A phenomenological model for cross-field plasma transport in non-ambipolar scrape-off layers

548

Journal of Nuclear Materials 176 & 177 (1990) 548-556 North-Holland

A phenom~nological model for cross-field plasma transport non-ambipolar scrape-off layers

in

B. LaBombard Plasma Fusion Center Massachusetts Institute of Technology Cambridge, MA 02139, USA

A.A. Grossman and R.W. Conn Institute of Plasma and Fusion Research, Department of Mechanical, Aerospace and Nuclear Engineering, University of ~al~ornia - Los Angeles, Los Angeles, CA 90024, USA

A simplified two-fluid transport model which includes phenomenological coefficients of particle diffusion, mobility, and thermal diffusivity is used to investigate the effects of nonambipolar particle transport on scrape-off layer (SOL) plasma profiles. A computer code (BSOLRAD3) has been written to iteratively solve for 2-D cross-field plasma density, potential, and electron temperature profiles for arbitrary boundary conditions, including segments of “limiters” that are electrically conducting or non-conducting. Numerical results are presented for two test cases: (1) a 1-D slab geometry showing the interdependency of the density, potential, and temperature gradient scale lengths on particle diffusion, mobility, and thermal diffusivity coefficients and limiter bias conditions, and (2) a 2-D geometry illustrating E x B plasma flow effects. It is shown that the SOL profibs can be quite sensitive to non-ambipol~ty conditions imposed by the limiter and, in particular, whether the limiter surfaces are biased. Such effects, if overlooked in SOL transport analysis, can lead to erroneous conclusions about the magnitude of the local ambipolar diffusion coefficient.

1. Introduction

Experiments have shown that cross-field particle transport in a tokamak scrape-off layer (SOL) plasma can be significantly altered by externally imposing an electrical potential to limiter and/or wall surfaces [l-6]. The SOL in these “limiter bias” experiments typically exhibit a modified density e-folding length that is longer or shorter than the “unperturbed” value, depending on the polarity of the applied potential [5]. The particle confinement properties of the core plasma also changes in most experiments, leading one to conclude that electric fields can affect plasma transport, not only in the SOL but also on closed magnetic flux surfaces near the plasma edge. It is speculated that E x B velocity shear may play an important role, reducing the radial correlation length of the turbulence [7]. It has been further suggested that the H-mode confinement regime may be a manifestation of such effects operating just inside the last closed flux surface [6,g]. 0022-3115/90/$03.50

A separate but related issue is the degree to which non-ambipol~ cross-field transport can be supported by the SOL plasma. In contrast to the situation on closed magnetic flux surfaces, cross-field electron and ion fluxes are not constrained to be ambipolar within the SOL. Currents may flow along magnetic field lines, through the limiters, and the vacuum vessel wall. Limiter bias experiments implicitly rely on and enhance this non-ambipol~ty, driving plasma currents in the SOL and closing the current path through limiter and wall components. However, currents in the SOL may arise even in the absence of an externally applied bias from a variety of means including: differences in electron temperature at divertor surfaces [9], differences in plasma potential at limiter surfaces (particle drift effects) [lo], and electric fields produced by magnetic induction [ll]. In turbulence-free plasmas, it is well known that non-ambipolar boundary conditions can modify the density profiles that would occur in a locally ambipolar SOL. Such a “short circuit effect” was first recognized

0 1990 - Elsevier Science Publishers B.V. (North-Holland)

