Contrib. Plasma Phys. 34 (1994) 2/3, 198-203

Finite Mean-F'ree-Path Effects in Tokamak S crap e- 0 ff Layers Ronald H. Cohen and Thomas D. Rognlien Lawrence Livermore National Laboratory, Livermore, CA 94550 USA

ABSTRACT We address the problem of calculating the parallel electron heat conduction

in a tokamak scrape off layer with non-negligible mean free path. We develop a model distribution function based on pieces of Fokker-Planck solutions, containing: a local Maxwellian at low energies, a modified Spitzer-Hkm distribution at energies below where the mean free path equals the distance to the nearest wall, a bounce- averaged distribution at higher energies up to the confining potential ener,gy, a filled- in outgoing loss-cone distribution at still higher energies, and finally an empty loss region at energies such that the mean free path exceeds the length of the field line. The model is compared with Monte Car10 simulations.

1. INTRODUCTION In diverted tokamaks, one often finds substantial variations of the e1e:ctron tem-

perature along a field line; for example, in DIII-D, the ratio of the midplane electron temperature = TeO to that at the divertor, Ted, is typically Teo/Ted 4 - 6 [l]. On the other hand, based on the classical electron parallel thermal conductivity, one would expect T,o/Ted - 1-2. It is generally acknowledged that the resolu.tion of this discrepancy lies in the fact that the electron mean free path Xmfpe, while small, is not negligible compared to the length L of a representative field line. For DIII-D pa- rameters, Xmjp,/L - 0.1 - 0.2 for thermal particles, but - l for energies comparable to the confining potential energy e@o of the combined sheath and presheath.

Finite mean-free-path effects are commonly handled in SOL fluid codes by im- posing a flux limit, in which the parallel electron heat flux 911 is limited to be no more than QFL = CflnTevte, where vte = (2Te/rn)l/' and Cjr is taken to be a con- stant of order 0.1. The flux limit is then often found to be in fact limiting the flux between the x point and the divertor. The use of a flux limit can lead to a sub- stantial computed temperature gradient [2]. This is because, in the codes, 411 varies appreciably in the divertor leg because of radial transport and radiative losses, while pressure balance tends to keep nTe roughly constant. Thus Te 0: 9;. While this produces a result qualitatively in accord with experimental results from moderate mean-free-path SOL'S, it is clearly wrong in the long mean-free-path limit, in which case the loss processes cannot be considered local. For X m j p e >> L , an appropriate description is based on a bounce-averaged Fokker-Planck equation; the temperature approaches a constant value along field lines; radial and radiative losses contribute

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to a lowering of this constant value. It is likely that the use of a flux limit produces erroneous results for moderate mean free paths as well; an example will be discussed below, in conjunction with Fig. 3.

There have been several attempts to calculate finite-mean-free-path effects start- ing from the Fokker-Planck equation. Refs. [3-61 all adopted a similar approach: attempt to solve the Fokker-Planck equation approximately by assuming f = fo(v)+ (vll/v)g(v), (formally justifiable in the limit of large effective charge 2) with varying additional approximations used to attain tractable equations for fo and 9. None of these published references took account of the significant feature that many of the carriers of heat were in collisionless contact with an end wall (and in a loss cone such that they can reach the wall). In fact, in the classical Spitzer-Hkm distribution, much of the heat flu is carried by particles with energies greater than 6Te; many of these particles are both in the loss cone and have mean free paths longer than the field-line length, and so are in fact absent from the distribution function! A recent calculation by Krasheninnikov addresses this problem by imposing absorbing-wall boundary conditions above the electrostatic potential barrier. In this latest work Krasheninnikov makes the same approximations to the Fokker-Planck equation as in his earlier paper, i .e. , f = fo(v) + (vll/v)g(v), with fo = Maxwellian x a slowly varying function of energy. These approximations necessarily break down near the loss cone in the long-mean-free-path limit. Krasheninnikov's solution may thus be regarded as a rigorous solution to a highly approximated form of the Fokker-Planck equation.

