Hutchinson, I.H.

DSpace@MIT

PFC/JA-95-32The Magnetic Presheath BoundaryCondition with E x B DriftsI.H. HutchinsonAugust, 1995Submitted to Physics of Plasmas.This work was supported by the U. S. Department of Energy Contract No. DE-AC02-78ET51013. Reproduction, translation, publication, use and disposal, in whole or in partby or for the United States government is permitted.The Magnetic Presheath Boundary Condition withExB DriftsI.H.HutchinsonPlasma Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts, USAAbstractIt is demonstrated rigorously that the effects on a one-dimensional, collisionless, magneticpresheath with oblique magnetic field B of a uniform tangential electric field, causing adrift VD = E x B/B 2, are equivalent to a transformation to a frame moving tangential tothe surface. Therefore, in particular, the Chodura solution with parallel velocity, vj = c,(the sound speed) is transformed into a solution with vjj = c, + VD / tan a where a is thefield angle to the surface.PACS: 52.55 Fa, 52.40 Hf1IntroductionThe magnetic presheath is a region of acceleration close to the boundary of a plasma incontact with a solid surface, in which a magnetic field B exists at an oblique angle a tothe surface. In obtaining detailed solutions of the plasma equations, the region is generallyconsidered to be divided into a Debye sheath region, where quasi-neutrality is violated,and a magnetic presheath region with thickness of order the ion gyro-radius, where theions are accelerated across the field to enter the sheath at the sound speed. For presentpurposes, we will refer to the combined sheath and magnetic presheath region loosely asthe magnetic presheath.Standard solutions of the magnetic presheath problem have been calculated. Whencollisions are negligible, the solution of Chodural applies. As Riemann has also shown2this solution requires the plasma ions to flow into the magnetic presheath with parallelvelocity greater than or equal to the sound speed. This inflow condition is then takenas the boundary condition to be applied to solutions of the external plasma region. Theboundary condition is extremely important in understanding the behavior of the divertorand scrape-off-layers of tokamaks. However, it is known that particle drifts caused byelectric fields are substantial in these situations. Therefore the extent to which theseperpendicular electric fields may change the boundary conditions is of vital importance.Stangeby and Chankin3 have discussed the effects of E xB drifts on the magneticpresheath boundary condition. On the basis of "intuition" they anticipate that the bound-ary condition will be modified such that the incoming parallel flow speed will be vg =c. + vD/ tan a. They also derive a generalized form of this condition, from an approximateanalysis of the ion fluid equations. However, the lack of rigor in Stangeby and Chankin'sanalysis and the complexity of their attempt to account for various extra effects (which2may well be important in practice) tends to obscure the result, and seems to leave doubtas to its applicability. Therefore a rigorous analysis is provided here of a more specificgeometric case, which nevertheless contains the essential effect. The present analysis isgeneral, applicable to any collisionless plasma, and not dependent, for example, on fluid orother approximations. The Stangeby and Chankin result is broadly confirmed but severalsubtle questions remain.AnalysisWe consider a planar, one dimensional magnetic presheath, in which z and y are ignorablecoordinates. The solid surface consists of an zy plane that may be taken as z = 0 andthe magnetic field is in the zz plane. We are interested in the effect of a uniform electricfield in the y-direction, Es. In elementary plasma analysis, it gives rise to a drift velocityE x B/B 2 = E,,Bx - EBj. Naturally there is, in addition, a self consistent electrostaticfield -VO in the z-direction arising from the presheath structure.Now we note that the particles of a collisionless plasma obey the equation of motiondT(mY)=q(E+vxB)where m and q are the particle mass and charge respectively. For non-relativistic velocities,Vt, the Galilean transformation of the parameters of this equation to an inertial frametravelling with velocity vt isV' = V - VtB'= B (2)E' = E+vxBand equation (1) is still satisfied by the primed quantities. The Maxwell equations, whichcomplete those obeyed by the plasma, are also invariant under this transformation (for3non-relativistic vs). If we assume, as is usually done, that the boundary