549

B. LoBombard et al. / Model for cross-field plasma transport

by Simon [12], who considered cross-field transport in a partially ionized, magnetized plasma column when the magnetic field lines terminated on conducting walls. In that system, the cross-field transport was observed to be at an “anomalous” level, two orders of magnitude larger than classical transport. This large level of transport was at first attributed to plasma turbulence by Bohm. However, Simon provided a simple explanation in terms of classical transport processes, i.e. non-ambipolar cross-field diffusion balanced by non-ambipolar parallel flows to wall surfaces. Here, neutral collisions provided the important mechanism by which non-ambipolar cross-field fluxes were generated. In a turbulent, nearly fully ionized tokamak SOL plasma, the transport physics is not so obvious. Clearly, non-ambipolar cross-field transport exists in a tokamak SOL, otherwise, little or no current would flow in limiter bias experiments. Similar to Simon’s plasma, neutral-plasma interaction can play an important role, particularly in SOLs where local ionization is important (e.g. high recycling or gaseous divertors). However, the degree to which plasma turbulence contributes to nonambipolar transport in a tokamak SOL remains unclear at this time. In this paper, we investigate the affect that non-ambipolar cross-field transport can have on SOL density, potential, and electron temperature profiles from a phenomenological point of view. For simplicity, we postulate that the non-ambipolar part of the cross-field fluxes arises through a mobility-like transport term. No attempt is made to argue what the absolute magnitudes of the mobility coefficients should be, although as a lower limit one may consider neutral-plasma interactions. Rather, we are interested in examining the effects that non-ambipolar transport can have on SOL profiles and the consequences of these effects on data interpretation. Furthermore, by matching numerical profiles from the transport model with experimental data, appropriate magnitudes of the phenomenological mobility coefficients may be suggested. Section 2 describes the 2-D SOL plasma transport model which is based on particle and electron energy conservation equations. The 3-D structure of the SOL is reduced to a 2-D system by writing the model in terms of spatial averages of densities, potentials, and electron temperatures along magnetic field lines. Boundary conditions on parallel electron and ion fluxes to wall surfaces are treated by a planar sheath analysis. Section 3 presents numerical solutions to the transport model for two test cases. The first case is a 1-D slab geometry in which the “limiters” are held at fixed potentials relative to the plasma at the last closed flux surface. In

this way, cross-field current is required to flow from the last closed flux surface into the SOL. Density, potential, and electron temperature profiles showing the effects of non-ambipolar transport are obtained for a range of diffusion, mobility, and electron thermal diffusivity coefficients. The second case is a 2-D geometry illustrating E X B plasma flows effects that can be induced by biased limiter, wall, or probe structures. Section 4 provides a brief summary.

2. Non-ambipolar scrape-off layer transport model 2. I. Scrape-off

layer geometry

The model geometry is shown in fig. 1. The SOL is assumed to occupy a volume of space described by a (r, 0, z) cylindrical coordinate system. The SOL is bounded along field lines running parallel to the z-axis by limiters, divertor plates, probes, or some other wall surfaces spaced at a distance of 2L. The interface between the main plasma and the SOL occurs on the z-axis at a radius, r = a. The SOL extends some finite distance in the 8 direction and a periodic boundary condition can be applied to this coordinate, as required. The “limiters” are symmetric about z = 0 and are assumed to be held fixed at some potential profile, @J,, which may be different than the local floating potential, @r. Here, Qrr is defined relative to the plasma potential, @, as Gr = @ -VT,, where ST, is the magnitude of the sheath-wall potential drop when no current flows to the limiter.

Main Plasma Zone

r=a

-

I t

rc- r// -

J//

I

I

I

0

t

Parallal Transporl

J// T

Fig. 1. Model geometry for parallel and perpendicular transport in a non-ambipolar scrape-off layer. “Limiter” plates are biased to a potential, a,.,, which may be different than the floating potential, @q.

B. LaBombard et al. / Model

550

The SOL receives particles and energy through cross-field fluxes (r, , qL) from the main plasma region and looses particles and energy through parallel fluxes (r,,, q,,) to limiter surfaces. Ionization, radiation, and electron-ion energy transfer provide additional particle source and energy sink mechanisms in the SOL. However, for simplicity, these contributions are neglected in the present study. Analogous to the particle flux, a cross-field plasma current ( JI) may flow from the main plasma into the SOL while a parallel plasma current (J,,) may flow to limiter surfaces. In general, all plasma parameters are functions of r, 0, and z. However, in the presheath plasma, the density and potential typically varies a factor of - 2 and OST, respectively along the magnetic field. (This paper defines all temperatures in units of eV so that Boltzmann’s constant, K, is defined as K = q.) High parallel electron thermal conductivity also tends, in most circumstances, to keep the electron temperature relatively constant along the magnetic field. In contrast, the density may vary by orders of magnitude in the cross-field direction. The model presented here concentrates on the cross-field variation of the density, potential, and electron temperature by considering the spatial average of these quantities along the magnetic field. 2.2. Model for cross-field fluxes and currents Cross-field fluxes of electrons and ions are modelled through phenomenological transport coefficients as

r:=-D,v,n+pe~nVI~+n~, ri=

-D,v,n-$lnVl@+ngI-.