Here we adopt a complementary approach: we construct a non-rigorous solution to the Fokker-Planck equation by patching together known pieces which recover key qualitative features of the true solution. The result is a model which requires surprisingly little in the way of adjustment of free parameters and nonetheless does a reasonable job of reproducing some benchmark Monte Carlo simulations. The model is constructed in Sec. 11, and comparisons with Monte Carlo simulations are given and discussed in Sec. 111.

I1 MODEL FOR THE ELECTRON HEAT FLUX A good treatment of finite-mean-free-path effects in the SOL must account for

non-locality, velocity-space loss regions, and the confining potential from the sheath and presheath. It is reasonable to start from a 3-dimensional Fokker-Planck equation:

x 2- [2- a (f + 2) + -- 2, ((1 - X 7 3 ] t u 1 i 2 aw dw w dX

where & and p are the total energy and magnetic moment, s is distance along a field line, X = vI~/v , w = mv2/2T, vt is the thermal-speed collision frequency, and 2, = (1 + Z e f f ) / 2 . Since the heat flux is carried by superthermal particles, it suffices to consider the linearized, high-energy collision operator of Eq. (Ib), which describes collisions of test particles with a spatially varying background Maxwellian, whose

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temperature and density will be determined by fluid equations. In principle, one should analyze a four-dimensional equation (including a radial coordinate); however, if we regard the radial transport terms as a source on a field line, prior experience with the Fokker-Planck equation [9] for a potential-confined species indicates that the solution is sensitive to the magnitude but not the velocity-space distribution of sources (unless they are in the loss cone). Hence radial transport will be handled as a low-energy source that affects the background Maxwellian temperature arid density.

Unfortunately, even the linearized 3-D equation is too hard to solve arialytically, and approximations used in Refs. 3-7, such as near-isotropy and f or e E I T f being slowly w i n g functions of energy, break down for 2, - 1 and high energy, particu- larly for electrons whose mean free path exceeds the distance to the wall ,and whose energy exceeds the confining potential energy. Hence, we adopt here an. approach complementary to those of Refs. 3-7. Instead of seeking a formal solution t o a further reduction of Eq. (l), we divide velocity space into regions of short and long mean free path, and add partial contributions to heat flux from each region., based on modifications of known solutions. We define “short” and “long” by whether mean free path (as function of energy) is shorter or longer than distance to nearest wall. Hence, me define a dividing energy l o such that X,,,f,, = v /ve (E) = L,, or

Eo(eV) = 8 . 1 ~ [(l + Z,ff)E(lO1g m-3)logXL,(m)]1’2 (2) where Q = 1 for the nominal definition of ue from Braginskii [8] , and K is the density averaged between the position in question and the nearest wall. Obviously, EO is an increasing function of distance to the nearest wall; see Fig. 1.

E l I

For energies below EO + T, we solve for q by expanding in powers of v t l as in the standard short-mean-free-path calculation (see, e.g., Ref. 8), but modified so that the O(v’l) correction to the distribution function, fi, approaches the zero- order hfaxwellian at energy Eo + T. The match is done at EO + T (rather than E 0 ) to compensate for approximating the X,jpe = L, boundary by a sphere in velocity space. -4s in Ref. 8, fi is constructed to make the total particle flux vanish. This calculation yields the modified Spitzer flux whose behavior as a function of &o is as shown in Fig. 2.

For energies above EO we apply long-mean-free path results. For ,? > l o but E < eiPo we “bounce average” the Fokker-Planck equation (sds/v l l average over

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the last mean free path for that energy), and calculate the first-order (in vl;/v) correction to the bounce-averaged distribution function to determine the flux

where 9 ( z ) = (22 + l)e-' - [ ( ~ z ) ' / ~ / 2 ] E r f ( f i . For E > €0 and E > e@o, we take the distribution function to be zero for electrons reflecting from the wall with €11 > Eo, but Maxwellian otherwise. The Maxwellian parameters are those of the background at either the local position or a mean free path away from the wall (at s sx) at energy e@o, which ever distance from the wall is greater. (This is an approximation to a more complicated model in which the distribution function of outgoing electrons is equal to the Maxwellian value at a distance equal to the mean free path for the energy in question.) Hence we obtain the filled loss-cone flux contribution:

qlC M 2-17r-1/2nvt+ exp(-Eo,/T,)(Eo. + 2 ~ ~ ) (4)

where * denotes evaluation at s, = s if s is more than one mean free path away from wall, or at s = sx otherwise.

