plasma density in the cold layer, VTi is the
ion thermal velocity, and KiN is the ion-neutral collision rate constant. Therefore, the
neutral influx into the hot temperature region, rN, is determined by a slow diffusive like
neutral flow through the dense cold plasma layer, which results in the inequality
FN "" rN,FS(%N-I/AC <<N,FS, where rN,FS is the free streaming neutral flux. We will
call this layer a "plasma baffle", since it effectively rejects a major part of the oncoming
neutral free streaming flux. The second explanation is that the plasma recombines before it
reaches the targets. Since plasma recombination only becomes effective at low temperature
and relatively high plasma density (11-13], in practice both these mechanisms probably
work together. Moreover, plasma recombination alone can not explain the rd decrease
when
(Qs - Qrad
- QN) -+ 0 and FN - const., since ionization events take some energy
from the plasma which can not be released back completely during recombination even in
the optically thick plasmas. Notice, that for a deep slot divertor the reduction of rN (for the
same neutral pressure at the bottom of the slot) could easily be explained by the shift of the
hot plasma region further up from the bottom, which results in a decrease of neutral flux
due to an increase in the vacuum resistance of the channel.
What can we say about the variation of plasma pressure near the target? Assume
a plasma flow velocity near the target of the order of the sound speed (along the magnetic
field line), Cd - Td /~M, where M is the ion mass, then we find
rd
"StndCdsinV
(3)
aStPd/vff
where Pd - ndTd, nd is the plasma density near the target, and St is the area of the target
exposed to the plasma flux, and V is the field line angle with the target. From Eqs. (1), (2)
for St - const. we find
4
Pd
(Qs -Qrad
-QN){Td /(EiOn +ypTd)
-
(Qs - Qrd - QN) -- 0, from Eq. (4) we
the plasma pressure near the target decreases as (Qs - Qad - QN) -*0 For a monotonic dependence of Td on
(4)
find that
In the preceding we discussed the importance of a plasma baffle for neutral
transport. Simultaneously however, a very significant change in the physics of the plasma
flow can occur in the plasma baffle region due to strong plasma-neutral coupling. Indeed,
we can estimate the ratio %iN/IC, where %iN= VTi/KiNNdj is the ion-neutral collision
mean free path, Nd is the neutral density in the plasma baffle, and 4c is length of the
magnetic field line in this layer. From geometrical considerations we find 14 a Ac/sinij.
The neutral density Nd can be estimated by balancing of the plasma flux onto the target,
Eq. (3), with the outgoing neutral flux NjdIFM/th anc4ff/Msin V, where Tc is the
neutral/plasma temperature in the layer. Substituting this estimate for the neutral density into
the expression for )4N we find
(5)
%iN/1c SXNi/Ac<l-
Inequality (5) shows that the ion-neutral mean free path is smaller than the length of the
magnetic field line in the cold layer and, therefore, plasma flow through the plasma baffle
region can be significantly affected by the neutral dynamics.
Next, consider the main features of energy loss due to impurity radiation and
plasma-neutral interactions. The energy loss due to impurity radiation can be roughly
estimated as
Qrad " imp(np)2 Vp,rad,
where n, and
(6)
imp are the plasma density and impurity fraction, and Vp~md is the
radiating SOL plasma volume. The estimate for the energy loss due to the neutral flux onto
the target gives
QN O SN NTWCW,
(7)
where N is the neutral density, SN is the area of the target and sidewalls exposed to the
neutral flux, Tw is the plasma/neutral gas temperature near the walls and target, and
C -- jw,/M . As can be seen from relations (6) and (7), impurity and neutral energy
losses have different natures; impurity radiation loss is volumetric, while neutral energy is
lost at a surface. However, since both are proportional to densities they could increase with
an increasing number of particles in the "closed box".
We introduce the average density of ions + neutrals in the "closed box", !i+N,
and make a "gedanken" experiment by going from low to high ii +N while keeping the
energy flux Qs fixed. At low average density, both impurity and neutral energy losses are
5
small and plasma temperature is high. Increasing nj+N increases impurity radiation loss,
which results in a decrease of the plasma temperature, and therefore in an increase of both
neutral density and QN. If we would increase ni+N more, it would result in a further
decrease of the temperature and an increase in neutral density, so that V p,
and/or np
would have to decrease to sustain energy balance. At higher i+N, temperature Tw
becomes very low and plasma recombination can be important. In the extreme limit of very
high !i+N, the divertor plasma cools down so strongly that neutrals penetrate into the core
(XN, ion a p) causing a global change of the discharge (MARFE, disruption, etc.) and our
"closed box" model can no longer be applied. Notice, that neutral penetration into the core
could be sensitive to the divertor geometry (length of the divertor leg, width of the divertor
channel, etc.)
The critical value of averaged density, ncrit, corresponding to the transition
from the high to the low temperature regime, can be estimated from dimensionless analysis
(see Ref. 14 for a more complete consideration). In a simple ID model of plasma and
neutral gas flow interactions in a "closed box" there are seven dimensional parameters:
length of the ID box in the direction perpendicular to the target, Lbox; averaged (ion +
neutral) density, ii; density of the energy flux into the 1D 'closed box', q.; charge, e;
mass M; magnetic field strength, B; light speed, c; and hydrogen ionization potential, I
(for convenience we take hydrogen ionization potential instead of Planck's constant, h, to
characterize the atomic processes). From these seven dimensional parameters one can only
obtain four meaningful dimensionless parameters
I2
qs
i
/
IM
_2
eBL b, ,,
e14
(8)
A fifth-parameter is the ratio of the velocities 4ri/c which is just a constant. From
expressions (8) we findqs - n t I 1-IM
, where
the function F can not be found from
dimensional analysis, F - F{2/(e4ncrit Lbox),ncrit
I) ,(c4fME)eBLJox)}. Notice,
that generally speaking F also contains other dimensionless parameters like imp, electronion mass ratio, inclination of the magnetic field to the target, etc. In the simplest case, the
critical density will have a power scaling
nxiit oc(q 8 )aq (Lb ox)aL BaB...,
(9)
where the numbers aq, aL, aB, ... have to be determined from a more detailed analysis.
For example, one can use the results which were obtained numerically in Ref. 15, where a
1D "closed box" model was considered for a pure hydrogen plasma. In Fig. 3 we plot ncrit
6
from [15] (defined as the average density resulting in Td= 2O eV) as a function of q.. From
Fig. 3 we see that ncit obeys a power law scaling, with aqmO.61.
We summarize the features of plasma detachment which we can qualitatively
explain with our simple arguments based on a "closed box" model. We found, in
accordance with experimental observations [1-4], high energy radiation loss from the SOL
region due to impurity radiation (see point 1 from Introduction) results in low plasma
temperature near the divertor plates (point 2), increasing neutral energy loss and neutral
density in the divertor (point 3), and (simply due to the energy balance in the "box"!)
decreasing plasma particle and energy fluxes onto the targets (point 4), Eq. (1).
Simultaneously, plasma pressure near the target decreases; recall Eq. (4). Notice, that in
agreement with experiments, detachment occurs at high plasma density and/or at high
impurity fraction, since both these factors result in increasing impurity radiation loss, Eq.
(6). We have argued that for tokamaks with relatively open divertor geometry, as in current
machines, plasma detachment can be accompanied by the formation of the cold dense
plasma layer (plasma baffle), Fig. 2, which decreases penetration of the neutrals into the hot
temperature region. Recent experimental data from DIII-D [16], obtained with divertor
Thomson scattering, support this conclusion. But the formation of this layer could be
sensitive to the divertor geometry. Simultaneously, the dynamics of plasma flow in this
layer can be strongly affected by plasma -neutral coupling, Eq. (5), which can result in a
parallel gradient of the plasma pressure (point 5). We also found that plasma recombination
can be an important mechanism for the reduction of the plasma flux onto the target.
Thus, we see that there are three key issues for plasma detachment: i) impurity
radiation and impurity dynamics in the SOL, ii) plasma-neutral interactions in the hydrogen
recycling region, and iii) divertor geometry. Notice, that neglecting the effect of plasma neutral interactions on the SOL plasma parameters results in the absence of a steady state
solution [17, 18].
3. Plasma-Neutral Interaction in Recycling Region at Low Energy Flux.
In this section we consider the main features of plasma and neutral flows in the
hydrogen recycling region assuming that the major part of the energy flux has already been
removed by impurity radiation. As we mentioned, this problem is rather sensitive to divertor
geometry, but some general conclusions about the character of this flow can still be made.
Recall that for a relatively open divertor geometry the width of the plasma baffle
region Ac should be much larger than the neutral-ion collision mean free path %Ni, Eq.
(2). Simultaneously there is a strong coupling of plasma and neutral flows, Eq. (5). Since
Ac>X Ni, neutral transport can be treated in a short mean free path approximation by fluid
7
transport equations. These fluid equations can be based on either neutral-ion coupling [15,
19], or on Navier-Stokes like equations [20, 21] retaining neutral-neutral collisions. In
these regimes the neutrals affect the coupled plasma-neutral flow by their relatively high
neutral viscosity and heat conduction terms, and by the gradient of the neutral pressure [15,
20-23]. Indeed, an estimate for the neutral diffusivity, DN, gives DN oc
Ti) 2 /nKiN
(neglecting the effect of neutral-neutral collisions). Assuming that perpendicular plasma
diffusivity is described by a Bohm diffusion coefficient, DB OC
(VTi) 2 /wiB,
where WiB is
the ion cyclotron frequency, we find that the neutral heat conduction coefficient,
X N - N DN, exceeds that of the plasma, xP - n DB, when
N/n> viN
iB ,
(10)
where viN - NKiN is the ion-neutral collision frequency. In the magnetized plasmas of
interest we have viN/WiB<<1 , therefore, even a small neutral fraction can significantly
alter plasma momentum and energy fluxes. For N -1014 cm-3 , KiN - 10-8 cm-3 /s, and
WiB 108 s- 1 , we find N/n> 0.1 from Eq. (10).
For a deep slot divertor, a plasma baffle need not exist, since the vacuum
resistance of the slot can prevent neutrals from penetrating into the hot temperature region.
Therefore neutrals in a slot divertor can be in the Knudsen regime, with a neutral mean free
path larger than the width of the slot, Asje. In this case, neutral transport is decoupled from
the plasma and affects plasma flow through particle, momentum, and energy sinks and
sources [19, 24-27].
In [15, 22, 23] plasma and neutral fluid approximations were applied for 1D
models (in a half space domain) of a hydrogen recycling region (no neutral-neutral
collisions and plasma recombination). In [20, 21] the Navier-Stokes like neutral models
were applied for realistic 2D divertor geometry and the impurity radiation losses were taken
into the account. Plasma-neutral interaction in the limit of a Knudsen type neutral flow was
studied in [22, 23-27]. In [28, 29] neutral transport was treated with the Monte Carlo
codes. The main conclusions which can be made analyzing the results of these
investigations are: i) a plasma baffle occurs and prevents penetration of the neutrals into the
high temperature region, and ii) plasma flux starts to decrease at a very low plasma
temperature near the target (s 1 eV), while significant plasma pressure variation along the
magnetic field line is already occurring (see Fig. 4, taken from [23], and Fig. 5, taken from
[27]). The physical reasons that the plasma flux onto the target starts to decrease at a very
low plasma temperature near the target are different for the fluid and Knudsen neutral
models.
8
For Knudsen neutrals the rough explanation of this effect is simple. The plasma
flux can be significantly reduced in comparison with free streaming one, if each ion
undergoes at least a few collisions with neutrals before it reaches the target. But,
simultaneously, due to the ion-neutral collisions the ion temperature tends to approach that
of the neutrals, To, which for Knudsen regimes is very low, and could be well below 1
eV. This strong plasma cooling can not be avoided since the plasma temperature upstream
of the ionization zone is rather low, T<Tj w5-7 eV, and the plasma heat conductivity is
not important. Thus, for the Knudsen limit of neutral transport the decrease in the plasma
flux onto the target is automatically accompanied by a strong reduction of the plasma
temperature to the level of the neutral gas temperature To< 1 eV.
For fluid neutrals the reason that the plasma flux onto the target starts to
decrease at.a very low plasma temperature near the target is a shallow angle between the
magnetic field lines and the target surface, sin V <<1. Indeed, the energy flux coming to the
target, qd, can be written as qd=Cd(y p Pd sin V + YNP N, d), where Y N is the neutral heat
transmission coefficient, and PN,d is the neutral pressure near the target. Since sin V <<1,
then even for the relatively low value of the neutral pressure PN, d, neutrals can dominate
the heat deposited onto the target. Recall the coupled plasma-neutral parallel momentum
balance. The only reason the total (plasma+neutral) pressure near the target can differ from
the upstream plasma pressure, Pu, is the influence of the neutral viscosity term. However,
this influence decreases with the decreasing plasma flux onto the target, which drives
viscosity. Therefore, if the plasma flux onto the target significantly drops, then the neutral
pressure near the target should balance the upstream plasma pressure, which (recall
sin V<<1) results in a drastic drop of the temperature near the target. In [22] the asymptotic
limit (PN,d - Pu) of the low temperature fluid neutral model was found to be
jd " Ar
4
, and nd - const..
In both fluid and Knudsen neutral models the plasma flux onto the target does
not decrease until a very low temperature is attained near the target (below 1 eV). As a
result, the plasma density in front of the target remains almost constant and can be very high
(a 1015cnr 3 for Td - 1 eV) even in current tokamaks. Then plasma recombination could
significantly alter the plasma parameters near the target For a plasma temperature Td - 1 eV
and density nd 10 15 cm- 3 the conventional three body recombination rate constant (see for
example [12]) is about 3 x 10-12 cm3/s. Recombination becomes important when the plasma
flux onto the target, jd, is comparable to the plasma particle sink due to recombination,
vmcndi=, where frec is the width of the low temperature region where recombination
occurs. For jd 1019 cm- 2 s-1 we find that recombination becomes important when
9
rec a id/vrnd -3 cm which is quite a realistic value. A strong effect of conventional
three body plasma recombination on the divertor plasma parameters was found in [20, 21].
In Fig 6. (taken from [21]) it is shown that for the C-MOD case the reduction of the particle
flux onto the target in the detached regime is caused by plasma recombination which almost
balances the plasma particle source due to neutral gas ionization.
In [12] it was pointed out that the formation of negative hydrogen ions H~ due
to dissociative attachment of the electron to ro-vibrationally excited molecular hydrogen,
H2 0,v)+ e -- H + H,
(11)
may enhance plasma recombination through charge exchange recombination
H + H -- 2H.
(12)
A more detailed (but still incomplete, because of unavailable atomic data) analysis of the
influence of molecular hydrogen on plasma recombination was performed in [13]. It was
shown that the main influence on plasma recombination is expected to be due to ion
conversion into molecular ions
H 2 0,v)+ H+ -- H2 + H,
(13)
and further dissociative recombination
H+ +e -- 2H.
(14)
An estimate of the effective plasma recombination frequency, vdg, was obtained in the
temperature range 1 eV<T,<4-5 eV and n+/N a 10-1, and gave
vf -3x 10-10 [cm 3 /s] [H 2 ],
(15)
where [H 2 ] is the molecular hydrogen density. For [H 2 Yn, - 0.1 and T. =1 eV, vgy is
30 times higher than the conventional three body recombination frequency and does not
exhibit the strong explicit plasma temperature and density dependence of three body
recombination.
4. Conclusions
Theory is able to explain qualitatively the main experimental features of detached
divertor regimes found in tokamaks and linear machines. There is reasonably good
quantitative agreement of the 2D modeling with experimental results. The key elements
affecting plasma detachment are: i) impurity radiation and impurity dynamics in the SOL, ii)
plasma-neutral interaction in the plasma baffle region, and iii) divertor geometry.
However, we do not understand quite well such important things as: i) impurity
and helium transport in detached plasma, ii) effects of molecules on plasma recombination
and energy loss, iii) kinetic effects for detached regime, which is characterized by a strong
plasma parameter gradients, iv) stability of detached regime and turbulence in detached
plasma, v) scaling for detached plasma, vi) effects of ExB drift and electric current on
10
impurity and plasma transport. There are some discrepancies between the results from
different codes (for example, between Monte Carlo and fluid neutral codes).
Also, probably, it is necessary to pay more attention to the interpretation of the
experiments on linear machines.
Acknowledgments
I wish to acknowledge useful discussions with 0. Batishchev, K. Borrass, P. Catto, S.
Cohen, J. Connor, P. Helander, I. Hutchinson, G. Janeschitz, D. Knoll, A. Kukushkin,
B. Labombard, K. Lackner, B. Lipschultz, A. Loarte, G. McCracken, J. Neuhauser, A.
Pigarov, G. Porter, D. Post, T. Rognlien, R. Schneider, D. Sigmar, T. Soboleva, R.
Stambaugh, P. Stangeby, S. Takamura, G. Vlases, and F. Wising.
Performed under the US DOE grant DE-FG02-91-ER-54109.
References
[1] Hutchinson I. H., Boivin R., et al., Physics of Plasmas 1 (1994) 1511.
[2] Janeschitz G., et al., 19th EPS Conf. on Contr. Fus. and P. Physics. Insbruck, 1992,
11-727.
[3] Petrie T. W., Buchenauer D., et al., Journal of Nucl. Materials 196-198 (1992) 848.
[4] Mertens V., et al., 20th EPS Conf. on Contr. Fus. and Pl. Physics. Lisboa, 1993,
1-267.
[5] International Thermonuclear Experimental Reactor (ITER) Conceptual Design Activity
Final Report, ITER Documentation Series No. 16 (IAEA, Vienna, 1991).
[6] Hsu W., Yamada M., Barret P., Phys. Rev. Lett. 49 (1982) 1001.
[7] Schmitz L et al., Physics of Plasmas, 2 (1995) 3081.
[8] Ohno N., et al., J. Nucl. Mater. 220&222, (1995) 279.
[9] Chiu G. S., Cohen S. A., J. Nucl. Mater. 196&198 (1994) 876.
[10] Kurz C. et al., Bulletin of the American Physical Society, 40 (1995) 1702-1703.
[11] Krasheninnikov S.I., Pigarov A.Yu. 11th IAEA Conf. on Pl. Phys. and Contr. Nucl.
Fus. Res. Kyoto, 1986, V1, p.387; Contr. Pl. Phys. 28 (1988) 443; 34 (1994) 442.
[12] Post D. B. J. Nucl. Mater. 220&222, (1995) 143.
[13] Krasheninnikov S. I., Pigarov A. Yu., and Sigmar D.J. these proceedings.
[14] Catto P.J., et al., Physics of Plasmas, 1996.
[15] Krasheninnikov S. I. et al, 12th EPS Conf. on Contr. Fus. and Pl. Physics. Budapest,
1985, 11-500; Nucl. Fus. 27 (1987) 1805.
[16] Porter G., Bulletin of the American Physical Society, 40 (1995) 1693.
[17] Borrass K., Nuc. Fusion 31 (1991) 1035.
[18] McCracken G.M., Pedgley J.M., Plasma Phys. Contr. Fus. 35 (1993) 253.
[19] Helander P. et al., Phys. of Plasmas 1 (1994) 3174.
[20] Knoll D. et al., Physics of Plasmas, 1996; these proceedings.
[21] Wising F. et al, Bul. of the Am. Phys. Society, 40 (1995) 1879; these roceedings.
[22] Krasheninnikov S. I. et al., Physics of Plasmas, 2 (1995) 2717.
[23] Soboleva T.K., et al., these proceedings.
[24] Nedospasov A.V., Tokar' M. Z., in Fus. Reactor Design and Techn., Vol. 2 IAEA,
Vienna (1987) 113.
[25] Stangeby P. C., Nuc. Fusion 33 (1993) 1695.
[26] Ghendrih Ph., Phys. Plasmas 1 (1994) 1929.
[27] Batishchev O.V. et al., these proceedings.
[28] Simonini R. et al., 21th EPS Conf. on Contr. Fus. and Pl. Physics. Montpellier,
1994, 11-694.
[29] Neuhauser J., et al., 21th EPS Conf. on Contr. Fus. and Pl. Physics. Montpellier,
1994,11-705.
11
s
bulk
plasma
plasmna
hot
pam
p
plasma
Pal
,AC
rd
neutrals
d
neutrals
Fig. 1 Schematic of plasma and neutral
fluxes in a diverted plasma.
