J. Plasma Fusion Res. SERIES, Vol.4 (2001) 194-198

Near-Wall Plasma in Magnetic Field and Bohm's Griterion
NASTOYASHCHII Anatoly* and MOROZOY Igor'"Troitsk Institute for Innovation and Fusion Research"
TRINITI 142190 Troitsk Moscow ReR., Russia(Received: 5 December 2000 / Accepted:27 September 2001)

Abstract
The near-wall plasma in a magnetic field and the Bohm criterion are discussed on thehydrodynamical model. A magnetic field is assumed to be in parallel to the solid surface.It is essentiallythat the methods used allow to consider the near-wall plasma without deviding it into separate sheaths.Ithas been found that in the case of a magnetic field directed in parallel to the surface the conditions for theBohm criterion validity (and formation of monotonous electric potential) depend upon the plasma
temperature, the magnetic field strength and the velocity of poloidal plasma rotation.It is marked that ion
scattering processes play a very important role in formation of the near-wall plasma structure. For
instance, the characteristic length of the quasineutral plasma where ions acquire a directed velocity equal
to the ionic sound depends on the electronic ionization rate of atoms, the magnetic field strength and thevelocity of poloidal plasma rotation.

Keywords:
near-wall plasma, electric potential profiles, particle scattering, the Bohm criterion(1) As a rule particle collisions are ignored in thetheoretical models used for the description of

1. Introduction
The investigations of physical phenomena in aplasma near a solid surface attract considerable
attention. Hundreds of experimental and theoreticalworks have been devoted to the subject. The greatinterest to the problem is explained in view of the factthat phenomena on the plasma boundary play an
extremely important part in lots of plasma applications:electric discharge devices, thermoionic converters,

the near-wall plasma.

(2) The thickness of a region, in which the plasma
neutrality is violated, is assumed to be comparable with the Debye radius.
(3) The fluxes ofpositively and negatively charged
particles to the wall are exactly the same, i.e.,
no charge transfer takes place in the system.

(4) There are three spatial regions [3]:plasma technologies in microelectronics, etc. Recently
the problem has become of great interest because of theinvestigations of the near-wall plasma in tokamaks and(a) the Debye sheath,(b) upstream of it, the Chodura magnetic presheath.

stellarators [1,2].

In the majority of the suggested up-to-date(c) upstream ofthat, the ordering pre-sheath.
(5) According to the pointed models, the monotonic potential distributions near the wall canbe realized provided that the Bohm criterion issatisfied [4] (see also [5]): the velocity of thetheoretical models an approximation of "two" or "threescale theory" has been used. A brief survey of existing
theoretical models describing the plasma boundary canbe found in [1,2]. The main assumptions underlying
these rnodels are as follows:@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research*Corresponding author's e-mail: anast@ triniti.ru194

Nastoyashchii A. et al., Near-wall plasma in Magnetic Field and Bohm,s criterion

directed motion at which ions penetrate into the area of

2. What is the Essence of the problem? TheProblem Statement
The problem is reduced as it is known to theinvestigation of the Poisson equation solutions 1wethe charged layer must exceed that ofthe ionic sound, ae> t = "[Til*r, where Z" is the electron temperature andm; is the ion mass. In the area of the discharged layer

ions gain an additional energy of the order of several

consider a one-dimentional case)electron temperatures.
In a number of works efforts were made to validate
or revise the Bohm criterion; references to some of these+6

d'tp/dx2 --4n2".,r, I f"(v,x)d3 v ,''"Jworks can be found in the reviews mentioned above(1)

where fo is the function of distribution in velocities ofparticles a with a charge eo. Significant difficulties

Lr,2l.

