non-ideal m.h.d. equations with application to the stability of a rotating plasma column

Physica 94C (1978) 251-258 0 North-Holland Publishing Company

NON-IDEAL M.H.D. EQUATIONS WITH APPLICATION TO THE STABILITY OF A ROTATING PLASMA COLUMN

Peter JANSSEN The “Stichting voor Fundamenteel Onderzoek der Materie” (I? O.&f.), Eindhoven, The Netherlands

Received 21 October 1977

In this paper we consider a low-p plasma confined in a magnetic field in which, in contrast to the F.L.R. ordering, for the core region the electrical force is more important than the pressure gradient force (M.H.D. ordering). Non-ideal M.H.D. equations are obtained from the moment equations. Considerations c.oncerning the outer region result, for example, in boundary conditions for the oscillations in the core repion. We investigate the stability of a cylindrical plasma column for some special cases and the agreement with experiment is discussed.

1. Introduction

Recently there has been considerable interest in low-frequency oscillations which arise in a cylindrical plasma confined by a magnetic field in the axial direction. The presence of an electric field in the radial direction gives rise to an E X B drift which drives these low-frequency waves unstable. A theory which describes low-frequency waves is given by Rosenbluth and Simon [ 11 in which they assume that the electric force is of the same order as the pressure gradient (F.L.R. ordering). In the rotating plasma experiment considered here [2] this assumption is no longer valid; in the core region of the plasma, where the oscillations are localized for the greater part, the electric force is considerably more important than the pressure gradient. Also, the observed frequency of the waves is larger than assumed in F.L.R. Thus, a M.H.D. description of the plasma seems more appropriate (section 2).

In section 3 we derive non-ideal M.H.D. equations for a collisional low-/l plasma from the moment equations and it is shown that to the required order ion viscosity is unimportant, e.g. F.L.R. effects can be neglected. Considerations concerning the region next to the core (the outer region) provide boundary conditions for the oscillations in the core: the outer region acts like a wall. A short comparison between F.L.R. and (non-ideal) M.H.D. is given.

Furthermore, in the M.H.D. ordering we investigate

the linear stability of a cylindrical geometry in which

the density gradient and the electric field point in the

radial direction and are functions of r only; in particular, we consider the special case of uniform rotation and a schematic example involving shear in the velocity. The last example clearly illustrates the stabilizing effect of the fictitious wall (outer region) on the oscillations (section 4).

In section 5 a summary of the conclusions is given.

2. Experiment and ordering

We shall consider low-frequency oscillations in a hollow-cathode argon discharge, confined by a uniform magnetic field B in the axial direction (for a schematic view see fig. 1). Here we take as standard conditions arc current I = 100 A, gasfeed Q = 4.5 cm3 NTP/s,

z,o -%

Fig. 1. Schematic of experiment.

251

252 P. Janssen/Non-ideal M.H.D. equations with application to the stability of a rotating plasma column

magnetic field B = 0.5 T, arc length L = 1.4 m and cathode diameter d = 1.3 X low2 m. Although it is found that the measured parameters of this experiment are z-dependent, we neglect this dependence; all

experimental results given in this paper are measured at z = 0.5 m, and taken from Boeschoten et al.‘s experiment (for a similar experiment see, e.g., Sijde et al. [3]).

The ion temperature is up to I N_ 4 X 1O-2 m constant over radius, namely T rr 4 X lo4 K. The electron density profile has a gaussian shape (fig. 2) where, for B = 0.5 T, the width of the profile r. = 2 X 1O-2 m. On the axis the electron density no equals 1020 rnh3. A characteristic feature of this kind of experiment is the rotation of the plasma column in the azimuthal direction. In fig. 3 the angular frequency a = Uo/r

(where U, is the azimuthal velocity), as calculated from experimental data, is plotted against radius.

-

Fig. 2. Electron density n, versus radius r.

25

0.5

n b rincm

2 4

Fig. 3. Angular frequency CZ versus r.

Fig. 4. E X B frequency WE and diamagnetic frequency WDi as a function of radius r.