Fig. 2. Cold plasma layer (baffler) in a
divertor.
---
2.,
---
0.4-
ID(n
--
1.5.
0.3-
-
Pd
PNd
~*Jd
0.2-4a~m
U-43
-1.5
-0.5
0.1
-
mx=8a
U'.
1.5
0
0.5
I
0.5
1.5
Td,
10q,)
Fig. 3. Dependence of ncit on q.,
calculated with data from [15].
-
Total ionization and
volume recombination (s 1)
IonIZO%C
W
6E+2
5E+2
40-
3E+2
--
--
-
--
%
SUN
*
4E+21
-d--Td
d/ u
30.
20.
1 E+2
A
o
10-
a
40
sOpM
I
CWTe"t pedk, O%C
60
lonIz. 0.5%C
a
Rec. 0.5%C
A A&
15
20
25
30
35
t Nose
Curreni peak. O.5%C
Field line number
Impurity density (a. u.)
Fig. 5. Plasma flux and temperature
dependence on impurity fraction [27].
Roe. O%C
A
~Ar
0b fo
20
0
1~an
-.-
2E+2
0
eV
Fig. 4. Plasma and neutral pressures, and
plasma flux at the target versus Td [23].
7E+2 1
50-
2
Fig. 6. Ionization/recombination balance
modeling in C-MOD [21].
12
Tokamak Edge Transport Theory
R. D. HAZELTINE (a) and PETER J. CATTO (b)
(a) Institute for Fusion Studies, Univ. of Texas at Austin, TX, U.S.A.
(b) Massachusetts Inst. of Tech., Plasma Fusion Ctr., Cambridge, MA, U.S.A.
Abstract
Kinetic equations commonly used to study the plasma core of a tokamak do not allow a
balance between parallel ion streaming and radial diffusion and are, therefore, inappropriate
in the plasma edge. Standard core tokamak transport orderings allow large divergence-free
flows in flux surfaces, but only weak radial flows. Alternate orderings are required in the
edge region where radial diffusion must balance large parallel ion flows to divertor target
plates or limiters. Similarly, core transport formulae cannot be extended to the edge region
without qualitative alteration. Here we address the necessary changes by considering a large
parallel flow ordering appropriate for the scrape-off layer of a tokamak. By deriving and
solving a novel kinetic equation, we construct distinctive transport laws for an impure,
collisional edge plasma. We find in particular a surprising form for parallel transport in the
scrape-off layer, in which the parallel flow of particles (and heat) are driven by a
combination of the conventional gradients, as well as viscosity terms and new terms
involving products of radial derivatives of the parallel mean velocity with density and
temperature gradients. The new terms are not relatively small, and could affect
understanding of limiter and divertor operation.
1. Introduction
In the core region of a toroidal plasma confinement system, flow on a magnetic surface is
not intrinsically associated with plasma loss; thus it can be first or even zeroth order in the
gyroradius-comparable to the ion thermal speed-as long as it is divergence-free.
However, diffusive flow across flux surfaces must be small to allow quasistatic evolution; it
is usually assumed to be second order in the gyroradius [1]. Because of this distinction, the
two types of flows are not directly coupled and do not appear in the same kinetic equation.
13
The situation near the plasma edge is quite different since the scape-off layer (SOL)
is affected by a collecting surface, either a limiter or divertor plate, which is intercepted lu
field lines. As a result, parallel flow becomes a rapid loss mechanism, which must be
balanced, by radial diffusion into the layer in the quasistatic state. The resulting near
equilibrium is often described phenomenologically by the particle conservation law [21
V -(NV) - Da 2 N/r 2 = 0,
(1)
where N is the ion density, VII the parallel ion velocity and D a diffusion coefficent.
Various versions of this balance have become familiar in the literature [31 where it is used in
particular to estimate the SOL thickness. The parallel flow is determined by conditions close
to the collecting surface (such as the Bohm sheath condition [2]) but is usually comparable
to the ion thermal speed or sound speed. Such rapid parallel motion can be balanced by
radial transport only because of the relatively steep gradients associated with the layer.
Kinetic equations commonly used to study the plasma core [1,41 are inconsistent with the
orderings implicit in Eq. (1), and therefore inapplicable in the plasma edge.
Here we consider an ordering appropriate for the SOL, and derive and solve a novel
kinetic equation to obtain distinctive transport laws for the plasma edge. For simplicity and
concreteness, we suppose that radial diffusion results from classical Coulomb collisions of
ions with a heavy impurity species. However it will be clear that the most striking features
of our results would persist were classical transport replaced by some anomalous or
turbulent diffusion process.
2. Edge Orderings, Kinetic Equation, Moments and Fluxes
The orderings emphasize two important properties of the edge: (i) the relatively fast flow of
ions along the magnetic field, VII - vt, where VII is the parallel flow speed and vt =
(2T/m) 1/2; and (ii) the relatively steep gradients in the radial direction, a/ar -11W, where r is
the radial variable (constant on flux surfaces), and W = the scrape-off layer width, is small
compared to the toroidal minor radius a, W <<a. We are interested in magnetized SOLs so
W remains larger than the ion gyroradius, p, resulting in three spatial scale lengths:
p << W << a,
(2)
where we take parallel and poloidal scale lengths to be comparable and of order a. The first
of these inequalities is used to define the basic small parameter, 8 = p/W << 1,
characterizing a magnetized SOL The second ratio, W/a, is treated as an independent small
parameter.
14
It is convenient to express the radial diffusion coefficient D in the form
D = vp 2 ,
with v
=
(3)
ThzNz~e~nA / m vi the ion-impurity collision frequency, p = vt/Q, Q=eB!mc
2
the ion gyrofrequency (Z=1), and where the subscript z denotes impurities. Then the basic
ordering describing the edge becomes
wt - v62,
(4)
where motion within a flux surface is measured by the conventional transit frequency, we =
vt/a. Consequently, the edge is magnetized only if it is collision dominated.
We employ a large drift velocity form of the drift kinetic equation [5]. To write the
Lorentz operator for ion-impurity collisions in the simpliest possible form, the frame
velocity V is chosen to be that of the impurities with V11 - vt >> V1 . Ion-ion collisions are
neglected for simplicity by assuming Z2N,>>N so that the standard results can be
recovered from the electron-ion Lorentz gas results by replacing the electrons (ions) by the
ions (impurities).
Using y and F = (v-V) 2/2 to denote gyrophase and energy, and letting v-V=s+u
with s=s(e2 cosy - e3 siny), u=nn- (v-V)=un, n=B/B, and n=e2xe3, the equation for the
gyroaveraged ion distribution function f=<4> for the edge orderings can be shown to be
(u+V)-V+()+
as
ss
\ar
--
f)
Or as)
(C(f)) + O(
.
(5)
\w
where <...> denotes gyroaverage, =fm -N(M/2xT)f 2 exp[-Mc/T] to lowest order, and
we neglect magnetic drifts (and, therefore, gyroviscosity) by assuming v6 >> ot. The
gyrophase dependent distribution function i =f -f satisfies
a
ay
-scosy(
m
k a-
aVafm"+C(i)+...,
Or as )
(6)
to the requisite order. Solving iteratively for the leading cosy tenn in i by assuming v << Q
and inserting the result in Eq.(5) gives the desired edge ordered kinetic equation in which
parallel streaming and radial diffusion enter to the same order.
(u +V)-Vfm +(c)
1
as
- C(f - fm) +D(fm),
(7)
with =u/(2e)11 2 W I n-VV- n -- V-V, E-= -V 1 D - (M/e)(V-VV) 1 ,
3
vv-s2 [rafm + 6uaV11(
0
ar jp,
Or Tse)l2(2)3/Z
DM\)D (r fMf
and
e(2c)1 '
(0) -ZeV-Y+
3
e(2E2 K+
E
2
3
-
M
15
eW,(1- 32(9)
(8)
Using the conservative form of Eq.(7),
Bu
B
lu
(+V)fm
+Of
IB
De
u
(c)f
=C(f - f.)+D(fsj)
(10)
with u - ±(2E - pB/M )112, we form the conservation of number, parallel momentum and
energy equations moment equations. Defining p = NT, the classical diffusion coefficient
De=4vp 2/3xl/2 , and the parallel friction F=Mfd3 vuC(f), the lowest order conservation
equations may be written as follows:
V-(nNV+) - -MD
dr[ kaN
N,
Or
n -Vp- eNE,,+ MNV,-VV,, - F +
ark5
36
and
NN2Tr)j
r
MDN
C
2T
r )
NYJA2
V-(n-3pV,)+pV-(nV,,)-MDN
2
5
+ MD 'V, (N -
r) +-
krr )
NN
Or)],
Or
a
Or
DT
N
Or
(ON
Or
+
((
T , (12)
)
2T Or)
N a'
2T Or)J
,
(13)
When Z1 2 N,<< N (in order to neglect impurity viscosity in the parallel impurity
momentum equation), F is small and all the terms in each of the preceding equations is the
same order and the three equations form a closed system for N, VII, and p=NT.
When the higher order moment equations are formed additional moments of f enter
so that Eq.(7) must be solved for the correction to the Maxwellian by expanding in
Legendre polynomials, solving a sequence of Spitzer [6] problems, and integrating to
evaluate the transport coefficients. Of particular interest is the relative parallel particle flux
betwecn te ions and impurities defined as Vli=N(Viii-Vilz)=fd 3 vuf which is found to be
r,= - 4
,ra my
In[VInN-(e/T)E+ (5/2)V InT]+ 9l
(ND
8v Or\k
3$F D. OV,
5 v Or
( ON
k Or
Or
0 +
61 N aT\
16T Or)
where all of the terms are the same order. The last term involves products of radial gradients
and is not part of the viscosity term that precedes it. Equation (14) can be used to find the
modification Fedge to the friction by writing F=FBrg+Feg, using the full parallel ion
momentum balance equation, and recalling the usual result from Braginskii [41 is
3
(15)
- waMvr,.
4
Fang -_ -NVIIT
2
The result is found to be
FeMN
[(3--Or
k 20
Or
+l
2
+
20
Or
(6--
5
a
32) ark
MND,-
l ,(16)
ar )
which is not obtained in standard orderings and modifies the usual F -0 result which drives
impurities relative to the ions toward higher temperature regions. Consequently, the new
term Fe*
is expected to play a role in modifying impurity penetration into the core.
16
3. Conclusions
The orderings and analysis carried in the preceding section results in a kinetic description in
which parallel ion streaming and radial transport enter in the same order in the kinetic
equation and the conservation of number, parallel momentum, and energy equations. New
terms involving products of radial derviatives are found in the expression for the relative
parallel velocity between the ions and impurities and parallel friction which may play an
important role in impurity penetration into the core plasma [2] and, if the treatment is
extended to include ion-ion collisions, flow reversal in the scrape-off layer [7]. In the
example considered Coulomb collisions of ions with heavy impurities provided the
dissipation, but the edge ordering methodology persists for anomalous processes as well.
Acknowledgments
This research was jointly supported by U. S. DOE grants DE-FG02-91ER-54109 at MIT
and DE-FG05-80ET-53088 at IFS. We are grateful to P. Helander and S. Krasheninnikov
for their helpful comments and useful insights.
References
[11 HiON, F. L., and HAZELTINE, R. D., Rev. Mod. Phys. 48, (1976) 239; and
HIRSHMAN, S. P., and SIGMAR, D. J., Nucl. Fusion 21, (1981) 1079.
[2] STANGEBY, P. C., and MCCRACKEN, G. M., Nucl. Fusion 30, (1990) 1225.
[3] HINToN, F. L., and HAZELTINE, R. D., Phys. Fluids 17, (1974) 2236; CHUNG, K.S., and HUTCHINSON, I. H., Phys. Rev. A 38, (1988) 4721; CATTO, P. J., and
HAZELTINE, R. D., Phys. Plasma 1, (1994) 1882; HELANDER, P., and CATrO, P. J.,
Phys. Plasma 1, (1994) 1882; and CATrO, P. J., and CONNOR, J. W., Phys. Plasma
2, (1995) 218.
[4] BRAGINSKI,
S. I., Reviews of Plasma Physics (Consultants Bureau, New York,
1965), Vol. 1, p.205.
[5] HAZELTINE, R. D., and WARE, A. A., Plasma Phys. 20, (1978) 673.
[6] SPITZER, L., and HARM, R., Phys. Rev. 89, (1953) 977.
[7] DE KOCH, L., STOTT, P. E., CLEMENT, S., et al., in Plasma Physics and Controlled
Nuclear Fusion Research 1988 (Proc. 12th Int. Conf. Nice, 1988), Vol. 1, IAEA,
Vienna (1989), 467; COOKE, P. I. H., and PRINJA, A. K., Nucl. Fusion 27, (1987)
1165; and KRASHENINNIKOV, S. I., Nucl. Fusion 32, (1992) 1927.
17
Transport theory in self-similar variables
for neutrals atoms in an edge plasma*
P. Helander (a,b) and S.I. Krasheninnikov (b,c)
(a) UKAEA Government Division, Fusion, Culham, Abingdon, U.K.
(b) MIT Plasma Fusion Center, Cambridge, MA 02139, U.S.A.
(c) Kurchatov Institute of Atomic Energy, Moscow, Russia
Abstract
A family of analytical solutions to the Botzmann equation is found describing neutral atoms
penetrating into a hot plasma whilst undergoing charge-exchange, ionization, and recombination.
It is shown that if the neutral mean-free path divided by the scale length of ion temperature
variation is constant, it is possible to introduce self-similar variables in the kinetic equation. Since
the mean-free path increases with increasing energy, a one-sided, high-energy tail arises in the
neutral distribution. This tail may contribute significantly to the heat and particle fluxes, which
are nonlocal in character. When this is the case, the fluid approximation of these quantities breaks
down at arbitrarily short mean-free paths.
1. Introduction
Neutral atoms play an important role in the tokamak edge, especially in the divertor. In a
hydrogen plasma, neutral atoms interact with the ions mainly through charge exchange (CX),
ionization and recombination. The dynamics of the neutrals is often (e.g., in plasma edge codes)
described by fluid equations taking these processes into account. Neutral fluid equations can be
derived rigorously [1-3] by a Chapman-Enskog expansion in r=/, the mean-free path divided
by the macroscopic scale length. However, the tokamak scrape-off layer (SOL) is generally only
about a centimeter wide, which may be comparable to the neutral mean-free path. Previous
kinetic treatments [4-8] of neutrals assume that the plasma is isothermal (except in Ref. 8), and
take a Krook operator for CX. We remove both these restrictions by introducing self-similar
variables in the Botzmann equation [91, which is possible if y is constant. This enables us to find
analytical solutions describing neutrals penetrating into a hot plasma.
* Work funded by the Swedish Natural Science Research Council, and by U.S. Department of
Energy Grant No. DE-FG02-91ER-54109.
18
2. Self-similar variables in the Bolzmann equation for neutral atoms
The Boltzmann equation for neutral atoms, interacting with a background plasma through CX,
ionization, and recombination is, in ID steady state,
v//V//f (v)=
fa(|v
v'I)|v - v'fi(v)f,(v')- f,(v)fi(v')]d 3 v' - Kzn if.(v)+ Vrfi(v) (1)
-
where f, and fi are the neutral and ion distribution functions, Kz is the ionization rate constant,
and Vr the recombination frequency. The CX cross section is usually taken to be inversely
proportional to |v - v'|, since the CX operator is then greatly simplified. Indeed, if
Kx=(lv - v'1)Iv - v'I is the CX rate constant, Eq. (1) gives
Sy
n +f
nKx+Vr
ni (K, +K,)
=/
,
(2)
where n and ni are the neutral and ion densities, respectively, and a/aym[n(K,+Kz)1- V// .
Let us seek self-similar solutions of the form
fT(YV)=
F(u),
(3)
where Uv/v-T is the velocity normalized to the thermal speed vT(2T/M) 1/2, a is a free
parameter, and N is a normalization constant chosen such that JF(u) d3u =1. The neutral
density then becomes n(y) = Nv4(y)/T'(y) - T3/2-0, and Eq. (2) takes the form
yxuu
+aF
F
-F=F
3 /nj,
where X=u///u, i a f vT
(4)
m (Kx + vr/n)/(Kx + K,), and y a vT dinT/dy are assumed to
be constant. Since n(y) falls off in the direction of increasing T(y), the solution to (4),
1
2Pxp
2(s-1)
sa
//''!~xp
ds , U//s
0,
F(u)=
2P
yu//
,I
2(s-u
s
, u1 <0
[ 7/_/s
describes penetration of neutrals atoms into an increasingly hotter plasma. If the mean-free path
is short, y-+O, the neutral distribution becomes proportional to the ion distribution,
19
lim F(u) = f(u) = p-3/2 exp(-u2), which we take to be Maxwellian. The distribution over
Y-+O
parallel velocities then becomes
F//(u//)a
F(u) d2U, _>
(
-
213 /
3
2
1
U-2z-2
where the asymptotic form is valid for large negative velocities (-u//>y/3>>1). Thus, there is a
power-law tail consisting of particles originating from the hot region y->eo. Note that the tail is
exponentially small in y, and thus cannot be found by means of any expansion in this parameter,
even if indeed y<<l. Let us now evaluate the particle flux and the heat flux associated with the
neutral distribution function. The former can be obtained directly from the kinetic equation (4) by
integrating it over velocity space, giving
/
Jf nv//d v = nvT F//u// du// =
10
)nvT,
regardless of the ion distribution. Only two of the three parameters a, P and y are independent
here since the integral over velocity space of the kinetic equation implies
a)=2 f du// f ex
(
21s- 11 u,/
2
ds
1
For y<<1, P(y,a)~I+(a-2)(a-5/2)j 2/4, and j,, y(a -5/2)nvT/2,
theory [2]. When a2,
in agreement with fluid
the contribution from the tail to the particle flux diverges, and the flux
becomes infinite regardless of the mean-free path! The figure shows a numerical evaluation of
the particle flux. The fluid approximation is not bad unless a is very close to 2, or the mean-free
path is very long. The effect of the tail is, of course, even more pronounced in higher moments
of the distribution, as in the heat flux
q// m f
V//d
VTnT71/
213f (I+
=
VTnT
Fu2 u// du//
ds
ds - fe~xp[2(1-s)U21
1 121
/u//du f1exp 2(sy//li 1)u
/ 1 2a5
/l/s _S2o -
For short mean-free path q//
Y-+2
T5(a -7/2)vTnT/4, again in agreement fluid theory. If y
is finite, the heat flux can be obtained by evaluating the expression above numerically, and is
shown in the figure. Again, the fluid result is not bad unless a is very close to the value where
the tail particles dominate, which now occurs at a=3, or the mean-free path is very long.
20
j,,
J/((a.
(
.
y)q,,(a.y)
0)
-,
q,,(a.y -+0)
44
ct-3.03
a-2.033
CE-2.05
a=3.3
2
2
-2.05=30
a-2.
.
0-
0.1
0-
10
a
10
The high-energy tail consists of fast neutrals that have escaped from the hot region y-o.
This is possible since the mean-free path increases with energy. Indeed, the assumption that
aoc /vj gives a mean free path that increases linearly with velocity, =<ava1 >vr(I=Kvr1 . In
reality, a depends much more weakly on the velocity. In a much more exact model for CX,
O(vi) = aOv'd, with 8=0.8, the kinetic equation again has self-similar solutions if simplifying
assumptions are made about the ionization and recombination rates. If K(T)= v.a(vT), ,
r=d(nT)/dys[nia(v)]-V/(lnT), and p a vr/nvTa(vT) are assumed to be constant, we can
make the self-similar Ansatz (3) and write (1) as
-YXu
+aF
+f
u -u'Iif(u)F(u') - ?(')F(u)
.
If we only consider tail particles, we may take u>>u' in the CX operator, and obtain
yxu(.
+aF) -(u' + )F=-(u +p)f.
For particles traveling in the negative (x<O) direction, the solution to this equation is
2
F(u)=---- (
yxu Jk.