As the Bohm criterion is widely used for physicalinterpretation of the phenomena close to the plasmaboundary (in the theory of probe characteristics, in lotsof numerical codes etc.) it seems expedient to discuss
the limits for Bohm criterion validity. This statement has
been considered in our papers [6,7] in the frame of astrict theory where it was shown that the above-saidarise at calculating charge densities in the right part ofeq.(l). In some works (see, e.g., [1,2]) an approximare
approach was used for deducing the Bohm criterion
based on two or three scale theory. In order to makeclear the notes given above let us use the mechanicalanalogy [6]. It follows from the Leuville theorem that in
the absence of particles scattering processes theircriterion was not universal. It was found that ion

distribution function depends only on mechanicalscattering processes play an important part in formation

invariants of the typeof electric potential profiles near the wall. In [7] aed =mv:12*

method of solving a kinetic equation for ions near theplasma boundary has been proposed and on its basis aeo(p=ss11s1,A)i.e. we have/o =fo(v,,9). Thus, the right part of eq.(1)numerical analysis ofthe problem has been done. It was
shown that under conditions when a charge exchange
frequency exceeds that of electronic ionization of atoms
monotonous regimes of electric potential distribution
take place regardless of the Bohm condition including

will depend only on g, i.e. equation (1) can be given in
the form:
d2rpdx'the case when a << s. At the same time, at a- _dU(rp)

(3)

dQ

wherepredominant role of ionization even in the region of aquasi neutral plasma ions can gain the velocity equal toatu= .l| F(oda

the ion-sound one (see also [6]). Thus, using
approximation models with ion collisions being(4)0

is "the potential" and F(e) = 4n(ni- n.).
So we deal with a "conservative problem" where thelaw of conservation of energy is satisfied:neglected can result in blunders. In particular, as follows

from the results of the numerical experiment [7] theextension of the wall plasma (areas of basic potentialdrops), the wall potential value, the electric field on theIt2

tl2ld(pldxl +U1tp1=6(5)

If there is no magnetic field and ion scattering, forsurface and the ion energy close to the wall substantiallydiffer from the results of the approximation models (seecharge densities n" and ni we can use expressions (see[,2]). Below the results of our works [6,7] arefor instance [6]):generalized for the case of a magnetic field directed infl.= fl6exp(etplT")

parallel to the surface. In the case of a magnetic field
directed almost perpendicular to the surface the near-n,= ,rrolr[1uf; -z"rplwall plasma structure has been considered by Chodura[8] (see also [3]) in view of the approximation model.

(6)

^;A)

where mi is the ion mass, lrs is the directed velocity andBelow we consider a theoretical model which does not
require deviding the near-wall plasma into different

7" is the electron temperature. Let us choose the origin-16 at some distancesheaths with essentially different characteristic scales.

from the surface , setting e@i = 0.

At the same time it is assumed that dEldx + O at x = xo,
since the field other than zero is necessary so that ionsThus in the frame of our theoretical description theunderstanding of "a sheath", "pre-sheath", etc is of noimportance.

could gain a velocity. The case, wherc dEldx = 0 in thecenter of the plasma layer symmetry, has been
195

Nastoyashchii A. et al., Near-wall Plasma in Magnetic Field and Bohm's criterionorders of magnitude which is not consistent with theobserved values. Nevertheless, with account of volume
ionization processes the ion velocity can attain u > s [6]'if the atom density in the boundary arca is rather high(recycling [l]).

3. The Influence of Scattering Processes
and Magnetic Field Effects
Thus, profile character for electric potentials isdefined by the velocity of directed ion motion. In its
turn this velocity depends on scattering processes inplasma. So together with monotonous profiles solutionsof the type of spatial potential oscillations prove to bepossible; a numerical study of the oscillations is not aneasy task because in this case it is necessary to allow for

Fig. 1 Potential curve U = U(q).

considered in [5].

charged particles, electrons and ions, captured in theFigure I shows an exemplary plat of function U =U(tp) at ue < s with charge densities of eqs.(6) and (7)taken into account. The constant in the right part of eq.
(5) is expressed in terms of a derivative of dqldx(xs):potential maximum (minima). A similar task can besolved using numerical technique.

(8)

3.1 lon Formation Rate Effect
Now turn to monotonous potential distributions.