Near the axis Q is approximately constant, while in the

outer region Q is strongly sheared and changes sign at r z 2.5 X 10d2 m. Also, the radial dependence of E X B drift frequency c+ and diamagnetic drift frequency ODi iS shown (fig. 4). From the experimental data it can be inferred that the plasma under discussion is a low-/3 plasma (p = 2r.lop/B2 y 10m3 < 1) for which on the axis ~2ciTii ” 1 (where 52, is the ion cyclotron frequency and 7ii is the ion-ion collision time) and the ratio E of ion cyclotron radius? ai to a typical gradient length L, e.g. a characteristic wave length or the width of the density profile, equals 4.

We finally note that low-frequency oscillations (V = 15 kHz) are found in the electric potential, the ion and electron density. Phase measurements indicate that we have to deal with an m = 1 mode and that the wave is azimuthally propagating in the positive @direction (see fig. 1). The waves are localized for the greater part near the axis (the “core” region) as can be seen from the radial dependence of the amplitude of the oscillations in the floating potential (see fig. 5).

As indicated in figs. 2 and 4 we have divided the plasma column into two regions: the core region and the outer region. By comparing the E X B drift frequency tiE with the ion cyclotron frequency fid we find that in the core aE = O(eQci) (f = 4) while in the outer region tiE = 0(&I,& The angular frequency of the wave o = 2nv is also of the order eati. All this

t Note, by the way, that in the literature the cyclotron radius is often wrongly called the Larmor radius.

P. Janssen/Non-ideal M.H.D. equations with application to the stability of a rotating plasma column 253

t

rcncm

Fig. 5. Amplitude of the low-frequency oscillations in the floating potential, Vfl, versus radius r.

suggests the following ordering in the two regions:

the frequency of interest is

w = O(OE) = Of&&,

implying for the electric field E

and the E X I$ drift V is of the order of the ion thermal velocity utIri,

This ordering is valid in the core region; in the outer region we can relax this ordering since then

E = O{WfJjiB); V = O(fUtlt,i)~

Of course the c~acteristic frequency o is the same as in the core region. To summarize we note that according to eq. (2) the core is a relatively cold plasma (i.e. the pressure gradient is ne~i~ble compared to the electric and Lorentz force), in contrast to the plasma in the outer region.

It is worthwhile to remark that the ordering

[eqs. (1) and (Z)] , which in fact is a M.H.D. ordering, can be used to derive a closed set of equations for the low-frequency waves found in the rotating plasma cohmm. This program will be worked out in the next section starting from the macroscopic equations. Rosenbluth and Simon [l] have done something similar for the finite Larmor radius ordering (F.L.R.). In F.L,R_ relatively low frequencies are considered, w = O(w& = 0(e2Q, yielding i = ~~V~/~~) and V= o(EUth,i). They have used this ordering scheme to solve the Vlasov equation order by order and arrived at the well-known F.L.R. equations (in addition the so-called flute ap~ro~mation was assumed}. Also, a derivation of the F.L.R. equations from the macroscopic equations is possible, at least if the correct expressions for transport coefficients like viscosity are taken (see e.g. Macmahon 1441).

3. Non-ideal M.H.D, equations

In the rest of this paper we use the flute approximation, namely

(a) a low-/3 plasma, i.e. V X E: = 0, moreover the magnetic field is a constant in time, and

(b) a two-dimensional problem; the magnetic field is constant in space and direction. It points in the direction of the +z-axis; we do not take into account field curvature effects.

These assumptions are more or less satisfied in the experiment under discussion. Note that the E X B velocity Y,

V=ZXB/@

is divergence-free,

v* v=o

by virtue of the flute approximation.

(6)

C-9

First of all we are interested in the core region where the M.H.D, ordering is valid. Next the outer region is considered. The basic equations we start from are the first two ion moment equations for a collisional plasma (no inelastic collisions), which can be obtained from Boltzmann”s equation. They read

254 P. Janssen/Non-ideal M.H.D. equations with application to the stability of a rotating plasma column

‘drr.=-Vpi-~‘4+eni(E+C:XB)+Ri_,, mini dt i

(9)

where d/dt E alat + Ui l V and where ni is the ion density, Ui the ion macroscopic velocity, pi the ion pressure, ni the ion viscosity and Ri_e the friction force between ions and electrons.

