ON THE OSCILLATIONS OF A WEAKLY INHOMOGENEOUS PLASMA1

SOM KRISHAN Departilteltt of Physics a n d Astrophysics, U7~iversity of Delhi, Delhi-6, I ~ z d i e

Received September 21, 1964

ABSTRACT

Oscillations of a collisionless plasma in equilibrium with a magnetic field which is weakly inhomogeneous in one dimension are studied. The calculations are based on longitudinal oscillations, i.e. the electric vector is parallel to the wave vector. The procedure enlployed is to use Maxwell's equations and expand the velocity distribution function a b o ~ ~ t its equilibrium value for finding the per- turbation in the distribution. The dispersion relation so obtained is different from that of Rosenbluth et al. (1962). The system is stable for the small Landau growth rate (Landau 1946), which might become appreciable for \vavelengths comparable with the Larrnor radius, provided k 2: e, where k is the wave vector and e is a small inhomogeneity parameter.

1. INTRODUCTION

Recently some papers (Rosenbluth et al. 1962; Krall and Rosenbluth 1961, 1962, 1963) on oscillations of a collisionless plasma in a magnetic field ~vhich is weakly inhomogeneous in one dimension have appeared which predict soille new types of modes. They make use of the self-consistent solution of the Boltzmann equation and Maxwell's equations. In particular, they study wave propagation perpendicular to the external nlagnetic field. Their main conclusion is the breakdown of the hydrodynamic approximation for ha << 1, where k is the wave vector and a , is the ion Larmor radius. At wavelengths comparable with the ion Larmor radius for inoderate values of /3 (i.e. the ratio of particle pressure to magnetic pressure), the systeill is ap t to have a sizable growth rate.

Very recently Siliil (1964) has studied plasma oscillatioils in the presence of an external magnetic field, which is wealcly inhomogeneous in three dimensions, by geometric optics. In the present paper we have replaced the solution of the collisionless Boltznlann equation by expanding the velocity distribution func- tion about its equilibrium value, and then finding the perturbation in the distribution with the help of the equation of motion for the particle and RiIax- well's equations. The dispersion relation predicts tha t the systeill is stable in the limit w / w , << 1 and ( k ~ ) ~ << 1, except for the small Landau growth rate, which might become appreciable for wavelengths comparable with the Larmor radius.

2. EQUATIONS O F T H E PROBLEM

We shall study a two-component plasma consisting of electrons and ions. We shall write the equations of motion only for electrons, since the equations for ions follow from them directly. The equation of motion of an electron is then

lThis work was done under the auspices of the Council of Scientific and Industrial Research ( India).

Canadian Journal of Physics. Volume 43 (April, 1DG5)

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KRISHAN: OSCILLATIONS OF WEAKLY IXHOMOGENEOUS PLASMA 641

(2. 1) md2r/dt2 = - eE - (e/c)v X H + mg,

where g is the gravitational acceleration. The magnetic field H = Ho(1 + ex) is taken along the z axis and has a sinall gradient e along the x axis. T h e component forill of equation (1) beconles

I d2x/dt2 = - (e/m)E, - we(l + ex)v, + g,

We write

where Gr(Sx, 6y, 62) is the perturbation in r, we is the cyclotron frequency, vll is the velocity parallel t o the external magnetic field, and vD is the drift velocity, which has two parts: the gradient drift evL2/2we and the gravitational drift g/we. Lilie VIHI, g also points in the x direction. If D stands for d/dt, then we inay write equations (2.2) in the following form:

D26x + we(l + ex)D6y = -(e/m)E,, (2.4)

D26y - we(l + ex) = - (e/m)E,.

These two equations further yield the following:

(2.5) D 9 x + weW6x = we(vDeawe sin 9 + a b e 2 e sin 29) + (e/m)w,~,

+ a2ewe3 sin 9 cos q5 + 2awe3e sin 9 xo,

In writing the solutions of equations (2.5) and (2.6) we shall neglect those terills which vanish on integration over velocity space. Then we obtain

where

ie J, exp (in 4) m4m 6x = - w e E v C

m C Jm exp(-im9), non+n- m=-m

n=+m JTL exp (in 4) a y = f ~ , C C Jmexp(-im9), non+n- ,,L=-m

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642 CANADIAN JOURNAL O F PHYSICS. VOL. 43, l9G5

In deriving the above expressions use has been made of the identity

+m

exp(ike sin 4) = C J. (k) ein: -m we

Jn being the Bessel fuilction of the first kind. Further, since we are interested only in the longitudinal oscillations, and for k > E' the x dependence of the perturbation is ~vealc (Bernstein et al. 1958), we have set in equations (2.5) and (2.6)

E, = Eo,, exp[.i(ky - wt)].

