J. Plasma Fusion Res. SERIES, Vol.3 (2000) 218-221

Revised Boundary Condition for the E x B Drift
Dominated Plasma Flow in the Scrape-off Layer
SHARMA Devendra and RAMACHANDRAN H.
Institute for Plasma Research, Bhat, Gandhinagar-382 428, India(Received: l8 January 2000 / Accepted: 19 April 2000)

Abstract
In the case of a strongly magnetized plasma interacting with a conducting surface, the E x B flow
generated by the self consistent presheath electric field plays a terminal role in determining the nature ofthe parallel flows at the magnetized presheath edge []. Presence of the cross-field drifts thus modifies
the boundary condition for the plasma flow to be used at the ends of the scrape-off layer [2]. Detailed
study of the presheath processes reveals a new kind of mechanism for perpendicular transport in theregion of magnetized presheath which is observed to have a resonant behavior close to the targets. Thisshows that the perpendicular and parallel plasma flows are strongly coupled and can not be assembled
non-interactively to achieve a condition for the entrance into the region of magnetized presheath. The
entrance condition should therefore be determined in terms of self consistent electric field which
effectively couples the parallel and the perpendicular flows. The form of such a coupling has beenobtained and the revised boundary condition for the plasma flow in the scrape-off layer is worked out
which is found to allow sub-sonic parallel flows at the magnetized presheath edge in presence of an E x

B flow.

Keywords:
scrape-off layer, boundary condition, parallel flow, oblique incidence

1. lntroduction

electric field. If the radial electric field is absent, thisboundary condition reduces in the its conventional form
ol = c, obtained by Chodura [2], which is true in an

The Boundary condition used at the ends of scrape-off layer (SOL), typically at the divertor and limiter
surfaces, generally have contribution from various flows

ideal case of perfectly magnetized plasma. In agenerated in parallel and perpendicular directions. Thesemagnetized presheath region, mechanisms are present
which are responsible for deflecting the parallel flows
into the direction normal to wall and provide an extra

flows include a parallel flow from the bulk plasma andE x B flows both in parallel as well as cross field
direction. At the entrance of the magnetized presheathcontribution to the boundary condition. This deflection
mainly occurs as there is a shear in the E x B flowwhich is typically in the region of SOlwhere the plasma
connects to the solid surface, a crucial role is played bygenerated because of a nonzero divergence of the self-

the inertia in the E x I direction, in determining thenature of flows parallel to magnetic field []. A moregeneral boundary condition was worked out by
Stangeby and Chankin [3] by incorporating an extra
effect of the cross field flow generated due to a radialconsistent presheath electric field E (not a radial orexternal field). The transfer of energy from a parallel tonormal flow is expected to be very effective close to thesolid surface and needs to be incorporated in theCorresponding author's e-mail: devendra@ plasma.ernet. in218@2000 by The Japan Society of Plasrna
Science and Nuclear Fusion ResearchSharma D. er a/., Revised Boundary Condition for the E x B Drift Dominated Plasma Flow in the Scrape-off Layerboundary conditions. In this paper this effect is included

by allowing'a finite derivative of the self consistentelectric field in the direction normal to the wall, which
results into an effective polarization drift of ions andmodifies the moment equations.

2. Modified Fluid Equations
According to the geometry shown in the Fig. 1,
there is a self consistent presheath electric field along y,Fig. 1 The geometry of the region, magnetic field I is ini.e. poloidal direction and a radial electric filed due tovariation in electron temperature along radial, i.e. x
direction. The magnetic field B lies in the toroidalpoloidal plane, making an angle 0 (sin0 = 8/lBal) withthe solid surface. A new set of unit vectors (Ax, A) canbe defined for the simplicity (see Fig. l), where 611 isalong the direction of magnetic field, in y-z plane, ?1 isperpendicular to ?11 in the same plane and d, inthe y-z plane. making an angle 0 with thetoroidally continuous target surface.

R- a'-(6)t^noand Cl is the ion gyro-radius and a prime denotes adifferentiation with respect to y.

unchanged.