Up to now the form of the viscosity tensor and friction force is unknown; it should, however, be noted that Braginskii’s 151 transport coefficients can be used in M.H.D. because the relative velocity between ions and electrons is still of the order euth i since for ions as well as electrons the E X B velocity V is the same in magnitude and direction. Also, the characteristic frequency w is of the same order as in Braginskii’s treatment. For the form of the transport coefficients we therefore refer to Braginskii’s article [5] .

Utilizing the assumptions in the M.H.D. ordering together with the additional assumption fiei TV = 1, the various terms in the momentum equation (9) can be estimated. The left-hand side of (9) is of the order e(times eni U*,iB), the pressure gradient is of the order e, the electric force and Lorentz term are of order 1, while the term involving the viscosity is of order e2, since tr = O(pi/2s1,iL). Furthermore, it can be estimated that the friction force Ri_e is not signifi- cant for the ions since this term is proportional to the ratio of electron mass m, and ion mass mi.

Expanding all macroscopic quantities (except E

and B) in powers of the small parameter E it is now possible to derive from eq. (9) an expression for the macroscopic velocity (correct to first order in E),

%-XB ml Dt

Vi = V-e B2

-v p. X B/en@ 2, (10)

where V = E X B/B2 and D/Dt = alat + V l V , so the macroscopic velocity consists of three parts: (a) the E X B velocity V; (b) the diamagnetic drift; and (c) a

term due to the inertia of the ions. Note that the E X B drift is of the order of the ion thermal velocity, while the other two terms are of order eUth,i* TO first order no contribution due to the viscosity is involved.

Substitution of eq. (10) into the continuity equation

in first order

~(n,+nl)+V.{nOu~+nlu~+rrgU1! =o (11)

finally yields after some algebra

D -ni + Dt

;E=O, (12)

where for the derivation of this equation the flute approximation is used. We emphasize that the rate of change of the ion density ni in the E X B drift frame is due to the E X B drift via the inertia term (usually called polarization drift).

Also, the moment equations for the electrons can be treated in a similar way; however, since the cyclotron radius of the electrons is much smaller than the ion-cyclotron radius, because of me/ml < 1, we only need an equation for the electron density in zeroth

order,

(13)

Combination of these equations with the two Maxwell equations

VX E=O and V.E=$(ni-3 (14)

finally results in a closed set of equations, which we call non-ideal M.H.D. equations, since first order corrections in E are taken into account.

It can be concluded that to first order ion-ion collisions do not contribute to the rate of change of the ion density, and we do not need a higher order equation for ni since already interesting physics is obtained, as will be seen in the next section where the linear stability of a cylindrical equilibrium is investigated. In addition we note that we have obtained the same results directly from the Vlasov equation (,n,i7ii * m) in the same way as Rosenbluth and Simon [l ] have for the F.L.R. equations.

The set of equations (12)-(14) can be simplified by taking the D/Dt of Poisson’s equation and then eliminating ni and n, via (12) and (13). The result is

EOB2D~.~+~.n.D~=~; mi Dt 1 Dt

I! Janssen/Non-ideal M.H.D. equations with application to the stability of a rotating plasma column 255

this equation introduces the parameter EoB2/m.ni.

NOW the parameter b2 f eoB2/mini= i23/~‘.,

where w pi is the ion-plasma frequency, is for tpe plasma under consideration typically of the order 10-7-10-6. So as a matter of fact two small parameters are involved in the set of equations (12)-(14) namely E =ai/L and b2 = foB2JmiYIi. However, since E % b2, it is sufficient to consider only the zeroth order in b2, in other words for these dense plasma’s quasi-neutrality, n, % ni = n, is a valid approximation.

Hence we arrive at the set of equations

D a) in =O,

b)V*n %,=o, 09

c) V X E = 0,

and by virtue of eq. (15b) we only need a zeroth order equation for the density n. For the low density case see, for example, Stringer and Schmidt [6].

The derived set of equations (15) applies to the situation in the core region of the rotating plasma column. In the outer region, however, we can relax the M.H.D. ordering, e.g. the E X B drift frequency increases one order: 0~ = o(f2 ati), thus v = O(eU,i).