0
u'2(c~-
L1
a
(
(u8 +Pexp 2
- -+
Yx uu
8'-I
1-
-i\~
(u')du'
for 0<8<1. Again, there is a power-law tail F(u,X)=Fo(X)/u 2 for sufficiently large u, the height
of the tail is non-expandable in y, and the moments corresponding to the particle flux and the heat
flux diverge for ct2 and cw<3, respectively. The shape of the tail is also important for wail
sputtering. The tail should be expected to be strongly enhanced over a Maxwellian, as was also
pointed out in [8].
21
3. Conclusions
If the density and temperature gradients are constant as measured in neutral mean-free paths,
there are self-similar solutions to the Boltzmann equation for neutral atoms in a plasma
undergoing CX, ionization, and recombination. These solutions describe neutrals penetrating
into a hot plasma. The neutral distribution function is markedly asymmetric with a peak at
positive velocities, and a power-law tail at large negative velocities. The tail is exponentially
small in y, and therefore eludes any short-mean-free-path expansion.
The self-similar solutions can be used to test the accuracy of neutral fluid models in
plasma edge codes and to benchmark Monte Carlo codes. The requirement that the mean-free
path be short is not sufficient for the fluid predictions of particle and heat fluxes to be correct. In
addition, the parameter a in Eq. (3), which is determined by the mean-free path and the rate
constants of the atomic processes, must be large enough. For instance, if a3 the heat flux
diverges. At short mean-free path (y<<l) this occurs when
$=
+ ,
KX +KZ
de model with
>1--y
8
l/vrei. Ionization is then sufficiently weak to allow a large number of hot
:trals to escape from the high-temperature region. Of course, the heat flux does not actually
2ome infinite since the temperature of the plasma is bounded. Some cut-off therefore appears
the diverging integral. The point is, however, that the heat flux q// - T3 can become
extremely large before this happens; since there are usually big temperature differences in the
GCC
plasma, Tnm>>Tmi,, the cut-off appears only at very high energy. On the other hand, if the tail
does not the dominate the fluxes, or if Tmax/Tmjn is moderate so the tail does not exist, fluid
results appear to be accurate even for relatively large values of y.
References
[1] Hazeltine, R.D., Calvin, M.D., Valanju, P.M., Solano E.R., Nucl. Fusion 32 (1992) 3.
[21 Catto, P.J., Phys. Plasmas 1 (1994) 1936.
[3] Helander, P., Krasheninnikov, S.I., Catto, P.J., Phys. Plasmas 1 (1994) 3174.
[4] Connor, J.W., Plasma Phys. 19 (1977) 853.
[51 Volkov, T.F. and Igitkhanov, Yu.L., Sov. J. Plasma Phys. 3 (1977) 668.
[6] Stakhanov, I.P., and Igitkhanov, Yu.L., Sov. J. Plasma Phys. 4 (1978) 559.
[7] Marushchenko, N.B., Pyatov, V.N., Turkin, Yu.A., Sov. J. Plasma Phys. 9 (1983) 717.
[8] Tendler, M.B., and Agren, 0., Phys. Fluids 25 (1982) 1040.
[9] Krasheninnikov, S.I., Sov. Phys. JETP 67 (1988) 2483.
22
Plasma Recombination and Divertor Detachment.
S. I. KRASHENINNIKOV (a,b), A. Yu. PIGAROV (b,c), D.J. SIGMAR (a)
(a) MIT, Plasma Fusion Center, Cambridge, MA, USA.
(b) Kurchatov Institute of Atomic Energy, Moscow, Russia
(c) PPPL, Princeton, NJ, USA
Abstract
We consider the influence on plasma recombination of the formation of negative and
molecular ions in divertor plasma. We take into account different atomic processes
(vibrational excitation of molecular hydrogen, electron dissociative attachment, ion
conversion, charge exchange and dissociative recombinations, dissociation and ionization,
etc.) and find the expression for the effective plasma recombination rate constant from a
suitably reduced set of coupled rate equations. We estimate the influence of the
recombination process on divertor plasma behavior and find that for AlcatorC-MOD like
parameters plasma recombination due to negative and molecular ions becomes significant in
the temperature range below a few eV. We conclude that plasma recombination is a strongly
contributing mechanism for the explanation of divertor plasma detachment in low
temperature plasmas.
1. Introduction
Plasma recombination in the divertor region can be considered as a simple explanation of
divertor detachment phenomena found in the experiments. But, conventional 3-body plasma
recombination (R3) only becomes important at very low temperatures (< 1 eV) and high
plasma densities. However, the presence of the hydrogen molecules which results in the
formation of negative ions H due to electron dissociative attachment (DA) [1,2]
H2 0,v)+e KDA ,H- + H,
and ion conversion (IC) [1,3]
(1)
H +H ,(2)
K
2,v)+ H
can strongly enhance the plasma recombination efficiency through the charge exchange
recombination (CER) [1,4]
H_ +W
H*
E -2H,
and dissociative recombination (DR) [1,5]
(3)
H++e KDR 1,2H .(4)
These reactions only involve two body processes, so that divertor plasma recombination can
be important for rather high temperatures and low densities. Here H2 0, v) is a rotational (j)
and vibrational (v) excited molecule, and K() are the rate constants of the corresponding
processes. The physical picture of the effects of the molecules on divertor plasma
recombination can be described as follows. Molecules originating at the target due to
hydrogen atom surface recombination diffuse through the plasma ions and atomic hydrogen
into the plasma volume. In the region where the temperatures is low enough and the
ionization of the molecules is not important (below a few eV) the main interaction of the
molecules with the plasma is due to vibrational excitation (VE) [1,6], DA, IC, and
dissociation (D) [1,7]. DA, IC, and D processes will lead to a decrease of H2 and increase
23
of H densities. Simultaneously, the ion saturation current onto the target will decrease due
to the CER and DR processes. Notice that in this region (where neutral ionization is small)
the flux of the atoms is directed towards the target. In the region located further upstream,
where the plasma temperature is higher, the ionization of the neutral component becomes
important and the flux of the neutrals is directed from the target thereby balancing the
incoming plasma flux. Some estimates of the effect of the reactions (1), (3) on divertor
plasma flow were given in Ref. 7.
2. Kinetics of chemical reactions and plasma recombination.
To make some estimates of the importance of the processes (1)-(4) for divertor plasma
recombination let us consider a simple O-D model. First we take into account that
endothermal reactions (1), (2) become efficient when the vibrational excitation exceeds
some threshold v a v, -5-7 (for T - Te -1 eV, where T and T. are the heavy particle and
electron temperatures). Then, we can write the following balance equations for vibrationally
excited molecules (v a v.), [H*], negative ions, [Hr , molecular [Hi], and atomic, [H+],
ions, and the total density of the positive ions n= H + H
:
d[H]+Vf, -- (H 2/t), -(KD +KDA)[H* ne-KIC[H4Hj+KIC[H 2 }H]. (5)
H
+vf - KDA[H2
H
+Vf2+ - KIC[HJH-*
j-
e -K
[H jne -KCER[-FJHr
IC[H +}H]-KDR[H ne,
,
(6)
(7)
-+Vf+
- -KCE4 IFH +r -KDIH 2 In,
(8)
dt
t
J
L
where (6H 2 /8t). is the source and KD is the rate constant of the dissociation of the
vibrationally excited molecules (v a v.); K(~) is the rate constant of the ionization of
negative ions by electron impact [1,8]; RIC is the rate constant of the conversion of H+ to
H +; and F,, F., I2+, F+ are the fluxes of the vibrationally excited molecules, negative
ions, molecular ions, and the total flux of the positive ions.
The reactions determining the densities of vibrationally excited molecules, negative
and molecular ions are very fast (typical rate constants are of the order of 10-8 cm3 /s 5) and
therefore we can neglect the left hand sides of Eqs. (5)-(7). Then, assuming that the neutral
density, N, is smaller or comparable to the ion density, n+, we find from Eqs. (5)-(7)
[Hi] -(8H 2 /t)./((KDA + KD);e + KIC[HD,
(9)
Hf
-WH2
KDA ne /
)
n+
n
IT
-2
KDA n
)
ne KE
KCER
+W,(10)
(11)
We neglect here the influence of the conversion of H+ to H+, since KIC M RIC while for
the temperature range below a few eV, KIC <KDR, which results in the inequality
H] < [H*+](recall that N s n+). Further simplification could be made by taking into
24
account the inequality KDA < KCER, which results in [H~< ne, so that from plasma
quasineutrality we have ne -n+
H
. From the expressions (8)-(11) we find
dn+
(12
(12)
dt+ V - -v, n+ ,
where the expression for an effective recombination frequency vff can be written as
vdr -
+
- (M/t.
)A~
+ KIC
n (KDA + KD +KIC.)Kje + KCER
-(13)
The first term in parenthesis in Eq. (13) describes the contribution of negative ions to
plasma recombination, while the second one determines the effect of ion conversion and
further dissociative recombination of the molecular ion. Thus the efficiency of plasma
recombination due to negative and molecular ions can be estimated as a ratio of these terms
4(H~/H +)
KAKCER
vff (Hr~)vf (H+) 2
2
KIC
(4
H + KCER
Taking into account that in the few eV temperature range KCER -3 x 10-8 cm 3 /s and
KDA = KIC - 3x 10-9 cm 3 /s [1,2] we find that for T,> 0.3 eV (when K(~) > KCER)
v(I- /Hj )<1 so that the main channel of the plasma recombination is the dissociative
recombination of the molecular ion.
3. Vibrational excited molecules
We see from Eq. (13) that the effective recombination frequency depends on the production
rate of vibrationally excited molecules with v a v, -5-7. There are two main mechanisms for
the population of high vibrational states [6,9]: a) the excitation by electron impact (VE)
H 2 (v)+e
KEv(vv') : H 2 (v')+ e ,
(15)
and b) vibrational-vibrational exchange (VVE)
H 2 (v)+H 2 (w) Kvv(v.w) * H 2 (v +1)+lH 2 (w-1) .
Vibrational excitation/deexcitation by impact of heavy particles (neutral or ion)
H 2 (v)+A KvT(v,v') *H 2 (v')+ A
(16)
(17)
for divertor plasma condition (N z n+) is not important since KEV(v, ') > KVT(v, v'). Here
KEv(vv'), Kvv(v,w), KVT(v,v') are the rate constants
of the corresponding processes.
Notice that the VVE process conserves the number of the vibrational quanta
(V -I (4H 2 (v)D/j[H2 (v)]) and only re-distributes the populations over vibrational
states so that the EV process is the only process which leads to the change of V and (as we
will see below) can limit the production rate of vibrationally excited molecules with
v a v. -5-7. Rather than solving the cumbersome set of equations describing processes
(15)-(17) in the general case we will make a simple estimate for the maximum of
(8H 2 /6t).. Let assume that chemical reactions (1) and (2) are the dominant processes at
v a v. (KDA, KIC - oo ). We will consider two extreme cases: a) the EV process is
dominant in the population of all vibrational states (KEv(v,v')>>KvT(v,v')), and b) the EV
process is only excites the v=1 state and, due to VVE, this excitation is momentarily
25
transferred into the v a v. range. In case (a) we can adopt a differential analog for the EV
kinetics in the v s v, range taking into account that KFy (v,w)-0 for Iv - w>1 :
0
KV(v,v + 1)n,(M(v) + (EO/Te)(aM(v)/av))
-
(18)
,
where EO is the vibrational energy. From Eq. (18) in the limit KDA, KIC
(M(v a v.)=0), we find
(8H 2 /6t),
-[H
2
] fdv f dv'(KE (v',v'+ 1)n)'exp{EO(v' -v)/Tr}
10
o for v av
-
.
(19)
J
0
Analyzing expression (19) we find that the maximum value of (8H 2 /8t), can not exceed
some limit
(20)
(8H 2 /8t), s (8H2 /6t)= - KEV (0, 1)ne[H2 Yv ,
which can be achieved for Eov,< Te provided there is a fast increase of the rate constant
KE with the increase of v. In case (b) the estimate for (bH 2 /6t), can be found very
easily. In this case one needs v. quanta to transfer the excitation into the v a v. region due
to the VVE process. Therefore, the corresponding effective rate of this process is v. times
smaller than the rate of the excitation on the first excited level. Finally, we arrive at the
has a very simple physical
estimate (20). Thus the expression for (8H 2 /8t)i
interpretation.
4. Recombination for the detached divertor plasma parameters
Substituting our estimation for maximum production of excited molecules (20) into the
expression for the effective recombination frequency (13) and taking into account our
estimate for v(Hf /H+) we find
KE (0,1)KIC(v.)[H 21
(21)
v.(KDA (*)+KD(v*)+KIC(v.))
As one sees from Eq. (21) at low temperatures, when dissociation of the molecules (T.
below 4-5 eV) is not important, and DR of H+ ions is faster than backward IC
H+ + H -+ H2 + H+ (occuring for n+/N a 10-1), veff does not exhibit the strong explicit
temperature and plasma density dependence which conventional three body recombination
does. At higher temperatures vfd decreases rapidly due to a strong increase of the
dissociation of vibrationally excited molecules and at low temperatures (below 0.5-1 eV)
veff decreases due to the decrease of vibrational excitation rate constant and a very small
population of the states with v a v.. For 0.5-1 eV < Te <4-5 eV assuming that KD(v.) <<
Klc(v.) - KDA(v*) - KEV(Ol) - 3x 10-9 cm 3 /s and v, -5 we obtain the following
estimate:
(22)
v.f -3 x 10-10 [cm3 /s] [H 2 ].
Let us compare estimate (22) with the conventional three body recombination frequency,
vR-3. For the typical divertor plasma regimes when Lyman radiation is trapped, from Ref.
8 we have vR3 (Te -1 eV, n. - 1015 cm-3 ) - 10-12 [cm 3 /s] ne. For the ratio [H 2 Yne
26
0.1 we find vf /vR3 - 30. Notice, that vR3 decreases drastically with increasing
Te,
while vdT varies slowly until the temperature reaches 4-5 eV.
To estimate the influence of the recombination on plasma flow for AlcatorC-MOD
like conditions let us compare vdT with the residence time of the plasma in the low
temperature region T. <4-5 eV. Let sr - 11/V 1 , where t, is the length of this region along
the magnetic field, and V11 is the parallel plasma velocity. For ill - 30 cm (but we notice,
that the corresponding real width of this region, in the perpendicular direction, would be
about 1-3 cm) and V11 - 3x 10 5 cm/s we find T w, 10-4 s. For [H2] - 1014 cm- 3 we
have vd m 3 x 104 s- 1 . Thus, for Alcator C-MOD like conditions -r x vdT >1 and the
effect of recombination on plasma density and plasma flow can be very significant.
5. Conclusions
i) We have estimated the influence of the hydrogen molecules on plasma recombination
using a simple model of the kinetics of the chemical reactions in hydrogen plasmas, for
divertor plasma conditions. ii) We have found that the main channel of plasma
recombination is ion conversion into molecular ions and further dissociative recombination.
iii) The estimate of the effective recombination frequency for the plasma temperature in the
range 0.5-1 eV < Te < 4-5 eV is vd m3 x 10-10 [cm 3 /s] [H 2 ]. For [H 2 J/ne - 0.1 and
Te =1 eV this frequency is 30 times higher than the conventional three body recombination
frequency. The difference is even more drastic for higher temperatures T. - 2-4 eV. iv)
Notice, that vdT does not exhibit the strong explicit dependence on the plasma temperature
and density (for 1 eV<T,<4-5 eV and n+/N a 10-1) as conventional three body
recombination does. v) An estimate of the effect of recombination for AlcatorC-MOD
detached divertor conditions shows that recombination can play a very significant role in the
reduction of plasma density near the target, and plasma flux onto the target, as observed
experimentally.
More accurate estimates of the influence of the molecular hydrogen on the plasma
recombination can be obtained from the collisional-radiative model which should involve
ground, rotationally, vibrationally, and electronically excited states of molecular and atomic
neutral hydrogen and ions, and negative ions H-. This investigation, generalizing the
results of [10], is underway.
Acknowledgments
Work performed under DOE grants DE-FG02-91-ER-54109 at MIT and DE-AC02-76CHO-3073 at PPPL.
References
[1] Janev R. K. et al., Elementary Processesin Hydrogen-HeliumPlasmas
(Springer-Verlag, Berlin, 1987).
[2] Berlement P., Skinner D.A., Bacal M., Rev. Sci. Instrum., 64 (1993) 2731.
[3] De Graaf M.J. et al, Phys. Rev. E, 48 (1993) 2098.
[4] Peart M. B., Hayton D.A., J. Phys. B, 25 (1992) 5109.
[5] Zhdanov V.P., Chibisov M.I., Sov. Phys. JETP, 47 (1978) 38.
[6] Cacciatore M.A., Capitelli M.,Celiberto R., Nucl. Fus., Suppl., Vol 3 (1992) 65.
[7] Post D.E., J. Nucl. Materials, 220&222 (1995) 143.
[8] Faisal F.H.M., Bhatia A.K., Phys. Rev. A, 5 (1978) 2144.
[9] Gordiets B.F., Mamedov Sh.S., Shelepin L.A., Sov. Phys. JETP, 40 (1974) 640.
[10] Sawada K., Fujimoto T., J. Appl. Phys., 78 (1995) 2913.
27
Effect of the Dynamics of the Impurity Distribution over the
Ionization States on the Radiative Plasma Instabilities and Shock
Wave Structure
S.I. KRASHENINNIKOV (a,b), D.Kh. MOROZOV (bc), D.J. SIGMAR (a),
J.J.E. HERRERA (c), T.K. SOBOLEVA (b,c)
(a) Massachusetts Inst. of Tech., Plasma Fusion Ctr., Cambridge, MA, U.S.A.
(b) Kurchatov Inst. of Atomic Energy, Moscow, Russia
(c) Instituto de Ciencias Nucleares, UNAM, Mexico D.F., Mexico
Abstract
We show that conventional quasistationary (coronal, or improved coronal) approximation
for the energy loss due to the impurity radiation in practice can never be applied to the
investigations of the impurity radiation driven instabilities in the edge plasmas due to
relatively slow evolution of the impurity population over ionization states. We show that
taking into account the effect of the evolution of the impurity population over ionization
states results in a very significant change of the growth rates of the radiative driven
instabilities and the structure of the shock wave in the radiative plasmas and leads to the
strict conditions for the existence of the shock wave.
1. Introduction
The energy loss due to impurity radiation plays a key role in the physics of the MARFE [1],
radiative divertor, radiative driven turbulence of the edge tokamak and stellarator plasmas as
well as in astrophysical objects [2]. Impurity radiation loss plays also important role in the
shock wave and heat front theory [3, 4]. To simplify the problem the quasistationary
(coronal, or improved coronal) approximation for the energy loss due to the impurity
radiation is widely used for the study of the time dependent processes (instabilities, shock
wave, etc.). This approximation assumes, in particular, that the variation of the impurity
radiation loss, Q, can be expressed in terms of the variation of the plasma/impurity density,
28
n, and temperature,T, Q(t) - Q(n(t),T(t)). However, actually the impurity radiation loss
is determined by the population of the impurity, nlz, over ionization states, z,
Q - ne I ni,zLz(Te), where ne and Te are the electron density and temperature, and the
function Lz(Te) describes the radiation loss associated with the impurity on the ionization
state z. The evolution of population ni,z is determined by the recombination/ionization
processes as well as by the impurity transport (see [5] and the references therein). Close to
the equilibrium the evolution due to ionization/recombination can be characterized by the
frequency vz. Quasistationary approximation for the radiation loss can only be applied for
the case when vz is high enough, in particular, higher than the characteristic cooling
frequency vR ~ Q/nT. However, the direct comparison of vZ and vR shows that for the
peak of coronal radiation loss for all impurities we have vz/vR
-
10~-4 h,
where
-=j nj/ne is the impurity fraction, and nj is the total impurity density. Therefore,
quasistationary approximation (vz/vR >1) can only be applied either far from the peak of
the radiation loss or for a very low impurity fraction (gi<10-4). In both cases the influence
of impurity radiation on the edge plasma behavior will be insignificant. Below we consider
two examples showing the importance of the non-quasistationary effects of the evolution of
the impurity population over ionization states for the typical edge plasma conditions.