3. DERIVATION OF T H E DISPERSION RELATION

The equilibrium distribution function f o , which satisfies Boltzmann's equa- tion, can be constructed from the constants of motion: x - v,/we and inzu2 - mgx. We choose a simple function,

The constant a is the inverse of the square of the thermal velocity, we = eH/mc, and the parameter E' is related to the magnetic-held gradient through the follow- ing equations:

Here j refers to the particle species and is to be summed over the values i and e for ions and electrons. I f f is the distribution function after the perturbation is switched on, then

Clearly then 6f is given by

Now using (2.7), (2.8), and (3.1), one can evaluate 6f. Thus

af 0 -.6v = - 2a(- vl sin 4fo)6vZ - 2a(v, cos 4 + vD)fo av

I t should be remembered that we are writing these expressions only for electrons, since the expressions for ions follow similarly. Next using the equa- tion V.E = 4ap, and integrating over the azimuthal angle, we obtain the dispersion relation

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KRISHAN: OSCILLATIONS OF WEAKLY INHOMOGENEOUS PLASMA 643

1 +" + I ne2 - x Sm 2c~,v,dv,e-~j~~ / 2ajw J,: ke' noj J ,~ m3 -, o 7 +--I. w j n+jn-,

Neglecting the Debye length in conlparison with the Larmor radius, the dis- persion relation reduces to

In deriving relation (3.6) we have made use of the identity J,,-1J,,+1 =

27zJ,,(z)/z and, since for k > E' the x dependence of the perturbation is weak, we have set x = 0 in expressions of the for111 e?". We have also assu~ned that the density decreases in the sense of gravity so that E' >> 2ug > 0.

I-Iereafter we shall be concerned with a special case, E'/E >> 1. W7e set E = 0 in the denominator of (3.6) except for the imaginary part which arises from the singularity of the integral; this term will yield a slow Landau damping or growth of our solutions. Working under the limit kal << 1, w / w l << 1, the dispersion relation can be reduced to a quadratic equation, the two roots of which are

I t call be seen easily that both the values of w are real. The negative value is untenable since it violates the approximation w/wl << 1. Thus the systenl is always stable.

Now we are ready to find the effect of the s~nall iinaginary part in the dispersion relation (3.6). Writing w = w l + i w ? , where w l is given by (3.7), and since w, > kg/we ive get the exponential dependence of the gro\vth rate as

Thus, a t wavelengths comparable wit11 the ion Larnlor radius the groivth rate may be considerable, provided k -v E.

This growth rate should also be compared with that obtained by Rosenbluth, who approaches the problem via the Boltzmann equation. The functional dependence of the growth rate in his case, although different, still implies that it might become appreciable for wavelengths coinparable with the ion Larmor radius.

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644 CAS.4DI.IX JOURNAL O F PHYSICS. VOL. 43, 1966

ACKNOWLEDGMENTS

The author is grateful to Professor F. C. Auluck, Dr. S. K. Trehan, and Dr. (Miss) B. Buti for their help and to Mr. Surindra Singh for reading the manuscript.

REFERENCES

BEI~NSTEIN, I. B., FKIE~IAX, E. A., I<UI<SKAL, M. D., and I<ULSURD, R. M. 1958. Proc. Roy. Soc. (London), Ser. A, 244, 17.

I<I<ALL, N. A. and ROSENBLUTH, M. N. 1961. Phys. Fluids, 4, 163. 1962. Phvs. Fluids. 5. 1435. -. .- . , - - -

LANDAU, L. 1946.' J. ~ h y s . U.S.S.R. 10, 25. ROSENBLUTH, M. N., KRALL, N. A., and ROSTOKER, N. 1962. Nucl. Fusion, Suppl. 1, 143. SILIN, V. P. 1964. Soviet Phys. JETP (English 'Transl.), 18, 33.

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