We are interested in the effect of the outer region on, for example, oscillations in the core region. To this end we consider the simple model of the plasma in which the properties of the plasma change discon- tinuously at the boundary between core and outer region, so that an interface exists. This interface can be described by

F(r, t) = 0. 06)

Since it is an interface, the velocity of the plasma normal to this interface must be equal to the velocity of the interface normal to itself, or

$F=(+J.V)F=O.

In lowest order in e we find

where V is the E X B velocity.

(17)

Applying Faraday’s law, V X E = 0, to this interface we find that the tangential component of the electric field must be continuous across the interface or the normal component of the E X B drift is continuous,

V * n is continuous,

where n is the normal of F,

n =VF/lVF].

Now since in the core V = O(U,i) and in the outer region V = O(EUth,i) we find as a boundary condition in lowest order

V*n=O, forF=O. (18)

In the next section, where the linear stability of cylindrical equilibrium is investigated, we shall utilize the (free) boundary conditions (17) and (18).

Finally, we remark that for F.L.R. plasmas some linear stability analysis on cylindrical equilibria has already been done in the literature. If, for instance, the special case of uniform rotation is considered, the azimuthally propagating low-frequency waves (usually

called flute waves) become unstable when the E X B

drift is sufficiently large compared to the diamagnetic drift and that these waves become stable in the opposite case [7, 1,8] , The stabilizing effect is pro- vided by the so-called F.L.R. effects which stem by the way from the “collisionless” part of the viscosity contribution to the velocity. Now, since a M.H.D. plasma is relatively cold compared to a F.L.R. plasma, viscosity is unimportant and in M.H.D. these flute waves will be always unstable (at least to the order considered here). In the next section we shall illustrate this with a few special cases.

4. A differential equation for low-frequency waves

We shall examine the linear stability of the following equilibrium. Consider a cylindrical plasma in a uniform magnetic field in the axial direction. The electric field E. and the density gradient Vn, are in the radial direction, and so the velocity V,, is directed in the azimuthal direction. All equilibrium quantities are only r-dependent. In addition the flute approxi. mation is assumed.

256 P. JanssenlNon-ideal M.H.D. equations with application to the stability of a rotating plasma column

Linearizing eqs. (12) and (13) for ions and electrons, then taking Fourier transforms in t and $J, using Faraday’s law and Poisson’s equation, we obtain an equation for the azimuthal component of the perturbed

electric field E@,

1 d -- rdr

dn T+ro2 2 \r,=o,

I (1%

9 = E,@ + mwE), vo - Eo WE=----> r rB

m = the mode number,

T = (o + mc+j2t10r3S,

S=l+b2, b2 = eoB2Jmino.

The term b2 which presents charge separation is, as already remarked in the previous section, small for the dense plasma considered here. Note that eq. (19) is of the same form as the wave equation in the article of Rosenbluth and Simon [l] , the stabilizing term -dp/dr is, however, missing in S because of our differ-

ent ordering. Next we have to specify the boundary conditions.

One condition is the finiteness of J/ at r = 0. The other

boundary condition is obtained by linearizing the interfacial conditions (17) and (18) and we find that J/ must vanish at that interface, i.e. the outer region acts as though it were a wall for the oscillations in the core. Denoting the position of the interface by lb, the boundary conditions read

G(O) is finite and $(rb) = 0. (20)

Eq. (19) and boundary conditions (20) constitute then the eigenvalue problem for w. Note that for complex w the equilibrium is unstable since w* is an eigenvalue too.

