THE UN IV E R S I TY O F M I C H I G A N COLLEGE OF ENGINEERING Department of Electrical Engineering! Space Physics Research Laboratory Scientific Report TS-1 ESTABLISBMENT OF STABILITY BY COLLECTIVE INTERACTIONS IN A PLASMA WITH CO'LLISIONS Ernest' G h. X"Qn:hm',:.:. 4,.4...~. ORA Projects 046:153'a:d" -6106 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NO. NsG-525 WASHINGTON, D.C. and OFFICE OF THE SECRETARY OF DEFENSE CONTRACT NO. SD-91 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR March 1964

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The purpose of this report is to show that a double humped initially unstable plasma will spontaneously approach stability if the high energy hump is held fixed. The frequently used Vlasov-Poisson equations represent a self-consistent system of nonlinear time-reversible equations. Because of the latter property, however, it is obviously unrealistic to apply those equations to problems involving the approach to equilibrium. An irreversible kinetic equation for plasmas has recently been independently derived by Balescul and Guernsey2'3. Guernsey carried out a double expansion of the distribution function in terms of the strength of the electromagnetic interaction 4ne2 and the parameter (4we2/V), where (1/V) = n is the number density of particles. The latter parameter is a measure of the interparticle correlations. By equating terms of the same order in 4we2 while summing over all powers of (4e2 /V) he obtained a hierarchy of equations. In this way correlations of the particles to all orders are taken into account at each step in the approximation with

respect to 4re2. Guernsey treated extensively the first 2 order equation in 4ne and proved that it describes an irreversible approach of the distribution function to the Maxwell distribution. In addition he showed that the Vlasov equation is the zeroth order equation in this hierarchy. Thus, the Vlasov equation correctly takes into account interparticle correlations to all orders. However, neither the Guernsey equation nor therefore the Vlasov equation include the effects of close collisions. In fact the interaction term in the Guernsey equation diverges as the momentum transfer becomes large. The reason for this omission is twofold. First, the Coulomb force does not hold down to arbitrarily small distances and secondly electron-neutral collisions are not considered since Guernsey treats a fully ionized gas. For the purpose of investigating the effects of close collisions a phenomenological term is frequently introduced into the kinetic equation. Such an equation has recently been 4 applied to plasma oscillations by Platzman and Buchsbaum.

These authors use a velocity independent relaxation model which is a good approximation, for example, in describing effects due to momentum transfer collisions between electrons and neutral particles. In our preliminary calculations the double humped distribution has been applied to the theory developed by Platzman and Buchsbaum. Following these authors we consider a system described by the following equations: (1) af + v af _ eE af = _ f at ax m av c (2) aE = _ e f f'dv, where (3) f = fo + f'' (4) fo = fol + fo2 n 2 fol(v) exp (2 ) wo(2w) 2w' 2 n1 (v - vd) (6) fo2(V) 1/ exp [, W (2= ) 2w12 Wo2 = - T/m,

Vc = constant collision frequency of electrons 2 of energy w2 /2m, e = electronic charge, m = electronic mass, co = dielectric constant in vacuo, Vd = drift velocity of high energy electrons, wI = velocity spread of high energy electrons, K = Boltzmann's constant, T = electron temperature with nI << no and wo << vd. The state of the system is described by the double-humped distribution (4), (5), (6) plus a superposed disturbance f' which is considered to be a longitudinal plasma wave. It has been shown5 that the kinetic equation (1) satisfies an H-theorem. Each time an electron collides, it is thrown out of phase with the plasma wave, This means that a certain amount of ordered energy of the wave has been converted into random

kinetic energy. The rate of energy transfer is given by the equation fv v2 f'dv dd v2 f dv c dt / vol dwo (7) = 2 nowo dtEq. (7) is not an additional assumption but merely a consequence of the collisional term in Eq. (1). It is well known that a double-humped distribution of the form (4), (5), (6) may be unstable. In that case any small perturbation will excite growing waves in a certain frequency interval. The larger f' becomes, the faster the temperature rises. It is for this reason that the system has been assumed to be describable at all times by a distribution function of the form (3), (4), (5), (6). The energy needed for this heating is supplied by maintaining fo2 constant. Without an external source of energy the hump f would diffuse o2 We express this explicitly by requiring that (8) afo2 = o at

Using Eq's. (4), (5), (7), and (8) one obtains afo arOl 2- Wo dw =_ = wol dt at at w o dt vc v2 - 2 fol c 0 2 (9) =..... v f' dv. 2 Wo no If Eq's. (3) and (9) are substituted into Eq. (1) there results (10) af V af eE af K L V2 _ +.2 folv2f'dv] As the temperature rises, the secondary hump fo2 becomes gradually absorbed by the main distribution fol, and the plasma becomes stable, In order to calculate the temperature values for which the system makes the transition from the unstable to the stable regime we observe that the boundary can be approached from the stable region, Hence it suffices to use the linearized form of Eq. (10) as long as one is careful not to draw any conclusions applicable to the unstable regime, The second term on the right hand side of Eq. (10) is nonlinear.* Hence, the linearized kinetic equation is (11) af' 3f' eE afo at + V m 5-av Vcf *The second term on the right hand side of Eq. (10) is of the order of f v2f'dv, which is proportional to the average kinetic energy of the particles in the disturbance. However, the average kinetic energy associated with the wave must be of the same order as the average potential energy which is of the order E2, le of the second order in the perturbation.