BXV@

-nx’*v,r,.

transport

Diamagnetic terms do not contribute to the particle and energy balance except through B x VB drifts, which are not included in the straight magnetic field geometry presently considered. In steady state, with no local particle or energy sources, continuity and electron energy conservation requires

V~

.r, = - v,,.r,,.

(4)

VI .J, = - v,, . J,,,

(5)

v,.qel+:r:.~~T~+nT,o~.V~ = - v,, 4; - nT,v,, . y;,

(6)

where the approximation, v,,T, = 0, has been used to eliminate that term, and contributions from electron viscosity has been ignored. In this model, ion temperature only weakly affects the transport fluxes through the local sound speed. Consequently, the complication of including perpendicular ion heat fluxes and the ion energy conservation equation has been avoided in favor of simply specifying the T, profile. 2.3. Model for parallel fluxes and currents at limiter surfaces Along the magnetic field, presheath electric fields draw ions to the limiter surfaces so as to satisfy the Bohm sheath condition. This situation persists under a wide range of limiter biases as long as the local limiter potential remains approximately O.ST, below the plasma potential at z = 0. Thus the parallel ion flux to the limiter surface can be modelled as r,; = a&, ,

BXV@

For simplicity it is assumed that cross-field diffusion is intrinsically ambipolar (0: = 0: = D, , as would be expected in the case of low frequency electrostatic plasma turbulence) and that the diffusion part of the transport is not sensitive to the local time-averaged electric field. Clearly, the change in particle confinement time that accompanies tokamak limiter bias experiments suggests that ambipolar diffusion may in fact change with applied potentials. Nevertheless, we separate these two transport effects, focussing here on only the contribution from non-ambipolar fluxes in response to an electric field. Cross-field electron heat conduction is modelled through a phenomenological thermal diffusivity, xel, as qy=

for cross-field plasma

where (Y is a parameter in the range 0.5 < a < 1 that relates the density at the limiter to the average density along a magnetic field line in the SOL, and C, is the ion sound speed. The superscript, w, is used to denote quantities evaluated at the limiter surface. The parallel plasma current and electron heat flux to the limiter can be estimated from planar sheath theory. Given the same restriction on bias potential mentioned above, J,;=aqnC,jl

-exp[

qi’“=anC,T,

‘“~“‘]),

2.5 +q+

(@WT

] xexp[

(3)

(7)

(@wi@f)],

’

@f>

1

B. LoBombard et al. / Modelfor cross-fieldplasma transport Secondary electron portant contributor this study.

emission, which can be an imin eq. (9) has been set to zero for

2.4. Scrape-off layer density, potential, and electron temperature profiles Upon lines, the expressed from eqs. procedure ferential potential,

averaging eqs. (4)-(6) along the magnetic field right hand sides of these equations can be in terms of the limiter fluxes and currents (7)-(9). In combination with eqs. (l)-(3), this results in the following three coupled difequations describing the cross-field density, and electron temperature profiles:

Bxv@

D,v:n+T.Vn

P: +pil exp

ant E-2

Pw- 9)

[

L

T,

P*‘I+ $1

the phenomenological

(10)

i

i

where

1,

transport

coefficients

of

D, , $1, $1, and x I, have been taken to be independent of space. 2.5. Effect of non-ambipolarity on SOL profile analysis Eqs. (lo)-(12) directly illustrate some important effects that non-ambipolar transport can have on SOL density, potential, and electron temperature profiles. In an ambipolar SOL, the local limiter potential, a,,,, is everywhere equal to the local floating potential, ar (nonconducting limiter case). Without currents in the SOL, the potential profile is flat and no E x B drifts occur. The characteristic density e-folding length, X,, is then simply related to the ambipolar diffusion coefficient through the widely used relationship, D, = ah&/L.