2. Radiative instabilities
In this section we consider the influence of the non-quasistationary effects of the evolution
of the impurity over ionization states on the radiative instabilities of the plasmas assuming
that either the wave vector of the perturbation is parallel to the magnetic field or plasma is
unmagnetized [1, 2]. We assume that all plasma species have equal velocities (i) and
temperatures (T ). Following [51 we adopt moment approach for the description of the
impurity population over ionization states and will treat z as a continuous variable. In this
approximation the impurity population over ionization states can be written in the form
n -=,Z(ni/42~a)exp{-(z - Z)2/}, where Z a Z(i,t) is the averaged charge state, and
G a o(i, t) describes the dispersion of the impurity population. Close to the
ionization/recombination equilibrium the functions Z(f,t) and a(T, t) can be determined
from the following equations [5]
at(niZ)+V(n1 ;Z) - -vznj(Z-Z,), at(ni a)+V(n 1 o)- -2v zn(o- a.), (1)
29
where Z, a Z,(T) and a, -a a(T) correspond to the coronal equilibrium, and vZ>O is
determined by the impurity ionization ( vzjOn - ne Kion(z,T)) and recombination
(vz-rC
ne Krec(z,T)) frequencies vz - az(vz+/ 2 ,r
-
vz-./2,ion z-Z., notice that
Kion(Z, -0.5,T) - K= (Z, +0.5,T). To close the set of the equations we consider the
continuity, energy, and momentum balance equations
at(ni)+V(nii)-O,
at(ni)+V(nlv)-O,
0.5at(3NT + pv2) +V(2.5NT + 0. 5pv2) V -jcV
(2)
} =
H-
Q,
at(p V)+V(pi- V) - -V(NT),
(3)
(4)
where N is the sum of the ion (ni), impurity (n1 ), and electron (ne - ni +Znj) densities;
p - Mi ni + Mi n, and Mi and MI are the ion and impurity masses; H =const. is the
heating term, and Q=nefdzniz(Za)Lz(T)neniL(T,Za); and K is the heat
conduction coefficient. Notice that the function L.(T) =LT, Z (T), F.(T)) corresponds to
the coronal approximation of impurity emission.
Linearizing Eqs. (1)-(4) near equilibrium i =0, ni - no=const., t 1 =const.<< Mi/Mi,
T - To=const., Z - Z,(To), and a - a.(To) we obtain the dispersion relation
im W2 -(5/3)(kc)
R 2 -(kc) 2
2
v,
V
2(kc) 2
2 _ (c)
(alnL + Vz (alnL) alnZ
klnT)+ vz -i ikAlnZ), alnT
(
2
where o and k are the frequency and the wave number of the perturbation, c - 2TO/Mi,
vR - Q/3nOTo, vc - ick 2 /3nOTO. We omit in the right hand side (RHS) of the Eq. (5)
the term associated with a variation, since it is in a./(Z)2 <<1 times smaller than the last
term in RHS of Eq. (5) so that L(T,Z*(T), o*(T)) -L(T,Z*(T),O).
Let consider the adiabatic sound, radiative, and radiative condensation branches of the
solution of Eq. (5), which correspond to the limits o J- 3kc>vR, k -+O, and
y 2 <<(kc)2 respectively (y - -iw). Recall the radiative condensation branch describes the
linear stage of the MARFE formation. For the radiative condensation branch of the
instability from Eq. (5) we find
30
Y
-(yf - vz)/2
(yf- vz)/2 2+ySvz,
(6)
where we can interpret the frequencies ys - (3/5){vR[2 -din L,(T)/<dT]- vK} and
yf
=
(3/5){vR 2 - aln L(T, Z)/aTIz-Z. I- v.
as the growth rates of the radiative
condensation instability for the quasistationary coronal approximation of Z, and for the
approximation of frozen Z (Z=const.) respectively. It is easy to see that the instability
occurs when either one of the inequalities y f> vz, y s>O is satisfied. However, the
Iy
conventional expression for the growth rate, y - y s, can only be recovered for vz >> S,
y f. Recall that these inequalities can only be satisfied for a very low vR. For more
realistic tokamak edge plasma condition of rather high vR, when vz/yfI, vz/y1
--
0,
we find y + - yf , y - - -ys(vz/yf ). The growth rate for the radiative instability can be
found from Eq. (6) by replacing y f on y f - -vR(aInL(T,Z)/aTJz.z,),
and ys on
y s-vR(dln L,(T)/dT). For the adiabatic sound wave for the limit vR>vZ we find the
growth rate y - -(2/5){vR[(3/2)+ 81n LT,Z)/OTiz.z.] + v
approximation we would get y
-
, while in quasistationary
-(2/5){vR[(3/2)+ din L,(T)/dT]+ v
.
3. Shock wave
In some cases the stationary, homogeneous solution of the Eqs. (1)-(4) (which actually
result in the equation H - Q) can be ambiguous and more than one stable homogeneous
solutions can exist. The simplest case of the interaction between these solutions (existing in
the different space domains) resulting in a propagating shock wave was considered in [3].
However, the analysis of the shock wave in [3] was made in the quasistationary
approximation of the impurity evolution over ionization states. Meanwhile it is easy to see
that in practice this approximation is not valid. (The more detailed investigation will be
published elsewhere, and here we only discuss some general points). Indeed, the width of
the shock is of the mean free path of the particle (coulomb mean free path in our case). That
means that the characteristic frequency of the plasma parameter variation within the shock is
of the coulomb collision frequency, vc. But, vc >>vz and, therefore, after the shock some
much wider region must exist where the impurity distribution over the ionization states
adjusts to the new equilibrium condition. Moreover, the transition from one equilibrium
31
state (characterized by Z - ZI) to another one (with Z - Z2 > ZI) can be accompanied by a
very strong kinetic energy loss of the plasma particles, ER, due to ionization of the
impurities and impurity radiation loss. It results in the restriction for the plasma velocity in
front of the shock, ul, u a CR 0 2ER/(Mi + iM) (we are using shock frame). In
general case the estimate for ER gives ER /T
-
(ZI
- Z2)(vR /vz). Therefore, only a very
strong shock wave can exist for vR /vz>>1 in general case, and only for some special
conditions for the shock wave parameters this strict requirement can be avoided.
4. Conclusions
i) Conventional quasistationary (coronal, or improved coronal) approximation of the energy
loss due to the impurity radiation in practice can never be applied to the investigations of the
impurity radiation driven instabilities in the edge plasmas due to relatively slow evolution of
the population of the impurity over ionization states. ii) Non-quasistationary effects of the
evolution of the impurity population over ionization states can result in a strong reduction of
the radiative driven instability growth rates and increase of their thresholds. iii) They also
strongly influence the structure of the shock wave in the radiative plasmas and leads to the
strict conditions for the existence of the shock wave.
Acknowledgments
This work is performed under the U.S. Department of Energy Grant DE-FG02-91-ER54109 at MIT, and under Mexico CONACyT project ES104393 DGAPA-UNAM.
References
[1] STRINGER, T. E., 12th EPS Conf. on Contr. Fus. and Pl. Phys. Budapest, 1985, 1-86.
[2] FIELD, G. B., Astrophys. J. 142, (1965) 531.
[3] ARANSON, I., MEERSON, B., and SASOROV, P. V., Phys. Rev. E 47 (1993) 4337.
[4] ZEUDOVICH, YA. B. and RAIZER, YU. P., The Physics of Shock Waves and High
Temperature HydrodinamicPhenomena. (Academic Press, New York, 1967).
[5] ARUTUNOV, A. B., KRASHENINNIKOV S. I., and PROKHOROV D. YU., Sov. J. Pl.
Phys. 17 (1991) 668.
32
Two-Dimensional Divertor Modeling and Scaling Laws
Peter J. CATTO (a), S. 1. KRASHENINNIKoV (a,b), J. W. CONNOR (c),
D. A. KNOL (d)
(a) Massachusetts Inst. of Tech., Plasma Fusion Ctr., Cambridge, MA, U.S.A.
(b) Kurchatov Institute, Moscow, Russia.
(c) UKAEA Gov. Div., Fusion, Culham, Abingdon, Oxon. OX14 3DB, U. K.
(d) Idaho National Engineering Laboratory, Idaho Falls, ID, U.S.A.
Abstract
Two-dimensional numerical models of divertors contain large numbers of dimensionless
parameters that must be varied to investigate all operating regimes of interest. To simplify
the task and gain insight into divertor operation, we employ similarity techniques to
investigate whether model systems of equations plus boundary conditions in the steady state
admit scaling transformations that lead to useful divertor similarity scaling laws. A short
mean free path neutral-plasma model of the divertor region below the x-point is adopted in
which all perpendicular transport is due to the neutrals. We illustrate how the results can be
used to benchmark large computer simulations by employing a modified version of UEDGE
which contains a neutral fluid model.
1. Introduction
We consider a two-dimensional (2D), fully equilibrated fluid plasma-fluid neutral model [1]
of the divertor region between the x-point and target which ignores anomalous processes,
recombination, the thermal force, plasma and neutral viscosity, and ion heat conduction,
and assumes parallel flows are much larger than perpendicular flows and that the ions,
neutrals, and electrons are all at the same temperature T. The target plate and entrance to the
2D rectangular box are located at y =0 and L, respectively, and the separatrix and sidewalls
are at x =0 and *A, respectively. The target and sidewalls are assumed to be fully recycling
so that the neutral density (Nn) vanishes at the entrance, and the magnetic field has
components Bt and Bp in the z and y directions with B the total magnetic field strength and
b = Bp/B = constant. Rather than make ad hoc assumptions about anomalous processes we
assume they enter only through the scrape-off layer (SOL) width Ap.
33
2. Fluid Plasma-Fluid Neutral Model
We specify the upstream pressure at y = L as the product of a peak value Pu and an order
unity shape function Sp(x/Ap) characterized by the SOL width Ap. Using Pu and I = 13.6eV
= ionization potential for hydrogen, we introduce the dimensionless plasma (Ne) and neutral
densities n = INe/Pu and 9 = IN/Pu, respectively, and the dimensionless temperature T =
T/I. We also use I to define the dimensionless poloidal and radial neutral velocities u =
(M/I)1' 2 Vny and w = (M/I) 112 Vnx and the ion and neutral pamllel velocity v = (M/)1 2 V,
with M the ion and neutral mass. It is convenient to introduce order unity temperature shape
functions associated with charge exchange (subscript x) and ionization (subscript z) by
defining <ov>x = KxSx(r) and <ov>z = KzSz(E) where Kx and Kz are true constants equal
to the peak values of the rate constants. We can then define a neutral penetration length f.=
l(IM)1/ 2 /pu(KxKz)1/ 2 and the normalized poloidal and radial variables p = y/l. and p =
x/t., respectively. Introducing the dimensionless parameter Y= (Kz/K)1/ 2 , the plasma and
neutral continuity equations, total parallel momentum equation, and the perpendicular
neutral momentum equations in the diffusive approximation, may be written as follows [21:
b -(v) - onIS,('), -(qu)
+ -(w)
nq(u - bv)S.() - -a
- -on
(nr),
S.(-r),
and
(2n + q)-+(n + 1)V
,
nilwS(r) - -o-(T).
(1)
Next we make the total energy conservation equation dimensionless by
introducing the constant k = 0.96M(KzKx)1/ 2 I3/2 /m1/ 2 e4 lnA = equilibration time over the
neutral penetration time for particles of energy I, and two new order unity shape functions
for hydrogen and impurity radiation via <ov>H = KHSH(t) and <ov>I = KISI(t), where KH
and K[ are true constants equal to the peak values of their respective rate constants and m is
the electron mass. Defining oH = EHKH/I(KzKx)1/ 2 and o0 = NiEiKi/Pu(KzKx)1/ 2 with N,
the impurity density, the energy equation becomes
a
ap lk2
+
v
2
(bn +qu)+ b
n -vkb
-221)1 ig
ap)J
k2
op k2
+
V2 1w
2)/
a f 5arir ft 1 a f SoTF t Iv
-ognS
()L2nST(')
2
032nS.(_r)
aP '
(C Op]
p - -onIS,()-aHnmSHr)
1
j
r).
(2)
Equations (1) and (2) are valid for large k and small b, a, u, w, kb2 , oH, and oi. The
magnetic field enters only through b so we are always free to choose B to satisfy a
gyroradius over scale length scaling or beta scaling.
34
There are five boundary conditions associated with this 5th order system of six
equations for the six unknowns n, i, v, u, w, and T. The first is the plasma pressure
specification at the upstream entrance
=Ue. denoted by a subscript u, 2 nuTu = Sp. The
s
next two are the Bohm sheath criterion for the plasma flow into the target, vd = -aT 2 with
a-I the Bohm factor, and the complete recycling condition at the target idud= -abnx v 2
and sidewalls lsws = 0. We use the subscript d to denote the divertor plates at p =0 and the
subscript s to denote the sidewalls at p =
±A/.
(we assume no plasma makes it to the
sidewalls). So far we have the three boundary conditions:
2nuu= Sp(p.Ap), vd= -aTr
qud= -abndrv
2
and
lsws = 0.
(3)
The fourth boundary condition is the energy flux to the target and sidewalls
5
[bd31
2_Cs12 + SCnMdrd lI
)312
dbndrd
+ kbt d2
2
and
where Yp -5 and Yn
-
+
-
- (ayPnd + Y Od)r
2np,d)
5S,r,
a/
-
2n1 S1 (',)OP
,(4)
0.2 are the plasma and neutral heat transmission coefficients and the
complete recycling boundary condition has been used. We denote the parallel electron heat
flux entering the divertor as -quSe(x/Ap) with Se an order unity heat flux shape function and
qu the peak value. Then the final boundary condition is on the entering parallel heat flux:
kbT 52
-(n!.)
.LS,
(pe./A).
(5)
The complete set, as given by Eqs.(1)-(5), contains the twelve dimensionless
parameters b, a, k, o1, ol, a, Yp, Yp, U I, A/I., Ap/e., and (M/I)"/ 2 bqu/Pu a QU. Connor
and Taylor [2-4] scaling techniques show that this system allows one scaling transformation
and that any unknown function can depend at most on p and P and the eleven independent
dimensionless parameters: a/b, bk, at/b, 31/b, a, Yp, Yn/b, Ul.er LPu, A/'.O APU,
Ap/l.c ApPu, and (M/I) 1I 2 qu/Pu n Qu/b. More importantly for our purposes, Eqs.(1)-(5)
have the important property that all geometrical lengths (L, A and Ap) are multiplied by the
upstream pressure Pu in order to keep collisionality fixed. As a result, if the lengths L, A
and Ap-are increased by some factor and the upstream pressure Pu and parallel heat flux qu
decreased by the same factor then the solution for n, ,, v, u ,w and r is unchanged! This
property is a general feature of two-body processes [3,5] and persists even when viscosity,
thermal forces, and ion heat conduction are retained. Three-body recombination can play a
role in high plasma density, low temperature (< 2eV) regions between the ionization front
and the target where it can alter this collisionality property. Equations (1)-(5) recover
Lackner's [5] P/R = constant scaling, where P is the power to the plates and R the major
radius, provided the eleven independent parameters are held fixed.
35
3. Modeling Results
A rectangular version of UEDGE [6] with neutrals and retaining anomalous transport was
modified to illustrate that doubling the SOL and divertor widths and the divertor depth, and
halving the upstream pressure and parallel heat fluxes leaves the solution unchanged in
agreement with our scalings. Preliminary results are shown in Fig.1 and provide an
important benchmark of UEDGE. The small non-similar features are caused by the
anomalous coefficients, grid effects, and details in modifications to the upstream boundary
conditions. The smaller geometry (L = 25cm, A = 4cm, Ap = 1cm) results are shown in (a),
(c) and (e) and the larger (L = 50cm, A = 8cm, Ap = 2cm) in (b), (d), and (f). An attached
case (qu =150,75 MW/im2 ; pu = 900, 450 Pa) is shown in (a)-(d) with (a)-(b) temperature
contours and (c)-(d) contours for the density and twice the density. The density and double
density contours are shown in (e) and (f) respectively for a detached case (qu =90, 45
MW/m2 ; Pu = 900, 450 Pa) with volume recombination turned on to illustrate that the
scaling still works reasonably well. The geometry shown in the figures differs from that of
the equations by having a private flux region with a width less than A (10cm for the smaller
geometry and 20cm for the larger). Collisionality imposes a severe constraint on the
similarity of present tokamaks to ITER [2]. Its large size makes it difficult to model using
enough grid points to resolve important physical effects. The results show that it might be
possible to use smaller geometries in divertor codes to model ITER relevant collisionalities.
Acknowledgments
This research supported by the U.S. DoE, MIT grant DE-FG02-91ER-54109 and INEL
contract DE-AC07-94ID13223, and the U.K Depart of Trade and Industry and Euratom.
References
[1] CATTO, P. J., Phys. Plasmas 1, (1994) 1936; and HELANDER, P., KRASHENINNMKOV,
S. I., and CATrO, P. J., Phys. Plasmas 1, (1994) 3174..
[2] CATrO, P. J., KRASHENINNIKOV, S. I., and CONNOR, J. W., to appear in Phys.
Plasmas 3, (1996) March.
[3] CONNOR, J. W., and TAYLOR, J. B., Nucl. Fusion 17, (1977) 1047.
[4] CONNOR, J. W., Plasma Phys. and Controlled Fusion 26, (1984) 1419.
[5] LACKNER, K, Comments Plasma Phys. Controlled Fusion 15, (1994) 359.
[6] KNOLL, D. A., MCHUGH, P. R., KRASHENINNIKOV, S. I., and SIGMAR, D. J., to
appear in Phys. Plasmas 3, (1996) January.
36
0.050
0.025
20
25 ~-------
0.000
E
5--
-- -
-3
15
-
0-
--
0.80
0. 85
0.90
0.95
Poloidal Distance (meters)
a:
0.10
S
0.05 :-
(b)
--
20
----
0-
is
25
-5
7ro
0.00
1.6
1.7
1.8
V9
Pooidal Distance (meters)
c
0.025
'a
IE19-
sfA9
6FOEig
,Sl
0.000
0.80
0.85
0.9009
Pololdal Distance (meters)
0.10
(d)
0.05
(D
4E19
1.8
0.050
7
1.8
Pololdal Distance (meters)
1.9
a(e
E
80.00
0025
-
5E19
0.0000.80
0.85
0.10Pololdal
-0.90
Distance (mneters)
0.95
'0L
80.00
-
010
g-
1.
---- -
-
100
1.7
Poloidal Distance (meters)
Figure 1: UEDGE temperature (a, b) and density (c, d, e, f) contours.
37
Effect of Perpendicular Transport on Edge Plasma Energy
Loss due to Impurity Radiation
S. I. KRASHENINNIKOV (a,b), D. A. KNOLL (c)
(a) Massachusetts Inst. of Tech., Plasma Fusion Ctr., Cambridge, MA, U.S.A.
(b) Kurchatov Inst. of Atomic Energy, Moscow, Russia
(c) Idaho National Engineering Laboratory, Idaho Falls, ID. U.S.A.
Abstract
We consider the effect of perpendicular energy transport in the low temperature divertor
region on impurity radiation loss from the SOL plasma. We show that the perpendicular
energy transport results in enlargement of the volume with relatively low temperature and a
very high density of the plasma. High plasma density causes a strong energy loss due to
impurity radiation peak in this low temperature region. For low Z impurity the energy
transport due to plasma convection and neutrals also can strongly influence the volume of
low temperature region.
1. Introduction
The energy loss due to impurity radiation plays a key role in the physics of the MARFE,
radiative divertor, etc. Roughly speaking, the impurity radiation loss, Qr, is determined
by the electron density, n., the impurity fraction, imp - nimp/ne, the local emissivity,
L, and the volume of the radiative region, Vrad, Qr -RimprV
= impL(ne) 2 Vr
(notice, that Vm can be determined by the peak of either jmp,' L, or (n,) 2 ). Each of
those parameters can strongly affect the magnitude of impurity radiation. Experimental
observations show that for some cases (MARFE [1], and radiative divertor [2]) a significant
amount of radiation (approximately a half) is coming from relatively small volume of rather
cold plasma. It is possible that this effect may be explained by the local increase of the
emissivity and/or impurity fraction. However, in this paper we show that even for
Simp - const., and L - const. these features of the MARFE and radiative divertor can be
explained by high value of (nc)2Vr in the low temperature region caused by the
perpendicular plasma energy transport.