As it is seen from the figure, no solutions exist for u6 <s, which could correspond to an "infinite motion of a

For simplicity we use the gas dynamic description forions (it is well-known the hydrodynamic description iseffective also in investigating rarefied plasma).

tt2

6=ld91*";lldx'"
Iparticle" (a monotonous potential profiles). Only
oscillation type solutions can be realized with adu, p- mi:Ja -vtu,- s;,Yrfr,ui'dr = L Lni

oscillation amplitude between Et and <pz; however thesesolutions do not meet the boundary condition on thesurface. Only at a6 > ,t possible are particles paths
corresponding with a monotonous profile. Thus to form

(9)

whire u1 is the velocity of directed ion motion and v; isthe frequency of ion scattering. In contrast to reference[6], here the ion gas pressure is taken into account (asmonotonous potential distributions near the surface theBohm criterion must be satisfied. However, to gain theion velocity, u) s, an electric field in plasma must be ofa significant value. At the same time there are noknown, in a boundary area oftokamaks electron and ion
temperatures are usually comparable in their magnitude
Ti - T"). Above ,, ='{yTJ^t where 7is the adiabaticindex. Equation (9) should be completed by the equationelectric fields without collisions.
In fact, in the frames of the considered statementof continuity for ions

!ox nrrr= d,"n,(T").

the problem is not closed since the particle flux fromplasma must be replaced, for example, through injection.
If the injection proceeds in the center of the plasma
layer symmetry at .r = 0, then the central plasma areaacquires a positive charge (electrons tend to quit thisarea). This results in a potential difference between theplasma center and the wall which equalizes electron and(10)where a" is the electron ionization rate of neutral atoms.
In the limit of a quasi neutral plasma, when n" = ni ar'ddn.ldx = dn1ld-t from eqs.(9) and (10) together with eq.
(6) we have (vi = 0):

f*i - r"il# + d,=o ,ion fluxes. As a result, the solution complies withboundary conditions. In tokamaks (stellarators) usuallyobserved velocities of plasma expansions n - I +- 1O m./s
So, in order toat a plasma density n - l0r3 -+ lQla

(1l)where s"12 = s2 + Ji2. Hence, J"f corlgs into the Bohm
criterion with the value of ion temperature taken into"--:.

account. From eq. (11) we find that the ion velocity
equal to s"1 is achieved at the distance x^= 213 s*/a..
Thus with due regard to ionization processes ions cankeep the plasma flux density and have the ion velocity
comparable with that of the ionic sound it would benecessary to reduce the plasma density by about 4-5196

Nastoyashchii A. et al., Near-walr plasma in Magnetic Field and Bohm's
criterion

gain a necessary velocity il - ser. The estimation isquoted. At a density of neutral atoms in the near wall
plasma nu - l}ta cm-3 and T" - 50 eV we have q.
101-s-r, so for the wall plasma extension where the main

rotation the latter can favour acceleration or decelerationof ions. If rotation is caused by a radial electric field,

then ion deceleration takes place. Using a set ofequations (6), (10) and (12)-(13) one can find the

3.2 Magnetic Field Effectfollowing equation for determining the velocity of thedirected ion motion u, which the ions can gain not

The results of the proceeding analysis hold true forthe case of the plasma in the magnetic field direction

leaving the area of a quasi neutral plasma:

which is almost normal to the surface. Now we consider
another case where the magnetic field is parallel to the,t(r' - r"ll4a* | a"s"f + ua,(at,x-r,rtfl =0 (16)"'dr t "'

surface. As before the electron distribution in thepotential field is described by the Bolzmann formula't'l

As in the tokamak plasma the Larmour radius of ions
usually exceeds the thickness of the area of spatial(eq. (6). The equation ofcontinuity for ions is kept also
the same (eq. (10)). The plasma parameters are assumed

charge separation, the influence of a magnetic field on

to depend only on the x-coordinate. At the same timethe ion velocity has two components (u*, uy) which aredescribed by a set of two equations:

ion motion can be regarded as insignificant. So thecriterion a > s"1 will be again a condition of formation
monotonous profiles of potential. To obtain analytical
data we substitute the second member in square brackets(r2)

in eq. (16) by a model member(r7)s,rDr(@rx -u,ro),^idu,., =-oJitrs,(13)

The solution of equation (16) with a model member (eq.
(17)) is of the form:

where ar1 is the ion cyclotron frequency. In the y_

4 - s"l + | F(x')clx'=
xP

coordinate direction the plasma is considered uniform,

Jand y-component of electric field is absent. Besides,u,.