(13)

551

However, in a non-ambipolar SOL, the exponential dependence on limiter potential in eq. (10) modifies this scaling. Consequently, the local density gradient can vary by a factor of 5 or more for the same ambipolar diffusion coefficient! Conversely, by neglecting non-ambipolar effects, it is possible in some situations to incorrectly estimate the ambipolar diffusion coefficient from eq. (13) by an order of magnitude! This “error” is reminiscent of the transport analysis “error” in weakly ionized plasmas pointed out by Simon [12]. The electron temperature profile may be similarly affected from the exponential term on the right hand side of eq. (12). Note that the local effect of non-ambipolarity on the density gradient is quite insensitive to the assumed magnitude of the mobility coefficients. All that is required is pil 2 peI, a situation which is typically expected since from momentum considerations, pi1 /pel m/m, (as would be the case for neutral collisions, for example). It is possible to imagine special situations in which the cross-field electron mobility is greatly enhanced over the ion mobility (as in an ergodic magnetic field structure) so that pi1 5 pel. In the case when term in eq. (12) vanishes, Pil K $1 1 the exponential and eq. (13) becomes valid for all non-ambipolar boundary conditions. Experimentally, one may readily check for the importance of non-ambipolar effects by simply looking at the magnitude of (@Jo - @+)/T,. As will be shown in the numerical results below, the spatial extent over which @, # Qr is some measure of the effective mobility coefficient in a tokamak SOL. In contrast to the n and T, profiles, the local gradient scale length of the potential profile depends directly on the values of $I and pel. (In fact, the potential equation becomes singular at pi1 = pel = 0, a limit which is not properly treated by this model. Near this limit, the quasineutral approximation breaks down and finite gyroradius effects become important, The use of this model must therefore be restricted to scrape-off layers in which the ion Larmor radius is smaller than the characteristic cross-field gradient scale lengths.) In order to examine the global influence of non-ambipolar transport on SOL profiles, it is necessary to self-consistently determine the density, potential, and electron temperatures from integrating eqs. (lo)-(12).

3. Numerical

results

A computer code, BSOLRAD3, has been written to iteratively solve eqs. (lo)-(12) for the 2-D geometry described in section 2.1. Arbitrary boundary conditions on n, 0, and T, and arbitrary “limiter” potentials can

552

3. LoBombard et ai. / Model for crosslfietd plasma transport

be specified. The code also includes ionization, B X VB drifts, and radiation terms, although these terms have been set to zero for the present discussion. Solutions are checked for accuracy by verifying that they indeed solve eqs. (lo)-(12) in an independent section of the code. Solutions were generated for a simple 1-D slab geometry, to explore the interdependency of density potential, and electron temperature scale lengths in the SOL on the transport coefficients. In this case, the plasma was assumed to be uniform in the 4 direction. The SOL interface radius, r = a, was set to a large value compared to the scrape-off lengths so that a Cartesian geometry was approximated. Variables in this problem include D, , peL, pll, and ~5, plus boundary conditions on n, Qi, and T, and the “limiter” potential profile in radius, GW. For the convenience of displaying the results, a normalized radial coordinate, p, was defined as p = (r - a)& with h, defined by eq. (13). Solutions were obtained in the radial zone 0 s p I 5. Boundary conditions were n = 10” cm-3, T, = 50 eV, T, = 50 eV (uniform), Qt= 0 at p = 0 and 6n/& = 6T,/6r = S@/Gr = 0 at p = 5. The latter boundary conditions were chosen to approximate conditions at p = co. However, they effectively place a reflection plane at p = 5, which does not appreciably change the results in the range 0 I p I 3. For the purposes of estimating X0, the values of T, = 50 eV and T, = 20 eV were used and C, was evaluated throughout the SOL using an isothermal ion model. The absolute magnitude of n is not important in the present model and was arbitrarily chosen. Results are shown in figs. (2)-(5) for various dimensionless ratios of the transport coefficients x:/Dlt ami $A T,/D, , under four “limiter bias” conditions: (a) electrically floating limiter (conducting, with zero net current), (b) limiter at -150 V, (c) limiter at -10 V, and (d) a nonconducting limiter. These profiles were computed for a fixed ratio of mobilities, $J& = 200!$ so as to roughly correspond to the mobility ratio resulting from neutral collisions in a hydrogen plasma. As discussed in section 2.5, the profiles and their dependence on non-ambipolar boundary conditions are relatively insensitive to $l/$,_ and qualitatively similar results can be obtained for pil/$, = 1. The ratio pJI T,/D, was evaluated at T, = 20 eV. The potential boundary condition in ~mbination with the limiter potential effectively forces cross-field current to flow from the main plasma zone into the SOL. The return current path is not specified here but it ultimately involves another non-~bipolar SOL zone of reversed polarity. The values of limiter bias potential (- 150 V, - 10 V) were chosen so as to validate the use of eqs. (7)-(9). However, stronger non-ambipolar ef-