To estimate impurity radiation loss a simple model based on the balance of the
impurity radiation loss and by the energy flux transported by the parallel heat conduction is
widely used.
d(KI(T)(dT/del))/dI - Rip,
(1)
where I is the parallel coordinate, T is the plasma temperature (Te=Ti), ic
1 (T) is the
parallel heat conductivity,. Assuming that plasma pressure is constant along the magnetic
field lines, L - L(T), and imp - imp(T) from Eq. (1) we arrive to the expression for the
maximum poloidal heat flux which can be re-radiated in the region with the temperatures
below T (see for example [31)
38
112
T
Qrad,11(T) - 4nRO&SOLP 2fdT',(T') iMp(T')L(T')(T')-
(2)
10
where P - neT, R0 is the tokamak radius, and ASOL is the SOL width. The parallel heat
conductivity in low plasma temperature region is small. Therefore, the main contribution to
the expression (2) is given by high temperature region of the SOL plasma. It is true even for
the low Z impurities like carbon [3] and coronal approximation for L(T), which is strongly
peaked at low temperatures. The reason for this is a small volume of the low temperature
plasma caused by the strong temperature gradient due to small K;;. But, this conclusion
means that the radiation loss for the low Z impurity from the plasma core should be much
higher due to larger volume and higher plasma density.
However, we show here that the perpendicular plasma transport is very
important for the estimates of the factor (ne) 2 V j in low temperature region. Actually,
one can see it just from Eq. (2). The width ASOL should be found self consistently from
the energy balance equation as a result of the competition of parallel and perpendicular
plasma transport. In the simplest case, when the heat conduction is dominant we have
V1 (K 1 V1 T ).+d(xni(T)(cr/de
d))/d
(3)
1
11 - Rimp,
where xK is the perpendicular heat conduction coefficient. From Eq. (3) we may conclude
that ASOL oc fi Li . Substituting this estimate in Eq. (2) we find
and Qad,1 does not depend on the magnitude of K;;.
To treat Eq. (3) more accurately let divide the SOL plasma in two regions: a) the
high temperature SOL mantle, and b) the low temperature divertor region.
2. Impurity Radiation from the SOL Mantle.
In the region of the high temperature SOL mantle we can only retain in Eq. (2) the radial (r)
derivative in the term V1 (K1 V 1 T), and represent the parallel term, describing heat
transport into divertor region, as a sink Ediv(T) - (Kl(T)T)/(ISOL )2 c7/2
d(K±(dr/dr))/dr - Ediv + Rip.
(5)
From Eq. (5) we find the radiation loss in the SOL mantle, QM,rad, and the energy flux
coming into divertor, Qdiv,
TS
1/2
T,
QM,rad - Ss fdT'ii (T')Rimp(T'
2fdT"Ki(T")(Ediv(T")+Rimp(T"))
0
/10
Ts
Qdiv - Ss fdT'Kj_(T')Ediv(T'
TP of
P
112
2fdT"iL(T")(Ediv(T")+ Rimp(T"))
,(6)
(7)
0
/10
where S. is the tokamak surface at the separatrix, and the separatrix temperature, T,, is
determined by the energy flux, Qs, coming into the SOL from the bulk plasma
Qs - 2 fdT'KI(Ediv(T')+ Rimp(T'))
0
39
.
(8)
Let us estimate the fraction of the radiation loss from the SOL mantle,
M,rad - QM xd /Qs, for the model function of the impurity radiation Rimp (T)R =
const., and Ki= const. Substituting Rijp (T)=R in Eqs. (6), (8) we find
bad(ts) - 0.5fdx/x9/ + x)1 t79/2+ t.
(9)
where ts - TsIrE, and Ediv(TE) -i. From Fig. 1 one sees that 68 a -1 for ts<1
and it decreases rapidly with increasing ts for t, a 1. The case t. < 1, when there is
practically a complete re-radiation of the energy flux incoming into the SOL requires a very
careful analysis of the radiation losses in the core plasma, since the energy balance can be
fragile. Below we will assume that t. 2 1 and the radiation loss from the SOL mantle is
small.
2. Impurity Radiation from the Divertor.
To consider the impurity radiation loss from the divertor we use (see Fig. 2) slab geometry
and rewrite Eq. (3)
al(I + b2 x1)(aT/ay)}/ay + 8(icj(aT/ay))/8x - Rimp,
(10)
where y and x are the poloidal and radial coordinates, and b is the ratio of the poloidal and
the total magnetic field strengths. Introducing the vector ii (see Fig. 2) and neglecting the
curvature of the temperature contours we can represent Eq. (10) in the form
a {(c + (-i. y)2 b2xj
)(aT/8a) a =-Rimp,
(11)
where 1. is the coordinate along i. As we mentioned above, the estimate of the factor
(ne)2V rad in a low temperature region based on the parallel heat conduction (see Eq.(2))
results in the radiation loss from the SOL mantle, which we assume to be small. Therefore,
we can neglect the term with icl in Eq. (11). Then, from Eq. (11) we find the estimate for
the impurity radiation loss from the low temperature divertor region, QD, d, caused by the
perpendicular heat transport
QD,rad - SD ,md 2fdT')Ci(T')Riwp(r')
,
(12)
0
where SD,jd is the surface of the radiating flame front. For the upper limit of the integral in
Eq. (12) we choose infinity, but practically it does not matter. Indeed, assuming that the
plasma pressure is constant along the magnetic field lines and imp= const. we find
2 (L(T)/T 2 ) and ic_ - nXi - P(X±/T), where Xi is the perpendicular
Rimp -jimpP
heat
diffusivity. One sees that in this approximation, the integral in Eq. (12) converges unless
Xi, mp, or L increases very rapidly with increasing temperature. Therefore, a strong
impurity radiation loss in the low temperature region can be caused by the perpendicular
plasma heat transport.
We adopt the estimates for the surface of the radiating flame front,
SD, rd =4Ldiv 2a Ro , and SOL width, ASOL -(LSOL/b) 'F(T-)/xi(T,),where Ldiv
40
is the divertor leg length, and LSOL /b is the half of the magnetic field line length. Then, the
ratio of the expressions (2) and (12) has the form
Q D,rad
O ,HmT,
-2Ldiv
SOL
Kflc
I-
l) (
)
lmL T-2/0r
m LT2T-2)
.
13)
0/
Let us consider the model functions L(T) = L H(T - Tmin )H(Tmax - T),
imp= const.,
and XI= const., where H(x) is the Heviside function and L= const. Recalling that
C11(T) o T5 2 from Eq. (13) we find
(14)
>>1,
Ts
)3/2,T 8
QD,ad/Q ad,v(Ts) -(Ts Jrmin )2 { /
I(T,/Tmax) ,T, >>Tmnx >> Tmin
where we assume Ldiv - I-SOL. The inequalities (14) justify the omission of the term with
x, in Eq. (11)
3. Effect of Plasma Convection and Neutral Transport
Above we have assumed that the energy transport is only caused by the plasma heat
conduction. However, low Z impurities like carbon can have a peak of radiation loss in the
temperature range below 10 eV. For this relatively low temperature region the effects of the
plasma conduction and neutrals on the energy transport can be important. To investigate
these effects we perform 2D modeling of carbon radiation loss (fixed fraction model,
improved coronal approximation for the radiation loss) with plasma code UEDGE coupled
with the Navier-Stokes neutral model (see [4] for details). In contrast to ID parallel energy
transport model [3] we find that almost 100% of Carbon radiation is coming from a low
temperature (s 10 eV) divertor region. In the radiative region a significant fraction of the
energy is transported by plasma convection and neutral energy transport (see Fig. 3).
Conclusions
i) We show that the perpendicular energy transport results in the strong enlargement of the
plasma volume with relatively low temperature and very high density. High plasma density
causes a strong energy loss due to impurity radiation peak in this low temperature region. ii)
Analytical estimates, accounting the perpendicular energy transport, show that the impurity
radiation loss from the low temperature region like MARFE and radiative divertor increases
drastically and can be dominant. iii) The results of the 2D modeling for the fixed impurity
fraction model and improved coronal approximation for the radiation loss support these
estimates and show that almost 100% of carbon radiation is coming from low temperature
(s 10 eV) high plasma density divertor region where the energy transport and, therefore the
volume of this region, is significantly affected by the plasma convection and the neutral
energy transport.
Work performed under US DoE grants DE-FG02-92ER-54109 at MIT and DE-AC0794ID13223 and INEL
[1]
[2]
[3]
[4]
References
Lipshultz B., J. Nucl. Mater. 145&147, (1987) 15.
Porter G., Bulletin of the American Physical Society, 40 (1995) 1693.
Post D. E., J. Nucl. Mater. 220&222, (1995) 143.
Knoll D. et al., Phys. Plasmas, 1996.
41
1 M, rad
0.5
n
01
0
2
1
3
ts
Fig. 2. Temperature contours in a divertor
Fig. 1.
0.040
0.030
x
a
to
X
0.010
0
YA
wine
M.
0-
w. i
0UM
0.sM
yame
Mn
y
0.04
(A~
b
0.02
0.00
0.80
0.85
0.90
0.95
Fig. 2. The results of 2D modeling: (a) carbon radiation contours, (b) ratio of the total
convective energy flux to plasma conductive energy flux.
42
1D Fluid Regime of Plasma-Neutral Interaction and
Divertor Detachment
T. K. SOBOLEVA (a,c), S. I. KRASHENINNIKOV (b,c)
(a) Instituto de Ciencias Nucleares,UNAM, Mexico D.F., Mexico
(b) Massachusetts Inst. of Tech., Plasma Fusion Ctr., Cambridge, MA, U.S.A.
(b) Kurchatov Institute of Atomic Energy, Moscow, Russia
Abstract
We show that self consistent decrease of both plasma flux and neutral ionization in current
tokamaks is only possible when neutrals can be treated in a short mean free path
approximation. We investigate these fluid regimes of plasma-neutral interaction with ID
fluid equations employing a neutral viscosity term to treat the neutral interaction with the
divertor plate. We have found that plasma flux onto the target starts to decrease at a very
low heat flux coming into the hydrogen recycling region, when the temperature near the
target drops below 1 eV, which seems lower than observed in the experiments. We
conclude that either 2D effects of plasma-neutral interaction or plasma recombination
processes should play a very important role in divertor plasma detachment.
1. Introduction
In Ref. 1 one dimensional physical model of the tokamak SOL was developed to investigate
one of the most interesting feature of divertor detachment: plasma - neutral interactions in
the recycling region of a tokamak divertor. Two opposite extremes of fluid and diffusive
Knudsen neutrals were considered there. Fluid approximation for neutral transport can be
applied for relatively high plasma density in the hydrogen recycling region, when the neutral
mean free path, XN, is smaller than SOL plasma width, A p. For a steady state condition
plasma particle sink and source must be balanced. Therefore, the decrease of plasma flux
onto the target (plasma sink) after detachment has to be accompanied by the reduction of the
43
global neutral gas ionization (plasma source). Notice, that experiments show that neutral gas
density increases for detached conditions. How to resolve this paradox? For current
tokamaks with relatively open divertor geometry the reduction of neutral influx into hot
temperature region with increasing sag density in the divertor is only possible (if plasma
recombination is not important) is hot temperature region is screened by the layer of cold
dense plasma where XN is smaller than plasma scale length. It results in a slow diffusive
like neutral flow and reduce neutral influx into hot plasma region in comparison with free
streaming flux. Note, that this very important feature of detached plasmas is usually missed
in the simple models of plasma detachment. However, near the target there is a thin layer
(with the width = %N) with the mixture of the neutrals just reflected from the target (with
almost zero velocity along the magnetic field line) and neutrals coming to the target after
neutral ion collision (with parallel velocity close to plasma velocity). Notice, that we assume
here no neutral-neutral collisions. The main goal of this paper is to investigate with a simple
1D model the fluid regimes of plasma-neutral interaction employing neutral viscosity term to
treat plasma interaction with first flight neutrals.
2. Equations, boundary conditions, and solutions
We will assume below that plasma only flow along the magnetic field lines, the electron, ion
and neutral temperatures are equal T, and all variables depend on poloidal coordinate y. We
will use coupled fluid equations for plasma - neutral gas flows (see for example Ref. 2 and
the references therein)
dji/dy - KI(T)nN
V-(MNYN 'N
,
(1)
+PNi+ fIN)
N,
(2)
d(MNVi 2 + Pp )/dl - -RN,1
(3)
dqy/dy - -EIKI(T)nN ,
(4)
where n, Vi, PIP a 2nT (N,
and pressure;
I
YN,
PN w NT) are the plasma (neutral) density, velocity,
is unit tensor; ll is the coordinate directed along the magnetic field,
d(...)/dfll - b(d( ...
)Mdy) ;
N
-
N+
NT(2/3)IV
N)
(5)
is neutral viscosity tensor, q - NT/nKiN; RN is the neutral-ion friction force
RN - -MnNKiN(N -
(6)
i) ,
qy is the total heat flux and ji a bnVi is the ion flux along the y coordinate; (...) 11is the
44
parallel component of the vector (...); and M is the mass of both the ions and neutrals. The
ionization and ion-neutral collision (charge exchange and elastic) rate constants are denoted
by KI(T) and KiN(T), respectively and El is the ionization "cost" of the incident charged
particles due to ionization of the neutral outflux and hydrogen radiation energy losses
El(T) - I+ ER(KR/KI), I=13.6 eV is the hydrogen ionization potential, ER the
characteristic energy loss, and KR is the electron impact excitation rate constants for
hydrogen. We neglect the influence of the thermal force on the plasma-neutral friction force
and the heat transport. Since we consider complete particle recycling at the target ("closed
box" model), the net mass flow along the y coordinate equals zero everywhere in the
recycling region, and the heat flux qy is only determined by plasma and neutral conduction
and electron flow
NT \dT5
dTr5
- +-T jim-n -+-T j,
(7)
MnKiN dy 2
dy 2
where ic (T) is the electron heat conduction coefficient along the magnetic field, c,, O2.4 is
(2
ce-(T)b + c
qy
the formfactor depending on ion-neutral collision cross section features [1], and
K
is the
global heat conduction coefficient. Using y component of neutral momentum equation (2),
zero net mass flow and neglecting small influence of y components of the inertia and
viscosity terms we find
ji - (M(N + n)KiN ) 1 (dPN/dy) .
(8)
From toroidal (z) component of neutral momentum equation and plasma
momentum balance equation along the magnetic field after some algebra we find
Mji (Vi - b' VN,z)+ b(Pp+ PN)-q b'(dV N, z/dy) - bPu ,
(9)
d(M ji VN, z + Tib'(dVN, z/dy))/dy - MiN nN(VN, z - b'V;),
where b' -
(10)
.
Thus, Eqs. (1), (4), and (7)-(10) describe the plasma-neutral interaction in the
fluid approximation, which account the effect of neutral viscosity. To close this system of
equations we impose the following boundary conditions:
qy(y-' oo)- -qj , qy(y- 0)- -Td{Ypjd + YNNdCd}I
N(y -+ oo) --b 0 ,
jd n -jI(Y- 0) - atbndCd,
VN,z(Y-0)-VN,z(y--*0)M,
where (...)d is the (...) value near the target (y=0), Cd - IFd-/M, a s1, yp and YN are
the plasma and neutral gas heat transmission coefficients, and q, is the specific heat flux
coming to the recycling region. One can see that the transformation 2nT/Pu .- pp,
NT/Pu - pN, dy -+(C
0
/nKo)d , qy/Pu -q,
ji/Pu -+ j, and V N, z - C U, (where
45
CO
=
j7M, KO -KiN(TO), and To is some fixed temperature) removes P, from Eqs.
(1), (4), (7)-(10), and the boundary conditions. Therefore, in the new variables, plasma and
neutral densities and temperature in front of the target only depend on upstream pressure
and heat flux through the ratio qu /Pu. To characterize the solution it is convenient to rewrite the relation between q, /Pu and Td as q. - Pu G(Td). We may also write the ion
particle flux onto the target as id - Pu J(Td) The results of numerical solution of Eqs. Eqs. (1), (4), (7)-(10), and boundary
conditions, written obtained by a shooting method for a=0.5, b=0.05, YN= 0 .2, and
y p =6 for the different temperatures Td (hydrogen ionization and radiation rate constants
was taken from Ref. 3). The function J(Td), and the dependencies of dimensionless
plasma (pp) and neutral gas (pN) pressures near the target on the temperature Td are
shown in Figs. 1, 2. One sees that plasma flux only starts to decrease at a very low plasma
temeprature below 1 eV, while plasma pressure near the target is already much low
upstream plasma pressure. Notice, that in agreement with [4], neutral gas presure starts to
grow quickly with decreasing Td (see Fig. 2). Since in all models discussed in Ref. 1 and
present paper, the plasma flux onto the target does not decrease till a very low temperature,
plasma density in front of the target should scale like nd c
and could be very high
/1VFd
(about 1016 cm-3 for Td - 1 eV) even in current tokamaks. Then plasma recombination,
could significantly alter plasma flux onto the target. Indeed, for the plasma temperature
Td - 1-2 eV and density nd - 1015 cm-3 the conventional three body recombination rate
constant is about 3 x 10-12 cm3 /s. Recombination becomes important when plasma flux onto
the target, jd, is comparable to the plasma particle sink due to recombination, veVnded,
where td is the width of a low temperature region. For Jd - 10 19 cm- 2 s-I we find that
recombination important when Id z jd/vre nd --3 cm which is quite realistic value.
3. Conclusions
i) We show that self consistent decrease of both plasma flux and neutral ionization in current
tokamaks is only possible when neutrals can be treated in a short mean free path
approximation. ii) We investigate these fluid regimes of plasma-neutral interaction with 1D
fluid equations employing a neutral viscosity term to treat the neutral interaction with the
divertor plate. iii) We have found that plasma flux onto the target starts to decrease at a very
low heat flux coming into the hydrogen recycling region, when the temperature near the
target drops below 1 eV, while plasma pressure near the target is already much low
46
upstream plasma pressure. iv) At low plasma temperature near the target the recombination
processes can significantly alter the plasma flux onto the target. To make more definite
conclusion on the importance of plasma recombination in detached divertor regimes a more
detailed investigation has to be done. Notice, that in [4] was shown that plasma
recombination can be enhanced by ion conversion H2 + H+ - H2+ + H and the foregoing
molecular ion recombination, which could be important even at higher temperatures.
Acknowledgments
This work is performed under the U.S. Department of Energy Grant DE-FG02-91ER-54109 at MIT, and under Mexico CONACyT project ES104393 DGAPA-UNAM.
References
[1] S. I. Krasheninnikov, P. J. Catto, P. Helander, D. J. Sigmar, and T. K. Soboleva,
Phys. Plasmas 2, (1995) 2717.
[2] P. Helander, S. I. Krasheninnikov, and P. J. Catto, Phys. Plasmas 1, (1994) 3174.
[3] R. K. Janev, W.D. Langer, K. Evans, and D. E. Post, Elementary Processesin
Hydrogen-Helium Plasmas (Springer-Verlag, Berlin, 1987).
[4] Krasheninnikov S. I., Pigarov A. Yu., and Sigmar D.J. these proceedings.
U.". 1
-
- 0
0
Pn x 10
0am
0.24+
0
1.0
1
2
3
4
0.11-.
0.1
5
Td, eV
Fig. 1. Function J(Td).
l.0
10
WO
Td, eV
Fig. 2. Functions pp(Td), and pn(Td) 47
Simulation of the Alcator C-Mod Divertor with
an Improved Neutral Fluid Model
F. WISING (a). D. A. KNOLL (b), S. I. KRASHENINNIKOV (a),
T. D. ROGNUEN (c), AND D. J. SIGMAR (a)
(a) MIT Plasma Fusion Center, Cambridge, MA 02139
(b) Idaho National Engineering Laboratory, Idaho Falls, ID 83415
(c) Lawrence Livermore National Laboratory, Livermore, CA 94550
Abstract
We have simulated detachment in Alcator C-Mod, using an improved neutral fluid model. It
features a parallel neutral momentum equation, fully coupled to the ions, allowing parallel
plasma momentum to be converted into neutral momentum, which is dispersed to the
material walls due to the high neutral viscosity. The simulations reproduce the pervasive
experimental feature that the plasma detaches gradually, starting at the strike point, while
always remaining attached above the nose of the divertor channel. The strike point heat flux
and current drop by an order of magnitude as 0.5% of carbon is introduced to induce partial
detachment. The plate temperature remains at or above about 1 eV all the way out to the
divertor nose, in agreement with the experimental data.