The type of eigenvalue problem (in general a non- Sturmian problem) considered here is difficult to solve since the eigenvalue w appears in a complicated way in the eigenvalue equation (19). We therefore consider only two special cases. First, we investigate the special case of constant E X B drift frequency WE. The result-

ing differential equation is of the Sturm-Liouville

type,

inor $ I) + [ (1 - m2)n0r + r2h 21 $J =o> (21)

where h = w2/(w + mwE)2. Note that if m = 0 then w = 0 (neutrally stable). Next we consider the case Im I> 1. For the finite region r E [O,rb] it can be proven [9] that the spectrum A of the eigenvalue problem (20) (2 1) is discrete and has an upperbound ho if dn,/& < 0. Now it is easy to prove that ho < 0 if dno/dr < 0 everywhere, since multiplication of eq. (21) by $* and integration over the interval [O,rb] yields

(22)

using the boundary conditions (20). The integral on the left-hand side is negative if dn,/dr < 0, the other two integrals are positive definite? and hence h < 0, e.g. ho < 0. Since w as well as w* is an eigenvalue we see at once, using the expression for X [eq. (21)], that the equilibrium is unstable for perturbations (Iml 2 1) propagating in the azimuthal direction if dnddr < 0. In the opposite case, i.e. dno/& > 0, the equilibrium is linearly stable.

The instability found here can be compared with the Rayleigh-Taylor instability in hydrodynamics. There, perturbations on the interface of a heavy fluid resting on a lighter one in a gravitational field are found to be unstable, corresponding to our case dn,/dr < 0. Of course the opposite case, a fluid resting on a heavier one, is linearly stable. The centri- fugal force in our case plays the role of gravitation in hydrodynamics.

We have seen that for uniform rotation all modes, characterized by the mode number m, are unstable if

t The exceptional case that for m = 1 (d$/dr) = 0 (except at the boundaries) yields h = 0.

R Junssen/Non-ideai M.H.D. equations with application to the sfability of a rotating plasma column 251

dn,/dr < 0, except the m = 0 mode. However, in general the rotation is not uniform and therefore the second special case is investigated.

In the second special case we consider the equilibrium sketched in figs. 2 and 4. Density no and E X 3 frequency WE are constant except at the point r = rk, where a diSCOntinUity in these quantities appears. Again, the outer region acts like a wall for the core region. Kent et al. [lo] have considered a similar case for the F.L.R. equations but they did not investigate the influence of a finite wall.

In regions 1 and 2 where density no and E X B frequency tiE are constant the wave equation (19) becomes

-$r3ij,+(l -m2)r$=0.

Requiring that J, is finite for r + 0 we obtain for m > 0 the solutions

pArm-l and J/ =Br”-l -I- Cr-m-k (23)

for regions 1 and 2, respectively. Using the boundary condition $I(&,) = 0 we find

B/C = -(+J-2m. (24)

The boundary conditions across the interface at r = rk of regions 1 and 2 are

(i) continuity of the normal component of the E X B velocity; this means that I& is continuous, implying

A =B+oi2m-1; (25)

(ii) another boundary condition is obtained by integrating equation (19) across the interface at r = tk

nr3(w + mwd2 dJ/ I 2 + (wrk)2(n2 - n&f!ofJ = 0. &I

(26)

Inserting (23), (24) and (25) into (26) finally yields the following dispersion relation [ 111 :

o*+2~[cY10~(m- l)(l-6)

f 42w2 {(m + 1) +- S(m - I))]

fap$I.l -S)(m-1)

+*,w~??2{(m+1)+s(r?2-1)}=0,

where

(27)

and oi= ? (1 I- S)n, + (1 - S)rQ ’

i= 1,2.

The parameter 6 represents the inthrence of the fictitious wall (formed by the outer region) on the frequency of the modes and one can see at once that, since rk/rb < 1, this influence is considerable for low mode numbers m ; higher m-modes hardly notice the wall. The quadratic dispersion relation (27) can easily be solved for w and for the typical case n2 = fnl, w2 = 10~ and 6 = ($&)%, which shows some resemblance to the profiles in the considered experiment, we have plotted Re(w) and the relative growth rate 7 2 ]Im(w)/Re(w)] against mode number m in fig. 6. The dotted line in the plots of Re(w) and 7 gives frequency and relative growth rate if no wall is present. From the plot of Re(w) the limitations of the used model can be seen since form = 7 the frequency w * 50, , and this value is already of the order of the cyclotron frequency. We have also done the calcula- tions with F.L.R. effects included, but they are only noticeable for m > 10-l 5. In view of the above remark it is clear that this observation is useless since our model is not valid for these high mode numbers m.