10

6 6 4 2 0 50 40 30 20 10 0 50

f?QWWs

Floating

Umiter

1

Limiter at -150V

Limiter at -10V

-150;

’

’ 1

I I I 2 (r-a)& 3

I 4

5

Normalized Dikance into SOL

Fig. 2. Scrape-off layer density, electron temperature, and potential profiles as a function of normalized cross-field coordinate (1-D test case) for coefficient ratios piiZ”/D, = 0.1 and xei /DL =3.

fects may be achieved with Iarger bias potentials section 3.3).

(see

3.1. Dependence of SOL profiles on kiA T,/ D, A comparison between figs. 2 and 3 shows the sensitivity of the SOL profiles to the cross-field mobility. Fig. 2 plots profiles for pi1 c/D, = 0.1 while fig. 3 displays results for pil TJD, = 10. In both cases, x’,/D, was fried at 1. A number of important points are illustrated by these figures, which are itemized be low. First consider the “floating limiter” case in comparison to the “nonconducting limiter” case: (1) As expected, the floating limiter receives a net flwr of electrons near p = 0 and a net flux of ions at larger radii such that global ambipolarity is observed (net current to limiter equals zero). (2) A consequence of this effect,

553

B. L.aBombardet al. / Model for cross-field plasma transport 10

SOL Profiles for XT/q= 1, tilTe/D,= 10

limiter. In contrast, (6) at a more positive relative bias of -10 V, electrons are efficiently removed, and the result is higher n and T, gradients. However, the effect diminishes quickly at large p because the floating potential rises rapidly to match the limiter bias potential. The rapid rise in the floating potential is due to a combination of rising plasma potential and falling T, profiles. The positive plasma potential gradient near p = 0 is required to block part of the ion flux into the SOL (net negative current into SOL).