condition imposedby the solid is simply that all particles impinging on it are absorbed, then the boundarycondition is also unchanged by this transformation provided that vt lies in the plane ofthe ignorable coordinates xy.Therefore, from any solution of equation (1), and the related plasma equations, we cangenerate solutions to related problems in which the electric field has an extra componentvt xB, by transforming to the moving frame. In particular, if there is in the lab frame anextra field E,, the required transformation velocity isV = iE- = iE (3)B, Bsina 'which will render E, = 0 in the transformed frame. The primed quantities may thereforebe taken to be a solution to the standard collisionless magnetic presheath problem. Thearchetypical solution of this drift-free problem' has as its asymptotic solution for the ionvelocity far from the solid surface (at large negative z):v=c,(i cosa+ sina) , (4)that is, parallel inflow at the sound speed, c, (although inflow at any supersonic speedgives a consistent magnetic presheath solution). The corresponding solution when theextra field E. is present is thereforeV = c, icosa + .) + i sina , (5)which may be writtenV=$ (C. + + (yb)VD ,(6)tan awhere VD has the sign of E,. Thus we obtain the standard perpendicular drift velocityplus velocity parallel to B altered by vD/ tan a.4DiscussionThat expressions (5) or (6) are a solution to the equations is clear. Whether it is the requiredsolution is more subtle, depending as it does on the boundary condition to be applied farfrom the surface. The usual condition applied is that the magnetic presheath solutionshould match smoothly to the outer plasma solution, which is presumed to have scalelengths much longer than the magnetic presheath. For the drift-free case, the magneticpresheath solution requires (super)sonic parallel inflow velocity, and consideration of thedynamics of acceleration in the plasma region usually dictates a (sub)sonic solution there;hence sonic parallel inflow at the join. We shall call this the sonic solution.The case with drift is less clear. The solution in the external plasma region cannot ingeneral be collisionless, because that would prevent acceleration from occuring4 . The mosttypical presumption is that collisions in the form of ionization are present in the externalregion; although sources such as momentum drag are also effective in providing a well-posedproblem2. In either case, the equations of the external plasma are not invariant under theGalilean transformation discussed, and the matching must therefore be considered in thelab frame.If -2c. 5 VD/ tan a < 0, so that the transformed sonic solution has subsonic parallelvelocity in the lab frame, it would appear that the external solution can be matched.There is, in fact, a range of supersonic magnetic presheath solutions to which they canbe matched and how to chose which solution is appropriate is not obvious. However, ifVD/ tan a > 0 or < -2c, then the transformed sonic solution requires supersonic flow in theexternal solution at the join. This is not necessarily possible. For VD/ tan a < -2c, thisproblem may be avoided by discarding the sonic magnetic presheath solution and insteadadopting a supersonic solution which transforms to vg ;. -c, in the lab frame. The case5of VD/ tana > 0 is more problematic since the requirement of supersonic lab-frame inflowappears unavoidable.In conclusion, solutions of the magnetic presheath equations in a plasma with unformelectric field, giving an E x B drift, can be obtained by a simple Galilean transformation ofsolutions of the drift-free problem. The resulting solution in the lab frame adds a parallelvelocity VD/ tan a as well as the cross-field drift. However, subtle questions concerning thematching of this solution to the external plasma remain, and can only be resolved by amore detailed description.AcknowledgementsIt is a pleasure to acknowledge stimulating discussions with P.C. Stangeby. This work wassupported by US Department of Energy contract No DE-AC02-78ET5103.References1R. Chodura, Phys. Fluids 25, 1628 (1982)2K.-U. Riemann, Phys. Plasmas 1, 552 (1994)3P.C. Stangeby and A.V. Chankin, Phys. Plasmas 2, 707 (1995)4 I.H. Hutchinson, Principles of Plasma Diagnostics (Cambridge University Press, NewYork, 1987) p6 7 .6

The magnetic presheath boundary condition with ExB drifts

http://dspace.mit.edu/bitstream/1721.1/95258/1/95ja032_full.pdf

The magnetic presheath boundary condition with ExB drifts

Abstract

Similar works

Full text

Available Versions

DSpace@MIT