The total flow velocity vector in terms of flows

2.1 Equation of eontinuity

along parallel 21, perpendicular ?1 and x, as below
v = {'u,

The equation of continuity, which comes from theconservation of flux along all the three directions cani +(arrsind+orcosd )f +(orsin0-a,'cos0;21(l)now be written as below, assuming that all theEvery ion gyrating and drifting in the y direction,

derivatives of electric field higher than E'1ate zetoshould experience an acceleration in the perpendicular
direction due to presheath electric field E, changing(nu,,)+dcosl ! @rn)+Fcosl
sin0$
qyq),

implicitly with time in the gyrocentre frame as there isx!dy"dy"dx{nuo)=-cosd ! {roo)-! {nu,) (7)no explicit time dependence in the steady state plasma.
This is equivalent to a polarization drift [4] when there
is a time varying electric field in the transverse direction(here E1). This drift is given byr dE, dy
-P OB dy dr

A general dependence of electric field E, andtherefore of uo =8,1131 on r, can be adopted directlyfrom the formulation of Sta-ngeby and Chankin [3]. TheE, here arises due to shear in the electron isothermsQ)

along the radial direction close to the separatrix, which
results into a variation of the floating potential along x
at the magnetic presheath edge. The op therefore isgiven byThus oo represents a coupling between an En x B
flow parallel to the wall and a drift generated in thedirection normal to the wall due to shear in this flow.
This is evident as the polarization drift is proportional to
the rate of change of ErlB with y. As the drift along y
has the contribution from o11, an E x B drift o, and theao(Q)=aol(D_ff'"=(t*)

tr''sind+(or+ oo)cosOJ (3)ar= dur, + Puo(8)

where @ is the electrostatic potential andpolarization drift a, (later two along ?1), the total
polarization drift becomes

ldEDr=#lh(e)The term dupldy can therefore be calculated, using theBoltzmann distribution of the electrons (n" - nn exp (eQ I(4)kT")) and the quasi-neutrality (nr = ne) to obtain an

where,

effective E, as follows| kT" dn(10)dQ dylBll, ne dy'The flux nu, along x, is written using a general r
dependence of density, n = no(l + xll,) and of electric
duo _dao 4O: _$ Ei sind"=[ I - $ E'rcos?

dy(s)

219

Sharma D. el al., Revised Boundary Condition for the E x B Drift Dominated Plasma Flow in the Scrape-off Layer

field Er. Thus

d-d; ."-_cos0nEr(O)-IE:U _--

cosekT" dn"Wdy(1 1)

where, lp = l(l)-t + 0,f11-1.
Substituting dapldy and df,/d-r from Eqs. (10) and(ll) respectively, continuity equation (7) yields

duu__[^|(f+l)sina ltf+l"n___!

dykT" dtrrrlt^e;lBl d.I(r7)

We now define, the Mach number, M = u11lc,, thestrength of drift,

rlo,=Lo2c, tan9

xlsin0ao +cosda, )+kr"cosol r

elBl \/p-8.'llor.
,, )lo,'(18)

and the other quantities(r2)

2.2 Momentum equation
We now write the momentum balance in the€ =M+2g(1e).'- - |or"el Bl l^ltF*l)tan0c,t'Q0)=l ^ \r",l)elBl=.d!u\dxlQr)'=[-h.rTfu]

Q2)

direction parallel to magnetic field B asb" -lane$+Andu,
sin9mnu,I"dy +(sin 0+acos91"-rid) -neEu=S^-muuSo (13),where S. and So are the momentum and particle sources

respectively, generated due to the cross field flow

Thus the revised equation which should be solved tofind the correct value of boundary condition (the valueof o11 at y = 0) while integrating the SOL equations incontributions, and given by

S....=-cosO !.. l*run(ao+ uo)ldy

each iteration, becomes-l'{ma,,nu,)d.r('z+2a(-b -c=O(14)Q3)

where, l, (i.e. /1 and lr) and danldx are the parameters toandSp =-cosg

ftt"Or+ ur)l_be iterated on while converging to an exact SOL

fr{nu,l.

(15)

solution.