5: -0 z =

4 : 50

s 4o 3Lm c 20 10 0

3.2. Dependence of SOL profiles on & ............................

Figs. 4 and 5 show the dependence of the SOL profiles on the magnitude of the electron thermal diffusivity, varying xe,/Dl between 0.1 and 10. For these profiles, pi1 T,/D, was fixed at 1. With the thermal diffusivity at the level of xe,/Dl = 0.1, the cross-field electron energy transport is essentially determined by convection alone. This leads to the very steep T, profiles

~~~~~

-150

/Dl

.................................................. Limiter at -lOV

_

-15oA m

1’

I

2

I

(r-a)&

I

I

3

1

4

5

Normalized Distancs into SOL

Fig. 3. Same as fig. 2 with p’IITJD,

= 10 and i’, /Dl

__ -1

I

= 1.

which shows up at large values of pi1 T,/D, , is a redistribution of the parallel electron heat flux on the limiter. In response, the T, profile steepens at small p, flattens, and remains at higher levels at large p. (3) At large values of pi1 T,/D, , the density profile also steepens at small p and flattens at large p, although this effect is slight. (4) Also as expected, pi1 T,/D, is important in determining the relative gradient scale length of the plasma potential profile. For large values, the non-ambipolar “perturbation” persists deeper into the SOL. Limiter bias is found capable of modifying the SOL profiles substantially, depending on the value of pi1 T,/D, and the magnitude of the bias. (5) The - 150 V bias case shows dramatic effects for large pi1 TJD, , extending the n and T, profiles deep into the SOL. This is because the excess cross-field ion flux, which supports the current into the SOL, can only be removed by the limiters at the sound speed velocity. Also, the negative limiter bias reduces the parallel electron heat flux to the

c

501

Potentralr: Floating Limiter

-I

Limiter at -15OV

Limiter at -1OV

-1500

m

I

I

1

2

I

I

I

(r-a)/ho3

4

Normalized Distance Into SOL

Fig. 4. Same as fig. 2 with pi1 T,/D,

= 1

and xel /DA = 0.1

5

B. L.aBombard er al. / Model for cross-field plusma transport

554

F 40 g 30 gx1 10 0 s

Z f

eotentlals:

5o 0

L

s

FloatingLimiter

1

-50

g-100

-150

to reduce the plasma density near sensitive plasma-facing components [13]. However, at limiter biases approaching and exceeding 0 V, eqs. (7)-(9) are no longer valid. This is because an “inverted” sheath begins to form near p = 0. As a crude approximation for this situation, the local parallel electron flux to the limiter was set equal to the random thermal flux (electron saturation current) for @_.> @. The ion flux was kept according to eq. (7). Fig. 6 shows SOL profiles that result from this model for the case of pil TJD, = 1, x:/D1 = 1, and limiter biases of - 150 V and 80 V. With 80 V of positive bias, the density and temperature profiles change dramatically. A factor of - 4 decrease in the density e-folding length over the “floating limiter” case is achieved. The T, profile essentially reaches zero around p = 1.5. This is the maximum impact that a positively biased limiter can have on SOL profiles, since

iTE -50“: S g-100

Limiter at -lOV

-150;

’

t

1

I

I

2 (r-a) A,

I

I

3

4

I 5

NormalizedDistance into SOL Fig. 5. Same as fig. 2 with p’* T,/D,

= 1 and xeL /DA = 10.

near p = 0 shown in fig. 4. In contrast, with x:/D,

= 10, the n and T, profiles have similar gradient scale lengths (fig. 5). The most dominant effect that xel has on the nonambipolar SOL is to change the floating potential profile. Consequently, for increased fJDI : (1) the -10 V bias case maintains more of a density reduction for larger p and (2) the - 150 V bias case m~ntains less of a density increase at large p. An interesting situation also arises for the case of the “floating” limiter in fig. 4. Due to the steep T, profiles, a large electron flux to the limiter occurs around p = 0 so that the remainder of the SOL behaves as a negatively biased Limiter. The density

A?

e-folding increased

p.lOO

length of the SOL is therefore over the case of a nonconducting

si~ficantly limiter.

3.3. SOL profiles with large positive limiter bias

Limiterat -150V

9

I

0 -50

-100 -150 100

Is: !j” g

0

f

-50

-150:

.

’

1

*

’

’

* (r-a) A. 3

’

4’

.

J 5

NormalizedDistanceinto SOL

Fig. 6. Effect of a large positive limiter bias on scrape-off layer The case of a large positive limiter bias is interesting since it may be possible to employ it in some situations

density, electron temperature, and potential profiles (1-D test case), Coefficient ratios are pii T,/L1, = 1 and xeL /Dl = 1.

B. LoBombard et al. / Model for cross-field plawna transport

the current-carrying electrons surfaces at thermal speeds.

travel

to the limiter