I. Introduction
The experimental program in divertor physics has made strong progress in recent years.
Increasingly advanced divertor geometries, with varying wall materials and different
combinations of puffing and pumping have been taken into operation or are planned, e.g.
Alcator C-Mod, DIII-D, JET, Asdex and TdeV. These experiments provide valuable data
with which plasma edge codes can compare and validate their different physics contents.
One particularly important and promising phenomenon which occurs in divertor tokamaks is
detachment [1], a state characterized by large drops of the heat and ion fluxes to the divertor
targets and low plasma temperatures. It is also particularly challenging to simulate, due to
the strongly nonlinear processes that are involved, e.g. volume recombination, hydrogen
and impurity radiation, and the sharp gradients of e.g. neutral density that develop. Recent
successful simulations of detachment, with a full 3-D Navier Stokes model for the neutral
fluid, developed for a rectangular geometry, have reproduced the strong drops of the heat
and ion fluxes to the target [2] for plasma parameters similar to those in C-Mod. The most
important physics effects were found to be the strong transfer of parallel momentum from
ions to neutrals and the rapid viscous radial transport of this momentum to the side walls.
In experiments on e.g. Alcator C-Mod it is observed that detachment depends
sensitively on the target geometry. In the vertical target configuration detachment is always
only partial, with the field lines above the divertor nose remaining attached [3]. In order to
simulate fully these nonorthogonal geometries, we have incorporated a 1-D parallel NavierStokes neutral model in the fully implicit code UEDGE. This model retains the important
physics mentioned above and is relevant and appropriate for high density, short mean free
path conditions such as in Alcator C-Mod and ITER. We refer the reader to Ref. [4] for a
48
more thorough discussion of the model. On an orthogonal C-Mod geometry, with divertor
plates normal to the poloidal field, we have shown that this model produces detachment as a
fixed fraction of carbon impurities is added to the plasma [51. We here report on fully
converged results obtained for the strongly curved vertical target divertor configuration of
C-Mod, reproducing the important experimental feature of partial detatchment.
I.
Simulations
The simulations described here are based on the C-Mod discharge #940623018,
with the strike point incident on the vertical target, below the divertor nose (cf. Fig. 1). This
discharge was on the limit of detachment and detached partially for some time. Plasma
parameters were measured using probes mounted permanently along the target plates as well
as with a fast scanning probe (FSP) providing radial SOL-profiles above the X-point. The
electron density at the core boundary was measured to be about 1020 m-3 . Using electron
and ion heat fluxes of 350 and 250 kW respectively, within the error bars of the values
inferred from bolometry, we obtained a core electron temperature of about 50 eV, in
agreement with the FSP. The upstream radial profiles of ne and Te closely match the values
from the FSP, for D1L =0.25 m2 /s and Xii = X.e = 0.4 m2/s.
Target heat flux (MW/M 2)
ELECTRON TEMPERATURE (eV)
.36. 34
.32
z .30
--
0%C
2.
0.3%C
.28
V)
.24
.20
0.5
/
01,
.16
14
.IC%
V:
11
V! W! V!
11
.0 '0
"R
R
MAJOR RADIUS (m)
Distance from separatrix (m)
Fig. 2. Outer target heat flux for
varying carbon fraction.
Fig. 1. Simulated Te contours at
detachment (0.5% C)
Using this attached discharge as a base case we proceed to induce detachment by
adding a fixed fraction of carbon, keeping core density and heat flux fixed. The main result
of impurity radiation is a decrease of the target heat flux, thermal detachment The
simultaneous drop of Te may lead to increased penetration of recycled neutrals as well as
volume recombination, eventually causing pressure detachment and finally plasma flux
detachment As we add carbon, up to nc/ne=0.5%, the plasma detaches gradually, starting
at the separatrix, while remaining attached above the nose. At 0.5% carbon, the total ion and
heat fluxes, as well as the peak values, have been reduced by about a factor of two, see
Figs. 2 and 3. However, the heat and ion fluxes at the strike point, and at the location of the
no-carbon peaks, are reduced by an order of magnitude, and the locations of the maxima
49
have shifted radially outward, close to the nose. Detachment proceeds slightly faster at the
inner target plate, but the asymmetry in the peak heat flux is less than 25%. A carbon
content of 0.5%, as well as the 185 MW of radiation that it produces, agrees well with
typical experimental values for C-Mod.
Divertor ion current (kA/r2)
0%C
--
2000-
1500-
-----
,~.
-
/
'\
Te (eV) at target
10-
0.3%C
0.5%C
a-
-5- Probes, med density
1000-
I
6-
- Probes, hi density
I'
/
4-
500-
2-
/
/
*1
/
A
0
Distance from separatrix (m)
Fig. 3. Outer target ion flux. Probe data
from semi-attached (medium density) and
detached (high density) shots are shown.
Fig 4. Te at outer target. Probe data from
semi-attached and detached discharges are
shown. Line captions as Fig. 3.
As can be seen in Figs. 2 and 3 detachment 7E+21
U
loniz,0%C
is only partial and does not extend all the
+21
o Rec, 0%C
way to the nose of the divertor, located 6
roughly at (.62,.25) in Fig. 1. Fig. 3 con- SE
+21
S
A loniz, 0.5%C
tains probe data for two different discharges
a Rec, 0.5%C
[3], and we note that the high density dis- 4E +21
charge, in contrast to our simulation, is parAA
tially detached all the way to the nose. Fully 3E+21
converged solutions above 0.5% C would
+21
AAA
require further grid refinement. Contours of 2
Te for 0.5% carbon are shown in Fig. 1. 1E+21
A
The separatrix Te remains low over an extended region between the target and the XS0prari 1
15
2D 25 3
35
point, in qualitative agreement with temperatures inferred from measurements of neutral
Current peak. 0.5%C
Current peak, OC
densities with a gauge located in the private
Field
line
number
flux zone. Comparisons between simulaFig 5. Total number of ionization and
tions of Te at the plate and probe data from
volume recombination events (s0)
the same discharges as in Fig. 3 are given in
integrated along field lines.
Fig. 4. Note that the simulated temperatures
remain above I eV, in agreement with the
probe data and higher than in orthogonal
simulations.
50
The important effect of volume recombination for plasma flux detachment is
demonstrated in Fig. 5. The former peak of the target ion flux is reduced by an order of
magnitude; almost all ions at this field line are recombined before they reach the target. This
underlines the need to include volume recombination in any code simulating detachment.
III. Conclusions
The vertical target divertor geometry of Alcator C-Mod has provided promising
results with respect to detachment. The strongly nonorthogonal geometry represents a
challenge for numerical simulations, particularly in combination with the additional
difficulties associated with detachment. We have reported on successful fully converged
simulations of partial detachment, where the plasma detaches half way to the nose of the
divertor. We used 0.5% carbon to induce this detachment, and it seems clear that the partial
detachment will proceed all the way to the nose for higher impurity fractions. The
simulations confirm the strong role of geometry on detachment; partial detachment initiates
at the strike point, both due to the long lengths of the field lines near the separatrix and to the
increased radial flux of neutrals to these field lines due to the tilted plate geometry. We have
also pointed out the strong role of volume recombination in removing the target ion flux plasma flux detachment - and the need to include it in simulations of detachment. The
amount of carbon radiation that we produce agrees well with experimental values, but future
extensions of this work include more refined impurity modelling as well as studies of
discharges in which detachment was induced by injection of neon.
Acknowledgements
This work was sponsored by the Swedish Natural Science Research Council and the U.S.
DOE Grants: DE-FG02-91-ER-54109 at MIT, DE-AC07-94ID13223 at INEL, and W7405-ENG-48 at LLNL.
References
[1].
[2]
[3]
[4]
[5]
MAmEws, G. F., 11th International Conference on Plasma Surface Interactions, Mito,
Japan 1994.
KNOIL, D. A., McHuGH, P. R., KRAsHENmKov, S. I., and SIGMAR, D. J.,
"Simulation of Dense Recombining Divertor Plasmas with a Navier-Stokes Neutral
Transport Model", To appear in Phys. Plasmas (1996).
LABOMBARD, B., Gooz, J., KuRz, C., et al., Phys. Plasmas 2 (1995) 2242.
WISING, F., KNoL, D. A., KRAsHENNmKov, S. I., RoGNuEN, T. D., and SIGMAR, D.
J., "Simulation of Detachment in ITER-Geometry Using the UEDGE Code and a
Fluid Neutral Model", These proceedings.
WISING, F., SIGMAR, D. J., KNoL, D. A., and RoGNuE, T. D., International
Sherwood Fusion Theory Conference, Lake Tahoe (1995).
51
Simulation of Detachment in ITER-Geometry Using
the UEDGE Code and a Fluid Neutral Model
F. WISING (a). D. A. KNOLL (b), S. I. KRASHENR;NKOV (a),
T. D. ROGNLIEN (c) AND D. J. SIGMAR (a)
(a) MIT Plasma Fusion Center, Cambridge, MA 02139
(b) Idaho National Engineering Laboratory, Idaho Falls, ID 83415
(c) Lawrence Livermore National Laboratory, Livermore, CA 94550
Abstract
We have simulated detachment in ITER, with the UEDGE code coupled to a Navier-Stokes
neutral model, for realistic toroidal geometry and plasma parameters, with a heat flux of 100
MW into the outboard SOL. For a fixed fraction of carbon, corresponding to Zeff-2.5 in
the SOL, the power to the outer plate is reduced by more than 90%, the peak plate
temperature is reduced to about 1 eV, and the total ion saturation current drops by 85%.
Volume recombination plays an important role in removing the current. Bifurcated solutions
and marfe-like structures below the X-point have been found under certain conditions.
I. Introduction
Thermal and momentum detachment is observed in many tokamaks, e.g. Alcator C-Mod,
JET, DIII-D and Asdex-U [1]. The thermal detachment is characterized by a large drop in
the target heat flux, at target-plasma temperatures of the order of a few eV, as compared to
the attached (high recycling) state. Momentum detachment is observed as an order of
magnitude decrease of both the plasma flux, as measured by target probes, and of the ratio
of the target to upstream total pressure.
For typical ITER conditions, 80% of the alpha power is assumed to go to the
divertor, resulting in local power loadings of 15-30 MW/rn 2 even for tilted divertor plates
[2]. As this is unacceptable from an engineering point of view, detachment is a very
promising operating scenario for ITER. It is therefore important to understand the various
physics mechanisms and geometry effects that come into play at detachment. A considerable
computational effort has evolved, aimed at interpretation of present experiments and
code/physics validation. As the improved physics models, including fluid and Monte-Carlo
neutrals, in some codes are now better able to reproduce the experimental results of present
machines, more faith can be put in the predictions for ITER. However, while thermal
detachment and pressure detachment are observed in many simulations, plasma flux
detachment (a large drop of the target ion flux) seems more difficult to reproduce, and is
less often reported on. Note that it is a common misconception that pressure and plasma
flux detachment are equivalent. In one simulation of ITER [3], the ion current was reduced
by a factor of 10 as a fixed fraction of impurities were added, but in another example in the
same paper the drop was much weaker, possibly due to the strong non-linear dependence of
current removal on radiated power. It appears that volume recombination is an important
factor for extinguishing the ion flux; for C-Mod like plasma conditions the saturation current
dropped by two orders of magnitude with volume recombination, while the peak current
52
changed very little without it [4]. Volume recombination has often been avoided in
simulations, due to convergence problems, but it was included in Refs. [4,5] as well as in
the work presented here.
II. Physical Model
The simulation of neutral particles in high density plasmas such as in ITER could be carried
out either with Monte-Carlo or fluid models. However, because of the short mean free paths
and importance of neutral-neutral collisions near the target, rendering fully non-linear
Monte-Carlo methods extremely expensive, we have adopted a fluid approach. Simulations
with a full 3-D Navier Stokes fluid neutral model, developed for a rectangular geometry
have shown that the interspecies momentum conservation and viscous transport of parallel
neutral momentum are two essential features which, in addition to the thermal detachment,
produce detachment of the particle current to the plate.
In order to incorporate these physics effects in simulations of realistic plasma
geometries we have added a new neutral fluid transport model [5] to the 2-D edge plasma
fluid code UEDGE. We solve an additional equation for the neutral parallel momentum,
fully coupled to the ions through ionization, recombination and charge exchange:
at
rMNv.11+
rMNn vnxvnl-
ax[
a
--
(ov)izMNeNnvni +(ov)r
I
+-
MNvyvnl -
Mvax
ay
[
avX11
ay
MNeNivil -(Ov)XMNnNi(vnii
V Pn -
- vill)
Here, Tn is the isotropic neutral viscosity; all other notations are standard. Neutral-neutral
collisions are included in in as well as in the neutral thermal conductivity in the combined
ion+neutral energy equation. The radial velocity Vny is diffusive, determined by the charge
exchange frequency, while the poloidal velocity vnx contains projections of the parallel
neutral velocity and the diffusive velocity in the third direction. The strong coupling
between the ion and neutral momentum equations allows transfer of parallel ion momentum
to the neutrals. The neutral momentum is dispersed to the walls due to the high neutral
viscosity.
I1. Results of Simulations
We have studied an ITER divertor benchmark geometry, with target plates orthogonal to the
poloidal magnetic field. Some characteristic parameters are R=8 m, a=3 m, BO=6 T and the
distance from the divertor to the X-point is about 2.3 m. In order to be able to converge
fully on this large device, measuring about 14 m from the outer plate to the symmetry point,
we have chosen to simulate only the outer half of the SOL. We use 174x24 grid points,
with a poloidal spacing of 3 mm at the target, to be able to resolve the very sharp gradients
at detachment We solve for a power flux of 100 MW from the core, a separatrix density of
4.1019 M- 3 , and take D1i = Xii = XIC = 1 m2 /s. For a pure DT-plasma at these densities,
hydrogen radiation alone is insufficient to reduce the temperatures and increase the neutral
density near the target enough to produce detachment. In our simulation, detachment is
induced by adding a fixed fraction of carbon (nc/ne is fixed), with a non-coronal radiation
53
function taking into account cx between carbon and hydrogen atoms [6]. As we add 4.8%
of carbon (Zej=2.5), producing 80 MW of carbon radiation, the peak heat flux to the target
is reduced strongly, falling from 48 to 4 MW/m 2 (Fig. 1). The total target ion flux is
reduced by 85% (Fig. 2), the peak electron temperature becomes 1.2 eV (Fig. 3) and neutral
densities increase by an order of magnitude to around 1021 M-3 . We could not converge
fully above 4.8% C, as the ionization front moves farther from the plate where the grid
spacing increases.
5,.
Heat flux to plate (MW/mA2)
Ion saturation current (MA/mA2)
45
2.5
0%C
35
0% C
2
3
1.5"
KA
15
10
0.5u-u
0
C
J
4.8% C
5-
4.8%
CD
9
00
9
0
9
Distance along plate (m)
Fig 2. Target ion current.
Distance along plate (m)
1.
Fig Target heat flux.
Plate Te (eV)
Ionization / recombination events
g loniz. 0% C
9E+25
30.
A Reoomb, 0% C
9E+25
0% C
7E+25
o loniz, 4.8% C
7E+25
A Recomb, 4.8% C
U
2D
no
4E+25
15
*
3E+25.
10
fy
2E+25.
1E+25.
5-
a
4.8% C
0
M
0
a
I
0
9
Distance along plate (m)
Distance along plate (m)
Fig 3. Plate Te, Attached/Detached.
Fig 4. Total number of ioniz. and recomb.
events (m-1 s-1) integrated poloidally.
Volume Recombination is a very important effect in reducing the ion flux incident on the
target. The total number of ionized particles along a field line, which roughly speaking
would stream to the target in the absence of recombination, are at detachment seen to be
almost completely balanced by volume recombination (Fig. 4). The 4 cm wide region
54
outside the separatrix where this is not so corresponds to what remains of the ion current
(Fig. 2). Globally, we find Srec/Sioniz - 0.79.
At low carbon densities the X-point region is hot, producing little radiation. For
intermediate carbon levels (nc/ne~0.4-1.6%) however, we have found fully converged
bifurcated solutions. In the "Hot" solutions the X-point region is still hot, with a
temperature above the peak in the carbon emissivity, and there is little radiation. In the cold,
"Marfe-like" solutions the temperature below the X-point is a few eV, below the peak in the
carbon emissivity, and there is considerable radiation. In this case Te peaks about 80 cm
further down poloidally and the pressure profile in the private flux region indicates
detachment (Fig. 5). The strike point energy and ion fluxes decrease by about 30% (Fig. 6),
while the peak values at the target are less strongly affected.
Plasma pressure (Pa)
700
Bifurcation at stike point
Ion Flux
45
0% C. Attachd
700
.......-
500
4oo
1% C, MaLL2
200
-40
~
10
4.8% C, Detached
12.7
13.2
Hea Flux
15E+24
63
123
E+23
-
-k
100
12.2
-
35 Hot X-point
-,
251E+240
23
1% C, Hot X-point
13.7
0
14.2
Poloidal distance (M) Target
Fig 5. Pressure 1 cm from
separatrix in private flux region.
X-point
001
.02'
Mare
003
.4'
'0
Carbon fraction, nc/n,
Fig 6. Bifurcation of strike point heat and
ion flux occurs at certain impurity levels.
IV. Discussions and Conclusions
These simulations have been carried out in an ITER geometry with orthogonal target plates.
Currently, the vertical target geometry, which has been highly succesful in C-Mod, seems
to be a stronger candidate for ITER. Its main advantage is a lower threshold for detachment,
combined with a projection of the incident heat/particle flux over a larger surface area.
Detachment may only be partial, but even so the peak fluxes should be strongly reduced.
Thus, lower values of Zef than observed here should suffice.
Simulations of JET [7] have shown that with multispecies impurities 70% of the
impurity radiation takes place above the X-point, while the same number for a fixed fraction
impurity model is 25%. However, the target parameters were similar in the two cases.
Thus, while the constant impurity fraction model that we use here may not give all details,
the general trends that we observe near the target should be correct A more quantitative
study of the marfe-like structures that we have observed should include an additional
continuity equation for the impurities. We have not yet analyzed the stability of the
bifurcated states, and it is not clear where the transition occurs. The stability and location of
marfes, in different radiation regimes and for different geometries, have been studied in
55
Ref. [8] who found a closed, vertical target geometry best for detachment and for control of
unstable marfes.
An important conclusion of our simulations is that volume recombination needs
to be taken into account to explain the strong removal of the ion current at detachment. It
also provides an extra source of energy in the plasma near the target, allowing higher
electron temperatures, in close agreement with measurements.
Acknowledgements
This work was sponsored by the Swedish Natural Science Research Council and the U.S.
DOE Grants: DE-FG02-91-ER-54109 at MIT, DE-AC07-94ID13223 at INEL, and W7405-ENG-48 at LLNL.
References
[1] MATHEws, G. F., 11th Int. Conf. on Plasma Surface Interactions, Mito, Japan 1994.
[2] JANEscHriz, G., BoRRAss, K., FEDECIc, G., et al., J. Nucl. Mat. 220-222 (1995)
78.
[3] WEBER, S., CoRiGAN, G., SImomm, R., SPEcE, J., and TARoM, A., Varenna, 1994
[4] KNolL, D. A., et al., 37th Annual Meeting of the APS-DPP, Louisville KY (1995).
[5] WIsjNG, F., SIGMAR, D. J., KNOL, D. A., and RoGNuEN, T. D., International
Sherwood Fusion Theory Conference, Lake Tahoe (1995).
[6] Posr, D. E., J. Nucl. Mat. 220-222 (1995) 143.
[7] TARom, A., CORIGAN, G., SIMomm, R., SPENCE, J., and WEBER, S., 11th
International Conference on Plasma Surface Interactions, Mito, Japan 1994.
[8] SCHNEIDER, R., REr=R, D., COSTER, D., NEuAusER, J., LACKNER, K., and BRAAms,
B., 11th International Conference on Plasma Surface Interactions, Mito, Japan 1994.