For the simultaneous solution of Equations (2) and (11) the customary Ansatz f', E a exp [i(kx - wt)] is being used. Then one obtains the well known dispersion relations) (12) Wr2 = W 2 + k2<v2. (13) V d (13) 2 [~ Wr dk W k2 n, a V=U where m =r + iy W 2 2 mp = noe /Soms u = Wr/k. The boundary separating the stable and unstable regions is defined by the equation (14) y = 0. Substituting Eq's. (4), (5), (6), (12), and (13) into Eq. (14) and simplifying one obtains

exp 1 2w0 no u w13 exp L- 2w12 (2)1/2 vc 1 3Wo Wp u3 U + ()l/[~ where it has been assumed that w 2/U2 << 1. This means that the thermal velocity of the electrons is small compared with the phase velocity of the plasma wave. For fixed fo2 the quantities nl, Vd, and wl are constant. In addition Vc and no (and therefore wp) are considered constants. Eq. (15) can, therefore, be considered as the relation between the phase velocity and the thermal velocity for which the imaginary component of the frequency vanishes. Fig. 1 shows a typical graph of Eq. (15). The area under the curve is the unstable region, and the area above the curve is the stable region. Ts = m woS/K is the temperature above which the plasma is stable. In the case of an initially unstable plasma, f' will grow but so will the temperature according to Eq, (7), As soon as the temperature Tsis reached, the system relaxes

toward the distribution fo given by Eq's. (4), (5), and (6) 2 with wo = KT /m. Since in the neighborhood of the boundary between the stable and unstable regions the system is adequately described by the linearized Eq. (11), the value of Ts is independent of the term af l/at and therefore of the dissipation mechanism of Eq. (7). However, the physical significance of this mechanism does not lie in its influence on the stability temperature but rather in the fact that it is responsible for the irreversible rise in temperature of the system to T if its initial temperature lies below Ts. As an example, Eq. (15) has been applied to the results of a plasma beam experiment recently performed by Singh and Rowe7. Fig. 1 shows the curve of w vs. u obtained from Eq. (15). The experimentally measured temperature7 was 28500~K which corresponds to a value of w of about 6.6 x 105 m/sec, 0 while the electron temperature corresponding to the calculated thermal velocity Wos = 7.9 x 105 m/sec is about 400000K. The main reason for this discrepancy is that in Eq. (7) it has been

assumed that all of the plasma wave energy dissipated is converted into internal energy, while in the experiJ-ient of Singh and Rowe heat is being lost to the outside. In summary the essential conclusions are as follows: Our equations describe a system in which part of the kinetic energy of the high energy particles is converted into the electric energy of the plasma waves of the main plasma by means of collective interactions. The electric energy in turn is dissipated into random internal energy through close collisions. As the plasma temperature increases, the growth parameter y decreases until it approaches zero and stability is achieved. Hence, both the collective interactions due to the long range Coulomb force and also close collisions play a significant role in the behavior of the plasma. In cases where a velocity dependent collision frequency is appropriate, the correct expression has to be substituted into Eq. (7). The temperature will increase again. In such a case the dispersion relation would, of course, be different 10

from Eq's. (12), (13). Hence, the stability temperature depends on the functional form of the collision frequency. But the essential conclusions are not affected. Finally, it is of interest to point out that the relaxation time for the heating of the main plasma up to the temperature T is determined by the collision frequency of the thermal electrons rather than that of the high energy electrons. AC KNOWLEDGEMENTS It is a pleasure to acknowledge some stimulating discussions with Professor A. F. Nagy and the cooperation of Professor J. E. Rowe in providing me with his experimental data before publication. This work was supported in part by the National Aeronautics and Space Administration under contract No. NsG 525 and in part by the Advanced Research Projects Agency under contract INo. SD-91. 11

References 1 R. Balescu, Phys. Fluids 3, 52 (1960) 2 R. L. Guernsey, Thesis, University of Michigan 1960 3 T. Y. Wu and R. L. Rosenberg, Can. J. Phys. 40, 463 (1962) 4 P M. Platzman and S. J. Buchsbaum, Phys Fluids 4, 1288 (1961) [There are two typographical errors in Eq. (6b) of reference 4. The factor I/no is missing in the first term and the factor 1/2 in the second term. Both have been corrected in Eq. (13) of the present paper.] 5 H. Grad, Handbuch der Physik (S. Flugge, ed.), Vol. 12, 205, Springer 1958, Berlin W. E. Drummond and D. Pines, Nuclear Fusion - 1962 Supplement, (Conference Proceedings Salzburg, 4-9 September 1961), p. 1049 7 A. Singh and J. E. Rowe (to be published) 12

9.0 Stable WoS 80_ _ 7. O I<~ ~~~~~ p / Unstable 6~5.0 In 4.0 3.0 2.0 2.5 3.0 3.5 u (106 m/sec) Fig. 1 Plasma stability as a function of the thermal velocity for a double humped distribution with the following parameters8: n = 5.0 x 1018-3 n 2.6 x 1016 d = 35 106m/sec, =, =2,Vd3.5 x 1016 m8/sec, 1 = 9 x 105 m/sec, vc 2.08 x 108 sec-1 13

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