Fig. 2. Modified Spitzer flux, normalized to Spitzer flux, as a function of the maximum energy.

E044 For still higher energies [E > €0, = max(Eo)], the mean free path exceeds

the longest connection length, and the loss cone actually becomes empty. This has two consequences: (1) there is a velocity-space flux of electrons into the loss-cone region; the accompanying spatidy local energy loss rate Q p (integrated over the velocity-space loss boundary) can be calculated be calculated following the method of Pastukhov [lo]. This will enter the fluid temperature equation in the same way as an effective parallel heat flux q p = - d s Q p . (2) The filred-in loss-cone distribution function discussed in the preceding paragraph should actually be a Pastukhov-type distribution (vanishing at the loss cone boundary) rather than Maxwellian. Both of these effects are significant only for (€0 , - &o)/T < 1, are unimportant for the examples discussed below, and so in the interest of simplicity are dropped for the remainder of the presentation here.

The net heat flux is then taken to be the sum of the above partial contributions,

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111. RESULTS Figure 3 shows results for the heat flux as a function of axial coordinate s for

model background profiles [T, 0: (1 - as in the Spitzer solution for a speci- fied constant heat flux, but with the thermal conductivity multiplied by 0.5; nT, = const., and 8 = 8wrl~ + 0.71(T - TW,11)]. Parameters chosen are representative of DIII-D. Because the model temperature profile is not obtained from a self-consistent cdculation, the heat flux need not be constant; rather, the assigned profile is con- sistent with a distributed heat source. The figure shows results from the MCPAT Monte Carlo code Ill], from Eq. ( 5 ) , and from the flux-limit formula. The wall is on the right. The heat flux from the Monte Carlo run initially rises steeply in the first meter on the left, where heat is injected in the simulation (by a model charge-exchange operator), and then rises slowly as the wall is approached. The analytic result, Eq. (5), was treated as having one adjustable coefficient, namely a in Eq. (2). The result shown in Fig. 3 was obtained for a = 2. This value was used for all subsequent calculations. The heat flux as a function of position for the model is nearly constant, which is close to but not quite the simulation result. For comparison, we $how the result for the flux-limited heat flux, with C f l = 0.15. The flux-limited form gives worse agreement for the spatial dependence of q. Figure 4 shows the comparison between Monte Carlo and model results for reprlesentative ITER (Conceptual Design Activity) parameters (using the same value of CY). Again, agreement is reasonably good.

Fig. 3. Comparison of Monte Carlo, model, and flux-limit heat fluxes vs. position for Dm-D parameters. Also shown (upper plot) are input temperature profile and Monte Carlo test-elecmn temperature profile.

Fig. 4. Comparison of Monte Carlo and model heat fluxes for ITER (CDA.) parameters.

Figure 5 shows the variation of the ratio of the parallel heat flux to t.he Spitzer

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value qsp in the middle of the simulation volume, as a function of the ratio of X m f p e to LII = T(dT/ds)-’ . The figure was obtained by varying the density at the plate for the DIII-D example. The agreement between the model and Monte Carlo results is within the Monte Carlo statistical error bars.

1.0 0.8

0.6 0.5 Fig. 5. Heat flux normalized to

Spitzer value as a function of the 3 0.4

ratio of the field-line length to the B 0.3 mean free path.

0.2

0.1

ManteCarlo ------ - - ’ Model ManteCarlo ------ - -

I

1 2 5 10 20 50 100

Uk* In summary, we have obtained a reasonably accurate, simple to compute model

for the parallel electron heat flux in the range of mean free paths relevant to toka- mak SOL’S. The model is, however, of necessity non-local, and this complicates implementation in implicit fluid codes.

This work was performed at LLNL under DOE Contract No. W-7405-ENG-48.

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