56
Kinetic Modelling of Detached and ELMy
SOL Plasmas
0. V. BATISHCHEV (a,b,1), X.
Q. XU
(c), J. A. BYERS (c),
R. H. COHEN (c), S. I. KRASHENINNIKOV (a,2),
T. D. ROGNLIEN (c), D. J. SIGMAR (a),
(a)
(b)
(c)
(1)
(2)
Massachusetts Inst. of Technology, Cambridge, MA 02139, USA;
Lodestar Research Corporation, Boulder, CO 80301, USA;
Lawrence Livermore National Lab., Livermore, CA 94550, USA;
Perm.address: Keldysh Inst. of Appl. Math., 125047 Moscow, RF;
Also: Kurchatov Inst. of Atomic Energy, 123098 Moscow, RF.
Abstract
Kinetic modelling of parallel plasma and heat flows in the scrape-off-layer (SOL) is
presented. Detached [1-4] and attached regimes are studied. A kinetic Edge Localized
Mode (ELM) [5] model is simulated by temporally varying the incoming heat power
from the core. Deviations of the plasma distribution functions from Maxwellian are
observed. Our simulations are performed using the collisional PIC code W1 [6], which
employs a non-stationary 1D2V fully kinetic model of a high recycling SOL plasma.
1. Introduction
A fluid plasma description can be inadequate for edge plasma modeling, since it assumes a short charged particle mean free path expansion. This expansion fails near a
plasma-material interface such as a divertor target. Absorption of plasma ions results
in a strong departure of the distribution function from Maxwellian. Fluid descriptions can also typically fail in the SOL because of strong plasma gradients along the
magnetic field lines. These gradients are associated either with neutral recycling near
the targets or with energy loss due to impurity radiation. It is known that Braginskii
plasma transport coefficients start to fail even for a relatively small ratio y of the
Coulomb mean free path of a thermal particle A, ~ 1012 T,(eV) 2 /n,(cm- 3) (T, and
n, are the plasma temperature and density) to the parallel temperature scale length
L. The typical y values for Alcator C-Mod, D-III-D, and JET as well as for ITER are
of order 1/10, for which one can expect a strong influence of the nonlocal features of
parallel plasma transport on the transport properties of the weakly collisional energetic particles. One of the uncertainties of a fluid plasma description is the modeling
57
of the detached divertor regimes observed on most diverted tokamaks. These regimes
are characterized by a significant plasma pressure drop in the divertor region of the
SOL, strong plasma parameter inhomogenity along the magnetic field, and very high
energy losses due to impurity and hydrogen radiation and charge-exchange, resulting
in a reduced heat load onto the target. This feature makes detached divertor operation
very attractive for the ITER divertor. In this work we describe the results of modelling
detached divertor regimes with the fully kinetic W1 code which does not require the
short mean free path expansion assumption of fluid plasma codes.
2. Physical Model, Equations and Geometry
Our model of SOL includes the following important features: Coulomb e-e, e-i, i-e, i-i
collisions; plasma recycling; neutral transport using various approximations; plasmaneutral interaction; self-consistent ambipolar electric field and sheath potential; impurity radiation model.
To describe the plasma evolution we solve the system of kinetic equations for ion and
electron distribution functions ft2(t, 11, v11, v±):
f_
--2 +
Of --t +fq-Ell
El- af- =
+
ah+
CN
Q=e,i
,
(1)
Here C2 is the non-linear Landau collision term in axisymmetric geometry, C, is a
heating term modeled by diffusion in the perpendicular velocity vj, and C.N represents
plasma-neutral particle collisions. For electrons we take into account hydrogen excitation and ionization, and impurity excitation, which are described by a Boltzmann
operator written in the small mass ratio approximation in energy space. For hydrogen ions we take into account the ionization and charge-exchange channels. For the
neutrals we adopt two long mean free path models. They consist of either a diffusion
equation for neutral density poloidal profile for a slot diverter or a balance equation
for the neutral density amplitude in a gas-box divertor. We obtain the ambipolar electric field from the parallel momentum balance equations, taking into account pressure
gradients and friction between plasma species. The sheath potential is evaluated using
a logical sheath boundary condition. The model geometry is shown in Fig.1.
mpurity
etasheating
divertor /
plate
symmetry
/-uta-s--------------0
plane
core
X
L
Fig.l. Geometry of the problem
Particles are assumed to be completely reflected at the symmetry plane. The divertor
plate absorbs ions and energetic electrons to maintain ambipolarity. The outgoing
58
neutral flux from the plate equals the incoming ion flux. The potential and electric
field are zero at the symmetry plane.
3. Results on Attached - Detached Transition
The following parameters remain fixed during the simulations: average plasma density
nP = 10 4 cm- 3 , connection length L = 5m, inclination angle of magnetic line to the
plate 0.05, and neutral diffusion coefficient DN = 10 5 m2 /s. Neutrals were assumed
to have a Maxwellian distribution with temperature TN = 2eV. The only parameter
varied was the heat flux q, coming from the core, associated with the heating term in
Eq (1). There were no impurity radiation losses in these simulations. Figs.2 to 5 display
the variation of the key plasma parameters versus G = q/Pmid, where Pmid(qoo) is
parallel plasma pressure at the midplane and G is measured in 10~ 9 MWm/eV units.
When we decrease q, (and hence, G), the electron temperature at the plate Td drops
from 100 to 1.7 eV, with a sharp step around a bifurcation threshold (Fig.2). The ratio
of midplane to plate pressure Pmid/Pd behaves in just the opposite way (Fig.3). The
transition from attached to detached flow is characterized by a sharp pressure drop at
the plate with respect to the upstream region. The thermal bifurcation threshold is
in a good agreement with the theoretical prediction [7].
2
A
A
2
A
AO
L
I
S
4
i
*
4
2
5
55
G
5
G
Fig.2. Td at the plate
Fig.3. Pmid/P during transition
The density at the plate nd and particle current onto the target Jd pass through
maximum values as q, is decreased. The plasma density first rises, then stays high for
a wide range, then drops almost linearly as G -+ 0 (see Fig.3). The particle current
onto target, shown in Fig.4, has a much sharper peak and at a somewhat higher G
value, due to the temperature dependence of jd ndTd-.
Fg4
nd agcmd
tr0aM"t-
A
AA
A
A
0
A
A
A
G
G
Fig.4. nd at the plate
Fig.5. jd during transition
59
During the transition, the ionization and excitation fronts move away from the plate
up to 1 meter along the magnetic field. The plasma flow velocity, estimated in the last
2 cm region near the plate, drops from .25C, to .1C, (C, is the local sound speed) due
to the rising neutral density near the target. The short neutral mean free path (below
1 cm near divertor) is responsible for the sub-sonic plasma flow towards the target.
It was also found that the parallel electron distribution function has an elevated tail
for both detached and attached cases near the plate, with the effective temperature
exceeding the bulk temperature by 2-3 times. This kinetic effect must be taken into
account when interpreting the electron temperature probe measurements in the SOL.
We also found that the ion distribution function is non-Maxwellian near the plate for
detached flows. The changes in the excitation and ionization rates are 20-50 %.
4. Influence of Impurity Radiation
In the next simulation the key parameter for establishing detachment was the impurity
density n,. Its spatial distribution is shown in Fig.1. We started from steady-state
profiles with q, = 2000MW/m 2 and then obtained steady-state profiles for nz =
10, 20, 50, 75 x 101 3 cM- 3 as shown in Figs.6,7. Evolution of the electron temperature
Te profile is shown in Fig.6. The particle flux onto the target to midplane pressure
ratio jd/Pmid is found to have a broad maximum at n. ; 30 x 1013 cm- 3 as can be seen
from Fig.7.
10,1
.................
02
idtmid. 8.u-
01
A0
:2
30
Z0
0
a07
.in03,-
m
Fig.6 Tfo
Fig.6.
T for different
n,
Fig.7. jd7Pnid vs n,
4. ELM Cooling Simulation
We simulated ELM cooling by switching off the heating term in Eq.(1). The initial
profiles are steady-state ones for q = 2000MW/m 2 . The temperature profiles at
different times are presented in Fig.8. Since the energetic particles stream towards
and are absorbed by the target in a short time, the parallel distribution function
fi = f fevdv1 is asymmetric and below Maxwellian on the outgoing wing (Fig.9).
60
24
T,,ev
*-
-
Mxwln-
-2
2 -
-
140.
X.
m
4-to divnrW
Fig.8. T during ELM cooling
Wr
Fig.9. Inf 1 during ELM cooling
We have found that for this fii the short mean free path heat conduction coefficient is
2 times smaller and the hydrogen ionization rate is 50% smaller than for a Maxwellian
distribution with the same temperature and density.
6. Conclusion
Realistic kinetic models are presented of the attached to detached SOL plasma flow
transition and ELM degradation. We found that a variation of the incoming heat
flux or its reduction by radiating impurities located between the X and strike points
can cause detachment. We found that the electron and ion distribution functions
depart from Maxwellian. During the heating stage the following kinetic effects occur:
hydrogen reaction rates 1.2-1.5 times higher than Maxwellian ones, divertor probe
measurements overestimating electron temperature by a factor of 2-3, and a sheath
potential drop of 1.8 - 2.5T. During ELM cooling stage when the energetic tail is
depleted, reaction rates and the heat conduction coefficient can be 2 times smaller
than Maxwellian values.
Acknowledgments
Work performed under US DOE grants and contracts DE-FG02-91-ER-54109 at MIT,
DE-FG02-88-ER-53263 at Lodestar, and W-7405-ENG-48 at LLNL.
References
[1]
[2]
[3]
[4]
[5]
I.H.Hutchinson et al., Physics of Plasmas 1, 1511 (1994);
T.W.Petrie et al., Journal of Nucl. Materials 196-198, 848 (1992);
G.Janeschitz et al., Proc.19th Eur.Conf.CFPP,Insbruck, 1992, 16C, Part II, 727;
V.Mertens et al., Proc.20th Eur.Conf.CFPP,Lisboa, 1993, 17C, Part I, 267;
H.Zohm et all, Nucl. Fusion 32, 489 (1992);
[6] O.V.Batishchev et al., Contrib. Plasma Physics 34, 436-441 (1994);
[7] S.I.Krasheninnikov et al., Physics of Plasmas 2, 2717 (1995).
61
Fokker-Planck Simulation of Electron Transport
in SOL Plasmas with ALLA Code
A. A. BATISHCHEVA (a), 0. V. BATISHCHEV (a,b,1),
S. I. KRASHENINNIKOV (a,2), D. J. SIGMAR (a),
M. M. SHOUCRI (c), I. P. SHKAROFSKY (c)
(a)
(b)
(c)
(1)
(2)
Massachusetts Inst. of Technology, Cambridge, MA 02139, USA;
Lodestar Research Corporation, Boulder, CO 80301, USA;
Centre Canadien de Fusion Magnetique, Varrenes, Quebec, Canada;
Perm.address: Keldysh Inst. of Appl. Math., 125047 Moscow, RF;
Also: Kurchatov Inst. of Atomic Energy, 123098 Moscow, RF.
Abstract
The recently developed Fokker-Planck code ALLA [1] permits us to simulate the temporal evolution collisional edge plasmas in 1D in space, 1V in velocity space. It was
used to simulate the SOL parallel electron flow in Alcator C-Mod (PFC MIT) and
TdeV (CCFM) tokamaks. Plasma parameters are taken as "best guesses" from available detachment experimental data [2,3]. We show that the actual electron distribution function is asymmetric in the direction parallel to the magnetic field and its tail
starts to departure from the Maxwellian at energies around 2T. This feature can substantially affect probe measurements and modify reaction rates and plasma transport
properties.
1. Introduction
The ratio of the electron Coulomb mean-free path A to the connection length L for
TdeV is around 1/10 and for C-Mod is around 1/40. This means that the electron
distribution function is almost Maxwellian for thermal particles with energies e ;:: T,,
the plasma temperature. This is nowever not true for suprathermal electrons because
of the known dependence A, ~ 1, where n, is the plasma density. But these electrons
influence the parallel heat conduction and impurity excitation rates. Simultaneously,
they determine the electron temperature probe measurements. For example, the floating potential for a deuterium plasma is 2.8T,. The slope of the probe VA characteristic
measured within a narrow interval around this value is then interpreted as the actual
temperature of the electron distribution. The effective temperature T eff of the electron
np
62
distribution function fe(s) evaluated as
(d in f(e)\
T4 f = - (
d
is equal to T, for the Maxwellian core distribution, but can be significantly bigger or
smaller for the suprathermal tail. As a result, probe measurements can under- or overestimate the electron temperatures. Also the non-Maxwellian tail can be responsible
for the experimentally found variation of upstream and downstream temperatures as
measured by reciprocating probes on C-Mod [4].
2. Equations and Geometry of the Problem
In our model the 3D electron distribution function fe(x, r,p) is described by the kinetic
equation:
af'e
5T
Ofe
I
10
eE
- -2 ( (V2/. -f)+Vp +vp~.
Ox
v
0V
10
m,
v8p
(1
2
A2
_i ] = CC
C
(2)
2
MI
here x E [0, L] - parallel to magnetic line direction, v = - v +v2 - modulus of velocity,
o /v - cosine of angle between particle velocity and x axis; e and me - magnitude
of the electron charge and mass, E, - parallel ambipolar electric field derived from
Braginskii parallel momentum balance equation:
eE, = -0.71
Ox
np, Ox
(3)
CF, - is the electron Coulomb collisional term taken in a form equivalent to [6] for
e-e interaction, but for e-i we are taking into account only pitch-angle scattering (no
energy exchange). The drag and diffusion terms in the Fokker-Planck operator depend
on the Rosenbluth potentials, which are calculated with fixed Maxwellian functions
both in time and space. The input plasma profiles are obtained from a conduction
model reconstruction of experimental data, provided plasma pressure is assumed to
be constant along the magnetic line:
T, ~ XT
,
n ~4 T
(4)
In our geometry, the divertor is located at x = 0 and the connection length L is different for various flux surfaces. Plasma temperature profiles match the divertor probe
measurements and fast scanning measurements taken between strike and stagnation
points. The initial electron distribution was assumed to be Maxwellian with fixed
n,(x), T,(x) and zero drift velocity. We assume that both Debye length and ion gyroradius are small, so we can not resolve sheath structure. Thus we use elastic reflection
boundary conditions for electrons at both target and symmetry plane. This system
was permitted to equilibrate in the presence of frozen electric field and self-consistent
collisions with background plasma.
63
3. Numerical Method
We performed simulations with the finite-element Fokker-Planck code ALLA [1] which
implements splitting schemes on non-uniform grids in real and velocity space with a
cubic spline technique [5] and implicit scheme for Coulomb operator. The time step
was comparable to the e-e Coulomb time for cold plasma near the target. Spatial
resolution near the plate was 1 mm, while near the midplane was 10 cm. The velocity
mesh varied from .01vt to 6v0", where indexes d and m denote divertor and midplane
electron thermal velocity vt = V2T/me. We used a grid size typically 100x129x33.
CPU time on HP/735 and SUN-10 was one to few hours till system equilibration.
4. C-Mod Results
We performed simulations for C-Mod open flux surfaces p = 1mm and p = 2mm (p
- radial distance from separatrix at midplane) using experimental detached data [2].
They were L = 12.6m, T = 3.2eV, T, = 48.6eV, nd = 6.9 101cm- 3 and L = 11.6m,
Td = 5.1eV, Tm = 44eV nd = . 10 1 cm- respectively. Figures 1 to 6 represent
obtained results for p = 1mm. As it can be seen from Fig.1 the parallel electron
distribution function fil f fevjdv± near the plate starts to depart from Maxwellian
at c = 2T,. Due to this, the effective temperature Tff (shown at Fig.2) exceeds T, by
40% at e = 3T, and by 80% at e = 4T,.
(Plate)
1mm
I
-1 p
max
2.8
-a..
-2 .
p1 mm (Plate)
TfT-bu.k -
2.6
2.4.
2.2
.-.-
1.6
14
1.2
t6
I
Fig.1. Inf1 at C-Mod divertor
Fig.2. Teff/T at C-Mod divertor
In Figs.3,4 we present fli and Teif obtained from simulation at point x = 7.2m, which
corresponds to fast scanning probe location for p = 1mm. Here the distribution is
clearly asymmetric in the parallel direction. Teff at downstream branch exceed T p,
while at upstream branch is significantly below. The magnitude of the temperature
variation at e = 3T, is around 30%, which is consistent with C-Mod experiment [4].
01.
.1
1m
,
p
w
1.2
p
I mm (FS probe)
2l
Tfr-bulk -
pre(FS
m F
probe)TA
13.
-3.
I
-5
Fig.4. T eff/
Fig.3. Infli at FS probes
64
T
at FS probes
Figs.5,6 show that our previous assumptions are relatively good. In the bulk, the 2D
distribution function f,(vij, vi) (Fig.5) has completely Maxwellized. Departure from
an equilibrium function occurs at v ; 2.5vt. As the Rosenbluth potentials are smooth
functions of fe, they can be taken as Maxwellian. The temperature profile stays almost
the same throughout the simulation (Fig.6). The density drops 20% only in the last
1cm region near the plate, and shows no changes along the rest of the 12 meters of
the magnetic line length. That means that the linearized collision term and Braginskii
electric field give us a consistent kinetic model of parallel electron transport in SOL.
70
4
p
1mTe
.
40
30
VNt
20
5
-.
0
10
h"* 4vt
U.001
Fig.5.
f,,( vl,
vi) x=10 cm
0.01X
10
Fig.6. Plasma profiles
We also found, that the heat conduction coefficient as well as the excitation and ionization rates can be significantly different with respect to similar Maxwellian quantities.
This means, that real plasma profiles should be different with used background profiles.
Showed in Fig.6 profiles are not changing much as we are solving liner problem.
5. TdeV Results
TdeV being one order of magnitude less dense and a more hot machine than C-Mod
shows even more pronounced kinetic features. We used detached profiles [3], but
because experimental data are not available for the whole magnetic line, we have
restricted our simulation from the x-point to the target region. The data used are
L = 2m, Td = 4eV, T," = 10eV, n, = 4 - 8 101 2 cM- 3 . On Fig.7 the parallel
distribution function fii at the divertor plate is shown. Even though the variation
along simulation domain of T is 2.5 instead of 16 for C-Mod, fli departs even more
from Maxwellian. But for the same reason Teff saturates 'faster' at high energies
(Fig.8).
1
.4..3
.
Fig.7. 1nfil at TdeV divertor
3
-3
1..
.1
Fig.8. Tei !/T, at TdeV divertor
Finally we estimated reaction rates and the heat transport coefficient and then compared them with Maxwellian values. Fig9 shows the ratio < 0v >f
65
/
< av >MAX
for H ionization, H and C 3+ excitation. Fig.10 shows short mean-free path heat
conduction coefficient K oc f v"fjjdv modification with respect to the Maxwellian.
1.53
H-ioniz
1.4
***
e.
K/K-m.
KK..
25
*
1.2
2.
1.1
1.5%
0.91
0..1
1
U.O1
X, MT
01
U.
X. Mi
Fig.9. Reaction rates
1
Fig.10. Ratio K/KmAX
We add note that for more steeper plasma profiles and for other processes (like high-z
impurity ionization) those ratios can exceed 100.
6. Conclusion
A consistent kinetic model of electron transport in C-Mod and TdeV tokamaks is presented. Fokker-Planck kinetic simulations with the ALLA code using plasma profiles
which are matching experimental data, show significant departure of the electron distribution from Maxwellian. This results in overestimation of plasma temperature by
divertor probes by an average of 60%. Simultaneously, fast scanning probes underestimate it by 20%. Variation of upstream and downstream temperature by a factor of
.3 measured experimentally by reciprocating probes is in agreement with our results.
Plasma reaction rates can be 50% higher near the target and 20% lower at midplane,
compared with the Maxwellian values. The heat conduction coefficient can be an
order of magnitude greater than Maxwellian in the divertor vicinity due to strong
dependence on suprathermal particle population.
Acknowledgments
Work performed for US DOE by MIT under Grant DE-FG02-91-ER-54109 and by
Lodestar under Grant DE-FG02-88-ER-53263.