3.4. E x B plasma flow effects The plasma potential profile can be significantly perturbed in the SOL by a biased limiter, probe, or wall structure. This may induce large E X B plasma flows across density and temperature gradients, perturbing the SOL profiles. Fig. 7 shows numerical results from a 2-D test case that clearly illustrates this effect. The 2-D test geometry was made identical to the 1-D slab geometry in the r coordinate. However, the “limiter” was no longer allowed to have symmetry in the 0 coordinate and was cut along the r axis into segments which were biased to - 125 V and -25 V with respect to the potential at the last closed flux surface (r = a). The limiter potentials were determined so that no net current flowed into the SOL zone. A periodic boundary

Plasma Potential (V)

Limiter Geometry & Bias

:u..l.:,,;lj

condition was applied to the 0 coordinate and boundary conditions in the r coordinate were made identical to the 1-D slab case. The ratios of transport coefficients were set to pi1 T,/D,= 1,xel/DL= 10. The size of the positively biased limiter segment (-25 V) in the B direction was chosen to correspond to ha, which is a factor of 4 smaller than the negatively biased limiter segment (-125 v). The magnetic field strength was chosen so that the ambipolar diffusion coefficient, D, , corresponded to the Bohm value. From the discussion in the previous sections, one may simply expect increased and decreased radial gradients in the respective negatively and positively biased limiter zones. However, the E X B drift, which is from top to bottom in fig. 7, convects plasma in the -8 direction and can modify the radial profiles at the interface between the two zones. The positively biased limiter also perturbs the potential contours and causes the E x B fluxes to have a radial component. The radial E x B flux is very effective in maintaining and extending the radial profiles in the upper interface zone and steepening the profiles in the lower interface zone. Far away from the interface zones, the radial profiles revert to the 1-D profiles shown previously. (This can be best seen in cases where the electrode size in the 8 direction is chosen to be large compared to X0.) Thus, the overall shape of 2-D SOL profiles arise from a combination of two effects: the particle and energy sinks in the SOL and E X B fluxes, both of which depend on the nonambipolarity condition imposed by the limiters.

(r-a)&

NormalizedDistfinceinto SOL

Density_ (1d2 cm? .

Electron Temperature (eV)

3 g ‘2

0

(r-a)&

555

(r-a)&

Fig. 7. Contours of constant density, electron temperature, and plasma potential in a 2-D scrape-off layer geometry, illustrating E X B plasma flow effects (2-D test case). Coefficient ratios are nilT,/D, = 1 and xel /Dl = 10. The vertical axis corresponds to the distance along the last closed flux surface (LCFS) normalized to the “ambipolar” density e-folding length, Xc. A periodic boundary condition has been imposed on this coordinate. The “limiters” are biased as shown so as to achieve global ambipolarity, i.e. no net current leaving the SOL zone.

4. summary A phenomenological model, which employs mobility-like transport terms, is used to investigate the effect of non-ambipolar transport on scrape-off layer density, potential, and electron temperature profiles. Cross-field particle “diffusion” is assumed to be intrinsically ambipolar (07 = D\ = D,, as would be expected in the case of low frequency electrostatic plasma turbulence), while cross-field ion mobility is assumed to be similar to or larger than cross-field electron mobility. The formulation of such a mobility-transport model is theoretically justified in scrape-off layers where plasma-neutral interactions are important, although there may be contributions to mobility from plasma turbulence and other momentum transfer mechanisms. It is shown that the SOL profiles can be quite sensitive to non-ambipolar conditions imposed by limiters, divertor plates, probes, or other wall surfaces, particularly if these surfaces are actively “biased”. In some situations, without properly

B. L.uBombard et al. / Model for cross-field plasma rransporr

556

considering non-ambipolar effects, it is possible to incorrectly estimate the ambipolar diffusion coefficient by an order of magnitude. Experimentally, one may readily check for the importance of non-ambipolar effects by simply looking at the magnitude of (@,,,- @r)/T, relative to unity and the spatial extent over which GWf @r. (The latter is some measure of the effective cross-field mobility). A computer code is used to simulate scrape-off layer density, potential, and electron temperature profiles for cases in which the limiters are floating and electrically biased. The sensitivity of the profiles to the phenomenological coefficients of diffusion, mobility, and thermal diffusivity is examined in simple 1-D geometry. E X B plasma flow effects are demonstrated in a 2-D test case.

Acknowledgement This

Energy.

work was supported by the US Department

of

References

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