3. Analysis and Discussion

Substituting Eqs. (14) and (15) in the right hand side ofEq. (13) and using the equation of continuity, quasineutrality and Boltzmann electrons, Eq. (13) can be

The Eq. (23) has some additional effects due to theinclusion of polarization drift. Since the factor B is
proportional to E'r, these effects are present if thewritten as

derivative of the electric field E, is finite at the magnetic
presheath edge. This can be readily seen that Eq. (23)AatnYt<9 + l) (sin da,,t + cosOa,u-)l =- 9ady'"
dyl,I I -r/T.\ cosflkT da, I
xlsinelci .P;).;;;#ltr
I

reduces into the equivalent equation obtained by
Stangeby and Chankin [3] in the limit (B -+ 0). In order(16)

to compare the results we adapt the similar procedure tofind the solution of Eq. (23) by redefining the quantities
a and b in terms of g using the definitions of up which isan E,x B drift given by

where all the derivatives ofelectric field E, higher thanE', are neglected.

,o=-4#rv*l

We can now eliminate duxld, from Eqs. (12) and(16), and equate the coefficient of d"ld, to zero, sinced,ldy i 0 as a finite electric field is required at thewhere, y, is the total potential drop between magnetic

magnetic presheath edge. This yields

presheath edge and the wall. The a and D therefore take220Q4)

Sharma D. et al., Revised Boundary Condition for the E x B Drift Dominated Plasma Flow in the Scrape-off Layer

the form'zo=ffiboQs)

u=_ffiffo,(a),(ao, bo)=(l; 1)

Q6)

where,oo=71;ilr,),'1,,

Q7)

and

bo= --+ (-+,-u(F+t)\ltf,ll, lV-lJ'--+).

(28)
420o0.2 0.4 0.6 0.8

10.2

1

We now write the solution of Eq. (23) giving therequired boundary condition.

\!tc"=- g(2+ao+bo)+fs'{oo+u)'*r*+oog'fiQe)

The value of u11yps6 is plotted in the Fig. 2 forvarious values of parameter B which signifies thestrength of the polarization drift at the magnetic presheath edge. This can be seen that the effect of thepolarization drift which is the mechanism responsiblefor deflecting the parallel flow into the direction normal
to the wall, is mostly to cancel off the effects produced-0.4 4.2

by the op and bring the value of allypss to a constantsub-sonic value. The two cases present in the Fig. 2(a)

0o0.4

0.6

0.8Fig. 2 The value of r.r,,r""r/c" plotted vs. the strength of Ex 8 drift velocity g, for the values of B = 0.0,2, 4,
10 and 100. Two cases presented are (a) (ao, bo) ={1. 1} and (b) (ao, 4) = (-1, 0}.

and (b) are for values ofparameters (ao, bd equal to (1,l) and (-1, 0) respectively, which represent maximum
contrast in their behavior in the analysis of Ref. [3].
This is interesting to note that in both the cases the roleof polarization drift remains unchanged and it keeps
canceling the effect of E x B drift by pulling it to aconstant sub-sonic value. This confirms that thepresheath, where the magnetic field is obliquely incident

presence of strong presheath mechanisms reduce theeffects of drifts produced by an external electric field.
This drift is modeled by an E x B drift present in theSOL due to a radial electric field E,. As the parameter B
has a resonant nature, it takes large values if the value ofE1 becomes larger at the magnetic presheath edge. In
both of the cases presented here the value of oyp56 is
sub-sonic at g - 0 except the case when 0 = 0 which isthe case of an ideal magnetic presheath where no drifts
are present or the ions are fully magnetized.

on the targets. The function of a polarization drift in amagnetic presheath is to deflect the parallel ion flows in

the direction normal to the target thus to make theparallel flow velocity required by the ions at themagnetic presheath edge sub-sonic.

Rerefences

tll K.U. Riemann, Phys. Plasmas,l,552 (1994).
t2l R.Chodura, Phys. Fluids,25,1628 (1982).
t3l P.C. Stangeby and A.V. Chankin, Phys. Plasmas,2,7O7 (1995).

4. Gonclusion[4] Dwight R. Nicholson, Introduction to plasma

The effect of an E x B drift may be reduced by thestrong presheath mechanisms present inside a magnetic

theory, (Krieger Publishing Company, Florida,1983) page 24.
221