.l0(18)0

there occurs plasma rotation perpendicular to theA maximum velocity is gained by ions in the plasma of

magnetic freld u, + 0 (with reference to tokamaks this is

thickness -r. which is determined by the following

poloidal plasma rotation). The magnetic field is directed

equation:along the z-axis.

^mr^
IJ F(x'\dx'=1s ]3"'0From eq. (13) for the y-velocity component wehave:llrr= urro -{|ix ,= 0.

As it is seen from eq. (14) the velocity of plasma
rotation reduces at approaching to the surface. We
restricted the consideration to an approximation of non_viscous liquid. In the near wall plasma there are electricfields of fairly high magnitude which play a dominanr

condition with a dominant effect of plasma rotation we
have x. - s"f luroai.In the second case, assuming s"s
107 cm/s, ati- 108 s-r we have r. - 0.1 cm.

-4. Basic Conclusions

part.

1. Processes of ion formation play an exclusive role inshaping electric potential profiles in the near wall

Equation (12) integrates, recording eq. (14) to thefollowing expression''t,(le)At al1 = 0 we come to the result obtained in [6]. If thelast member in the left part of eq. (19) predominares(strong magnetic fields) we have x^ - s"s/ai. ln(r4)where airo is the velocity of plasma rotation at,+ =-t,5rP+ a'(u,,ox-depends on, beside g, the electron and ion temperatures,the velocity of poloidal plasma rotation and themagnetic field magnitude. Depending on the direction ofpotential drop occurs we obtain x^ - | cm. Thus theprocesses in a fairly extended plasma area (e.g.
compared to the Debye radius) have an effect onshaping the potential profile structure.

u,.!=-"
44g*r,u,,;
'" dx
s2 dx

In this case the boundary conditions ux (p 0 at -r 0
= ==were used. As it follows from eq. (15) the ion velocityplasma.

2. Monotonous distributions of electric potential are
possible only at a fairly high ionization rate of neutral
atoms, when ions can acquire the velocity of directed

(1s)

r97

Nastovashchii A. et dL, Near-Wall Plasma in Magnetic Field and Bohm's Criterion

motion equal ionic sound In the absence of any
scattering processes only the oscillation-type solutions
can exist.
3. In the magnetic field directed parallel to the surface,conditions for realization of monotonous potential
distributions depend upon the magnetic field strengthand the velocity of poloidal plasma rotation as well as

t2l K.-U. Rieman, J. Phys' D:24,493 (1991)'
t3l P.C. Stangeby, A Note on the Bohm-Chodura

Plasma Sheath Criterion, Preprint JEI-P(94)52'
JET Joint Undertaking, Abingdon, Oxfordshire'oxl4 3EA, UK (1994).

t4l D. Bohm, The Characteristics of Electical Discharges in Magnetic Fields. Eds' A' Guthrie andR.K. Wakerling (N.-Y. MacGraw-Hill) ch' 3, p'17 'on electron and ion temPeratures.
4. In the paper the theoretical model was used which

t5l O.B. Firsov, J. Tekh. Physics (in Russian) 26,308

does not require deviding the near-wall plasma into
separate sheaths and consequently the understanding of"a sheath", "a pre-sheath", etc is of no importance.

t6l I.N. Morozov and A.F. Nastoyashchii, Physicaltll(1956).

Conditions at the Interface Between Plasma andMaterial Surface, High Temperature 33, 179(19e5).

References
A.V. Nedospasov et aI., Near-Wall Plasma inTokarnaks (in Russian), in Voprosy teorii plasmy

t7l I.N. Morozov and A.F. Nastoyashchii, Plasma
Phys. ReP. 22, 595 (1996).

Vol. lE, M, Energoatomizdat (1990).

198