References
[1]
[2
[3]
[4]
[5]
[6]
A.A.Batishcheva et al., MIT Report, PFC/JA-95-23 (1995);
I.H.Hutchinson et al., Physics of Plasmas 1, 1511 (1994);
B.L.Stansfield et al., Proc. 22 Europ.Conf.CFPP, Bornemouth, 19c pIII-101;
B.LaBombard, private communication;
M.M.Shoucri and G.Gagne, J.Comput.Phys. 27, 315 (1978);
A.A.Batishcheva et al., "Massively Parallel Fokker-Planck
Code ALLAp" (this issue).
66
Kinetic models of ELMs burst
D. J. SIGMAR (a), A. A. BATISHCHEVA (a), 0. V. BATISHCHEV (a,b,1),
S. I. KRASHENINNIKOV (a,2), P. J. CATTO (a,b)
(a) Massachusetts Inst. of Technology, Cambridge, MA 02139, USA;
(b) Lodestar Research Corporation, Boulder, CO 80301, USA;
(1) Perm.address: Keldysh Inst. of Appl. Math., 125047 Moscow, RF;
(2) Also at: Kurchatov Inst. of Atomic Energy, 123098 Moscow, RF.
Abstract
Fluid descriptions can fail for fast processes such as Edge Localized Mode (ELM) [1]
bursts because plasma transport properties may be affected by incomplete Maxwellization of plasma particles. In this case only kinetic plasma descriptions of ELMs can give
qualitatively correct results. In the present work we propose three different models
of ELM propagation in the scrape-off-layer (SOL). We also present the first results
of the kinetic modeling of the propagation of ELM bursts along the magnetic field,
towards the target in a SOL plasma. These simulations are performed with a 1D2V
finite element Fokker-Planck code ALLA [2] and with a 2D PIC code W2 [3].
1. Introduction
Plasma fluid codes are widely used for SOL plasma modeling. However, a fluid plasma
description can be inadequate for edge plasma modeling, since it assumes a short
charged particle mean free path expansion. Fluid descriptions also fail for transient
regimes if the typical time scale is less than the one needed for equilibration. For relatively fast processes such as heating and cooling bursts propagating into a tokamak
scrape off layer plasma caused by ELMs, a fluid plasma treatment might not be applicable since the plasma particles may be unable to completely relax to a Maxwellian
thereby affecting the plasma transport properties. Another situation in which the
short mean free path assumption fails (that is, results in a strong departure of the
distribution function from Maxwellian) is in the vicinity of a material surface which
absorbs plasma ions. We do not consider the creation of ELMs, but rather study their
propagation in the SOL towards the plate. We present here results from 3 models of
ELM bursts, which have different levels of complexity. The hierarchy of models is as
follows.
(i) 1V model. The initial ELM propagation stage (heating) is modeled by increasing the background plasma temperature used to evaluate the Maxwellian Rosenbluth
potentials. The final ELM propagation stage (cooling) is modeled using a Krook ope-
67
rator, which mainly affects energetic particles.
(ii) 1D2V model. Parallel electron transport is modeled using experimentally obtained plasma profiles (C-Mod data). Heating is performed by temporally varying the
temperature in the Maxwellian Rosenbluth potentials in the upstream region.
(iii) 2D2V model. An extra (radial) dimension is added. Electrons can diffuse
across magnetic field lines due to an anomalous perpendicular diffusion coefficient as
they experience collisions with the background plasma in a parallel ambipolar electric
field.
2. 1V Model of ELM Burst
This simplified model is used to estimate the kinetic features of the SOL plasma
transport properties during ELM propagation. We investigate the influence on the
SOL plasma transporf coefficients of incomplete relaxation of the distribution function
to a Maxwellian due to the time dependence of the SOL plasma parameters during
ELM propagation. We consider 2 stages of ELM propagation:
1) the initial stage, when the SOL plasma is heating up due to the energy flux
coming from the bulk plasma.
2) the final stage, when the strong energy flux from the bulk is turned off as the
SOL plasma is cooling down due to fast particle loss to the target.
For simplicity we consider a distribution function homogeneous in space and isotropic
in velocity, so pitch angle scattering by ions can be neglected. We describe the transient
effects of the SOL plasma heating by the kinetic equation:
of
.
) +
1 aAfv
BL2 +
v(Af
T(D(v,t)
Of
)
(1)
The cooling stage is described by the kinetic equation with a sink term on RHS:
Of a- Af + B
)
-
dV2,f
(2)
Here v, vd and # are adjustable parameters. The initially Maxwellian distribution
function at different moments of time during heating stage is shown in Figs.1,2
0
Fig~l duing
7
f
eatin
.
30
...-
2-
Fig.2. f during heating vs
68
The distribution function evolution during cooling stage is shown in Figs.3-4.
-2.-
3
.12
Fig.3.
f
during cooling vs v
EJT(T
The effect of incomplete Maxwellization is estimated by the short mean free path heat
conduction coefficient K oc f fjlv"dv. Figs.5-6 below show ratio K/KM, where KM is
calculated for Maxwellian distribution with the same T and n as f.
,a
2
0.7
0
0
:.:
040
0-
01
0
a
4
0
0
1
Fig.5. K/KM during heating
Fig.6. K/KM during cooling
From the 1V model we can conclude that during ELM bursts f can be significantly
non-Maxwellian and the heat conduction coefficient is affected.
4. 1D2V Model of ELMs Burst
We simulate plasma flow in C-Mod along the magnetic field, 2 mm from the separatrix at the midplane. The assumed 1D model geometry is shown in Fig.7 below:
Frozen Ions
T(t)
Midplane
Plate
0
Magnetic line
/
IL
III
Fig.7. Geometry of the problem
We solve the following non-stationary kinetic equation for 3D electron distribution
function f,(t,11, v, p):
-f+
VI -f
t1
=
aV 2jU-
2T
69
a
Tp
v
_
(3)
(3)
The fluxes on RHS of the kinetic equation are the sum of the Coulomb collisional flux
and flux due to parallel electric field:
(m, apg829
Mg e+
2
m O, 8vC
e Ei e+
Jv ju=pP me
3M -
1- i
-
8v
2fe +
mm
-v
J
+ i
g o pv
(4)
a ofef
o"vM*
(5)
Collisional fluxes are expressed through Maxwellian Rosenbluth potentials Om and Om
and the electric field is obtained from Braginskii momentum balance equation
e E = -. 71VT,
V(T np)
nP
(6)
where T, n, = plasma temperature and density. The parallel distribution function
fii = f fevjdvi obtained from this model is shown in Figs.8-9 versus a Maxwellian.
At both spatial locations fil has asymmetric wings with the downstream one having
the more energetic tail.
.2
-4
-6
-6
-u
.e
-4
-2
p
~
4
~
*.
ft.
U
@
-
-
.40 ivl .
Ofr
Fig.8. fjj 1 cm from the plate
Fig.9. fii 0.5 m from tb e plate
Due to this non-Maxwellian tail, the heat conduction coefficient exceeds the equilibrium one by factor of 30 in the region between thermal front and the divertor plate
(Fig.10). Temperature wave propagation during an ELM burst is shown in Fig.11.
1u'o
*
600
*
400
1.L
0
200
'S
100
0
U
L _I1.
m
2
4
~
UI. M
Fig.11. Te, during ELM burst
Fig.10. K/KM for ELM burst
5. 2D2V Model of ELMs Burst
In this model the electron distribution function also depends on the perpendicular
coordinate lI and is described by the following kinetic equation:
+ VP
+
70
D
= C,,p
(7)
where C,, is a collisional term similar to (3) and D ; 1m 2 /sec is the perpendicular
diffusion coefficient. The term on the RHS of (7) is taken in its linear approximation
and calculated for frozen background plasma profiles np( 1 1 , l±), Tp( 11, l). The parallel
ambipolar electric field is derived from a Braginskii moment balance equation for each
l-L. The geometry of the problem is shown in Fig.12.
A
X
Sr mmetry
p ne
n,, Tp
Divertor
0
X
Tcore
L
Fig.12. 2D geometry used by W2
Here A ~ 2cm is the width of the region and L ~ 10m the connection length. We
assume that electrons intersecting the lII axis between the x-point and symmetry plane
are reflected back with a high temperature To,, ~ 10 2 eV. We balance losses to the
target by a an incoming flux of hot electrons through the [x, L] surface.
Simulations of the 2D2V problem are in progress and results will be presented later.
6. Conclusion
In this work we presented three different kinetic models of ELM propagation in SOL
plasma. We found that during ELM bursts, the distribution function is unable to
equilibrate. During the initial heating period of the 1D2V ELM burst model, the
distribution function has an elevated tail and the short mean free path heat conduction
coefficient exceeds the Maxwellian value by ~ 10 times in the region between the heat
front and target. At the midplane the distribution function falls below Maxwellian,
leading to a decrease in the heat conduction coefficient of about a factor of 2 as found
in 1V. We also have found that the parallel distribution function has asymmetrical
upstream and downstream wings.
Acknowledgments
Work performed for US DOE by MIT under Grant DE-FG02-91-ER-54109 and by
Lodestar under Grant DE-FG02-88-ER-53263.
References
[1] H.Zohm et all, Nucl. Fusion 32, 489 (1992);
[21 A.A.Batishcheva et al., MIT Report, PFC/JA-95-23 (1995);
[3] O.V.Batishchev et al., Contrib. Plasma Physics 34, 436-441 (1994).
71
Massively Parallel Fokker-Planck Code ALLAp
A.A. Batishcheva(1), O.V. Batishchev(1,2,a),
S.I. Krasheninnikov(1,b), D.J. Sigmar(1),
A.E. Koniges(3), G.G. Craddock(3),V. Djordjevic(3)
(1)MIT Plasma Fusion Center, Cambridge, MA 02139 USA
(2)Lodestar Research Corporation, Boulder, CO 80301 USA
(3)Lawrence LivermoreNational Laboratory,Livermore,CA 94550,USA
(a)Permanent address: Keldysh Institute,125047 Moscow,Russia
(b)Permanent address: Kurchatov Institute,123098 Moscow,Russia
Abstract.
The recently developed for workstations Fokker-Planck code ALLA simulates the temporal
evolution of 1V, 2V and 1D2V collisional edge plasmas [1]. In this work we present the
results of code parallelization on the CRI T3D massively parallel platform (ALLAp version).
Simultaneously we benchmark the lD2V parallel vesion against an analytic self-similar
solution of the collisional kinetic equation[2]. This test is not trivial as it demands a very
strong spatial temperature and density variation within the simulation domain.
1. Model Equations and Geometry.
In this work the evolution of the ion and electron distribution functions is described by the
system of kinetic equations :
afeji + vR L"'-I
a V2g qe'jEx
Mm e
a X V 2 v2
afet i
1 ' (l - g2 ) qe,jEx
m e__
1
i )q ve
_ _
=fy,
where q and m are the electric charge and mass, E is the parallel electric field,
=
vj / v
v = Vv + v2 , x is the coordinate directed along the magnetic line, and Cv, is the FokkerPlanck term, which represents the divergence of corresponding components of particle flux
in velocity space due to collisions,
=
V2jcol)+
-
i
p.112)
.
(2)
The electric field is ambipolar and obtained from the Braginskii momentum balance equation
eEA (X)=.71VT - V(Tn)
np
where T, and np are plasma temperature and density (Te = Tj
= Tn).
We impose the following boundary conditions:
fe(t,0,v, ) = fe(t,0,v,- )
for
0 vg U*,
fe(t,,v,9)= 0
for
v > U*,
72
(3)
fj(t'O'v'R) = 0
for
< 0,
for
f)= (TL,nL)
fe,i(t,L,v,
where x=0 corresponds to the target and x=L - to the midplane, fK"is the Maxwellian
distribution function. The electron cut-off velocity U* is determined from the ambipolarity
condition of plasma flow onto the target ( the so called logical sheath boundary condition):
J
_ v3
= J Af=(t,0,v,
V3 fe(t,0,v,
e)dvd )dvd.
(4)
The sheath potential is determined as 2eApsh = meU*2
The initial plasma profiles correspond to constant conductive parallel heat flux and plasma
pressure along magnetic field:
Tp(x,t = 0) = TLO(+ax) 2 1 7 ,
np(x,t = 0)T(x,t = 0) = const ,
ne(t = 0) = ni(t = 0),
Te(t = 0) = Ti(t = 0).
(5)
The initial distribution function is assumed to be Maxwellian with local plasma parameters
TP = T(x,t = 0) , np = n(x,t = 0) and zero drift velocity.
2. Numerical Scheme
We cover the simulation domain [0,L] x [0,Uma] x [-1,+l] with a non-uniform 3D mesh
of NX x NV x N. size. Here Umax is chosen to match the condition f(t,x,Umx,g)= 0.
The time integration of the evolution equation (1) is performed using a splitting scheme. We
use a fractional step method involving three steps corresponding to spatial transport T. and
the Fokker-Planck term coupled with an acceleration in the electric field. We use a timecentered splitting scheme that is unconditionally stable and gives O(r 2) accuracy (1/2
means half time step):
T1/C2, gTI/
(6)
The free streaming term Tx corresponds to a spatial shift of the distribution function. The
required interpolation of f(t,x,v,g) is performed by cubic spline technique[3]. The
distribution function is characterized by strong spatial gradients (which may be located at an
arbitrary point of the simulated region) and the energetic tails as high as 100 Tmin. We
employ modification of the numerical technique for the Tx operator which involves a
nonuniform adaptive spatial grid and the maintenance of a positive sign of the interpolated
function via linear correction at critical points. As the Fokker-Planck operator Cv, has a
divergence form we write down a conservative differential scheme:
j jY1 2 --S
- .2
f -- -. f 'a=
Uj
+
Ij j Y 1/ i9j
Aj
_ S9
-1
jP
/2
,
(7)
where
=
+2'
and
j
=
I,3
22,'
,"'with U.
and Sy,, S". the corresponding cell
,
volumes and surface areas. In general, we have a 9-point implicit differential operator with
second order accuracy. The 9-diagonal system is resolved using a fast sparse matrix solver.
For the cases of explicit non-diagonal terms or Maxwellian Rosenbluth potentials this
reduces to a 5-point implicit scheme.The 5-diagonal system is solved by a 5-diagonal solver
73
based on Jacobs strongly implicit procedure. A completely conservative scheme [1] is not
used as it requires extensive calls on the sparse matrix solver.
3. Massively Parallel Version.
As the Fokker-Planck term is the most time-consuming step, taking 95% of the run time,
we perform a domain decomposition in the spatial dimension x so that each processor stores
only one vg cross-section of the distribution function. Thus Coulomb collisions at each
spatial grid point can be calculated independently. As a parallelization method we used
Work Sharing and Data Sharing [4]. In Table 1 we present the CPU time required for a test,
10 time step run. We varied the number of T3D processors (NPES) and vg-grid size
used. Note that the number of spatial grid points is proportional to the number of
processors. Therefore, the relatively small increase in run time with increasing processors
implies improved run efficiency for larger number of processors.
Table 1.
NPES
vp -grid
4
8
16
129x33
13.29
15.11
18.27
25.0
39.0
74.2
256x65
55.3
58.3
69.6
94.6
142.6
255.4
513x129
215.5
237.4
282.6
376.5
581.4
1000.4
32
64
128
For 32 prosessors and vg-grid 129x33 , wall clock time was 10 times less than for work
station HP/735. A full code parallelization using the Message Passing Interface
programming model is in progress.
4. 1D2V Benchmarking Problem.
We benchmark the 1D2V version of the ALLA code with an exact analytical solution for the
collisional electron kinetic equation found in [2]. It was shown there that the electron kinetic
equation, retaining the complete e-e Coulomb collision term, pitch angle (g) scattering due
to e-i collisions, and the ambipolar electric field results in a nonuniform solution, fe (x, V),
which can be represented in self-similar variables as fe(x,V) = F(*)(Teff(x))-a, where
Teff(x) is the effective electron temperature, * = / Teff (x)/m, and a is an adjustable
parameter. This class of the solutions corresponds to a constant ratio of electron Coulomb
mean free path, Xe(x)=(Teff) 2 /(21re4Ane), to the temperature scale length,
LT(x)=(dTeff(x)/dx)",
Y= ke(x)/LT(x) = const.; and corresponds to the following
temperature and density profiles: (Teff)a-1/ 2 (dTeff/dX) = const., ne -(Tff) 3/2-.
Notice that for a=3, Teff(x) corresponds to a constant heat flux for the Spitzer electron
heat conductivity (- T 5 / 2 ). In [2] it was shown that the tail of the distribution function has
a power law dependence F(*) c 4<(s)/w 2 a (for Ii'>f7-1/2). However, in a computation
where we have maximum (Tm.) and minimum (Tmi) temperatures, this tail is cut off at
74
a power law dependence F('v) . cD(g)/w 2ca (for *I> 7-1/2). However, in a computation
where we have maximum (Tma) and minimum (Tmin) temperatures, this tail is cut off at
energies above Tma, and so power law dependence only exists in the low temperature
region if Tmin/Tma < 75. Results of Benchmarking.
For ID, IV and 2V problems we performed the following benchmarking against exact
analytical solutions:
i) Free streaming of an arbitrary function with the solution f(x,t) = f(x - vjt,0)
ii) Acceleration in an electric field. The analytical solution is a shift of the distribution
function f(V,t) = f(V +ett / m,0) in velocity space.
iii) Rosenbluth potentials for Maxwellian distribution function (to test potentials solver).
iv) Relaxation to a Maxwellian of an initially non-Maxwellian function. Mass conservation
is within 15 digits, relative error is 10-3 up to particles with energy of 400T (see Fig 1.)
v) Simulation of the IV self-similar solutions [5].
U.4IWJeAIMm aiuj. oeuii
fu-'1 s
*T
0
.v
-.,00
-5
Maxwel(Tmin)
Maxwell(Tmax)
-10,
-
-20
-25
-30
,-35
-40.
0
1so
ice
IN
Fig 1. Relaxation to a Maxwellian
of an arbitrary distribution function.
US 50 1
O
1
2
Efr
2530 35 40
Fig 2. Parallel distribution function
for y =0.01, 6=30
We perfomed a simulation of lD2V self-similar solutions with the folowing parameters
8 = Tma / T min = 30,100 , y =0.1, 0.01.We could not observe the self-similar behavior
for y =0.1 due to low collisionality (expression (3) for electric field valid if y -> 0). For
y =0.01 and 8=30 the parallel distribution function is shown in Fig 2. It clearly consist of
2 Maxwellian parts connected by a different temperature through the self-similar region
around E - T / -N= lOT. Due to the small temperature variation this region is narrow.
Figs.3a,3b represent parallel distribution function for y =0.01 and 8=100. As can be seen
from Fig.3a the bulk and tail parts are Maxwellian with completely different temperatures.
The self-similar part is now more broad and corresponds to a linear segment on the In-in
plot of Fig.3b.
75
4 I~
-5 -. ..
-10-
-15-Ti
-20
-25
.30
Maxwellian - -
Maxwellian --
0-
T-min
'1
s
Tmax ....
-20
%Unear
segnm it
.30
-
-35
T-max
%1-0,1
.40
-45
50 6
lb It 20 !ft 3U
E/T
4M
,04
3 -0h.
1'0
Ef T
100
Fig3a.
Fig 3b.
Parallel distribution function for y =0.01 and 8=100.
6. Conclusion.
The massively parallel version using Work Sharing and Data Sharing, ALLAp, of the
Fokker-Planck code ALLA for T3D is complete. The perfomance of the new version versus
number of processors and grid size is measured and compared with the workstation
version. The results of benchmarking against analytical 1V, 2V and 1D2V self-similar
solution are presented.
Acknowledgments.
The work perfomed for US DOE by MIT under grant DE-FG02-92-ER-54109, by
Lodestar under grant DE-FG02-88-ER-53263, by NERSC-LLNL under Contract W-7405Eng-48. T3D time provided by NERSC under the Massively Parallel Processing Access
Program.
References
[1]
[2]
[3]
[4]
[5]
Batishcheva A.A. et al., MIT Report, PFC/JA-95-23 (1995).
Krasheninnikov S.I., Sov. Phys. JETP 67, 2483 (1988) .
Shoucri M. and Gagne G., J.Comput.Phys. 27, 315 (1978).
Cray MPP Reference Manual, SR-2504 6.2, Cray Research, Inc.
Batishcheva A.A. et al., "A Kinetic Model of Transient Effects in Tokamak SOL
Plasmas" ( submitted to Physics of Plasma).
76