TWO sources for instability are present, according to the dispersion relation (27), namely

(i) different ni values; if this effect is dominant we call the instability a Rayleigh-Taylor one; and

0.7 .

Y

03

.2 Reb-? in unit5

wt 1

I d 5

Fig. 6. Re(w) and relative growth rate 7 versus mode number m: n&2 1 = ), c+/wt = + and r&b = 4 4.

258 R Janssen/Non-ideal M.H.D. equations with application to the stability of a rotating plasma column

10 El InT

01 @3 0.5

Fig. 7. Magnetic field dependence of frequency of the m = 1 mode. Comparison of theory (-) and experiment (0).

(ii) different wi values; we call the instability a Kelvin-Helmholtz instability if this effect is dominant,

The m = 1 mode certainly is a Rayleigh-Taylor

instability since

Re(w) = -2~30~ and y = (28)

Higher m-modes have both a Rayleigh-Taylor and Kelvin-Helmholtz character, the latter becomes more important for increasing m.

From the y-curve one can see that the relative growth rates are high. We conclude that for realistic perturbations within a few periods we have to take into account non-linear terms in our wave equation which we have of course neglected in this linear treatment .

Finally, we discuss the agreement with the observed m = 1 mode in the rotating plasma column. To this end we have calculated the B-dependence of Re(w) (m = 1) from eq. (28); o2 and w2 are deter- mined from least-squares fits of the jump-profiles with the experimental results for various values of the magnetic field B. Fig. 7 gives theoretical as well as experimental B-dependence of the frequency of the m = 1 mode.

5. Conclusion

We have shown for two special cases that in M.H.D. all azimuthal propagating “flute” modes (except m = 0) are unstable in contrast with the F.L.R. ordering where a threshold for instability is present 17, 1,8] . The reason for this difference is that in M.H.D. the stabilizing effect due to the viscosity is

one order higher than the destabilizing effect due to the E X B drift via the inertia term of the ions.

Furthermore, the outer region of the plasma column is investigated resulting in a boundary condition for the flute modes in the inner region. The outer region acts as though it is a wall for the flute waves, the second special case illustrates the stabilizing effect of this fictitious wall for lower mode-numbers m. The found instabilities have a combined Rayleigl-Taylor and Kelving-Helmholtz character.

Although it is clear from the curve of the relative growth rate y that soon non-linear terms become important, the agreement between theory and experiment seems reasonable as has been illustrated by the second special case in the previous section. Also, the theoretical phase difference between fluctuating density and potential [which can be readily obtained from the linearized version of eq. (15a)] is in agreement with the experimental found phase difference [2].

Acknowledgements

The author is indebted to M. P. H. Weenink for his introduction into this branch of Physics and for useful suggestions concerning this subject and he thanks F. Boeschoten for providing the experimental data prior to publication.

References

[l] M. N. Rosenbluth and A. Simon, Phys. Fluids 8 (1965) 1300.

[2] F. Boeschoten et al., T. H.-Report 75-E-59, Eindhoven University of Technology, The Netherlands (1975).

[3] B. van der Sijde and P. A. W. Tielemans, Proceedings 10th ICPIG (1971), Oxford.

[4] A. Macmahon, Phys. Fluids 8 (1965) 1840. [S] S. I. Braginskii, Reviews of Plasma Physics, vol. I (1965)

(Consultants Bureau, New York-London) p. 205. [6] T. E. Stringer and G. Schmidt, Plasma Phys. 9 (1967) 53. [7] M. N. Rosenbluth, N. A. Krall and N. Rostoker, Nucl.

Fusion, supplement, part I (1962) 143. [8] T. D. Rognlien, J. Appl. Phys. 44 (1973) 3505. [ 91 Ph. M. Morse and H. Feshbach, Methods of Theoretical

Physics, Part I (McGraw-Hill, New York, 1953) p. 719. [lo] G. I. Kent, N. C. Jen and F. F. Chen, Phys. Fluids 12

(1969) 2140. [ 111 P. A. E. M. Janssen, Proceedings XIIIth ICPIG, Berlin,

(1977) p. 769.

non-ideal m.h.d. equations with application to the stability of a rotating plasma column

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