stabilization of a current-carrying plasma by wave-particle interaction

2
VOLUME 14, NUMBER 18 PHYSICAL REVIEW LETTERS 3 MAY 1965 sition temperature is approached. Future improvements in thermometry and temperature control should allow refinements of about an order of magnitude in this experi- ment. Eventually one is limited by the slope of the A line which will produce a variation of T^ over the dimensions of the container due to the hydrostatic head of liquid helium. The author wishes to acknowledge discussions with M. Stephen and C. T. Lane, and to thank James Clow for technical assistance. The electron-velocity distribution in an elec- tron-proton plasma subjected to a uniform stat- ic field E 0 was calculated by Spitzer and Harm (SH), 1 using the linearized Fokker-Planck equa- tion in the Landau form, i.e., with static De- bye shielding. Linearization of this equation is valid when E Q is much less than the runaway field E run = n e ea e (a^ji)'' 1 . Here asH is the linear conductivity obtained by SH, and a e is the electron thermal speed. Analysis of the stability of the current-car- rying plasma, using the SH distribution, indi- cates that if the temperature ratio 9 e /6i is large, instability with respect to longitudinal ion waves occurs when E 0 exceeds a critical value £ cr it, which is small compared to £ run . 2 As the field E 0 approaches the critical value, the fluctuation energy of those waves approach- ing instability increases. 3 This results from their excitation rate (due to spontaneous emis- sion by particles) remaining approximately constant while their Landau damping (the net effect of absorption and induced emission by particles) approaches zero. There is thus an increase in the effect these electric-field fluc- tuations have upon the motion of electrons. This can also be described as an enhancement of the electron interaction due to wave exchange, i.e., due to wave emission and absorption. *Work supported by the Army Research Office (Dur- ham) and the National Science Foundation. l J 0 D 0 Reppy and D. A D Depatie, Phys. Rev. Letters 12, 187 (1964). 2 J. R 0 Clow, D. A. Depatie, J. C. Weaver, and J. D. Reppy, Proceedings of the Ninth International Confer- ence on Low Temperature Physics, Columbus, Ohio, 1964 (to be published). 3 J. B. Mehl and W. Zimmermann, Jr., Bull Am. Phys. Soc. 10, 30 (1965). 4 J. G. Dash and R 0 Dean Taylor, Phys. Rev. 105, 7 (1957). When the effect of wave-exchange interaction becomes significant relative to the statically shielded Coulomb interaction, the appropriate form of the Fokker-Planck equation is that of Lenard and Balescu (LB). 4 The LB equation incorporates wave exchange as a feature of dynamic shielding. We expect that as E 0 approaches E cr ^, the enhanced electron-electron interaction produces a change in the velocity distribution in such a direction as to make the plasma more stable, i.e., so as to increase the damping rates of those waves about to become unstable. As a result, E 0 can be increased beyond £ C rit> with- out instability occurring. This expectation has been confirmed by an explicit numerical solution of the LB equation. For several values of 6 e /8j, the velocity distribution has been determined as a function of E 0 . For E 0 «E cr it, the distribution is nearly identical to that of SH. But as E 0 approaches £ C rit> tne enhanced wave exchange causes the electron distribution to become more isotropic in the ion frame, thereby preserving the plasma stability. In fact, as E 0 increases further, the distribution keeps adjusting itself so as to remain stable. This effect is illustrated in Fig. 1, which shows the damping rate of that ion wave which would be the first to go unstable on the basis STABILIZATION OF A CURRENT-CARRYING PLASMA BY WAVE-PARTICLE INTERACTION* Gary A. Pearsont Lawrence Radiation Laboratory, University of California, Berkeley, California and Allan N. Kaufman Department of Physics, University of California, Los Angeles, California (Received 1 April 1965) 735

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Page 1: Stabilization of a Current-Carrying Plasma by Wave-Particle Interaction

V O L U M E 14, N U M B E R 18 PHYSICAL REVIEW LETTERS 3 M A Y 1965

sition temperature is approached. Future improvements in thermometry and

temperature control should allow refinements of about an order of magnitude in this experi­ment. Eventually one is limited by the slope of the A line which will produce a variation of T^ over the dimensions of the container due to the hydrostatic head of liquid helium.

The author wishes to acknowledge discussions with M. Stephen and C. T. Lane, and to thank James Clow for technical assistance.

The electron-velocity distribution in an elec­tron-proton plasma subjected to a uniform stat­ic field E0 was calculated by Spitzer and Harm (SH),1 using the linearized Fokker-Planck equa­tion in the Landau form, i.e., with static De-bye shielding. Linearization of this equation is valid when EQ is much less than the runaway field Erun = neeae(a^ji)''1. Here asH i s t h e

linear conductivity obtained by SH, and ae is the electron thermal speed.

Analysis of the stability of the current-car­rying plasma, using the SH distribution, indi­cates that if the temperature ratio 9e/6i is large, instability with respect to longitudinal ion waves occurs when E0 exceeds a critical value £ c r i t , which is small compared to £ r u n . 2

As the field E0 approaches the critical value, the fluctuation energy of those waves approach­ing instability increases.3 This results from their excitation rate (due to spontaneous emis­sion by particles) remaining approximately constant while their Landau damping (the net effect of absorption and induced emission by particles) approaches zero. There is thus an increase in the effect these electric-field fluc­tuations have upon the motion of electrons. This can also be described as an enhancement of the electron interaction due to wave exchange, i.e., due to wave emission and absorption.

*Work supported by the Army R e s e a r c h Office (Dur­ham) and the National Science Foundation.

l J 0 D0 Reppy and D. AD Depatie, P h y s . Rev. Le t t e r s 12, 187 (1964).

2 J . R0 Clow, D. A. Depatie, J . C. Weaver , and J . D. Reppy, Proceedings of the Ninth Internat ional Confer­ence on Low T e m p e r a t u r e P h y s i c s , Columbus, Ohio, 1964 (to be published).

3 J . B. Mehl and W. Z i m m e r m a n n , J r . , Bull Am. P h y s . Soc. 10, 30 (1965).

4 J . G. Dash and R0 Dean Tay lor , Phys . Rev. 105, 7 (1957).

When the effect of wave-exchange interaction becomes significant relative to the statically shielded Coulomb interaction, the appropriate form of the Fokker-Planck equation is that of Lenard and Balescu (LB).4 The LB equation incorporates wave exchange as a feature of dynamic shielding.

We expect that as E0 approaches Ecr^, the enhanced electron-electron interaction produces a change in the velocity distribution in such a direction as to make the plasma more stable, i.e., so as to increase the damping rates of those waves about to become unstable. As a result, E0 can be increased beyond £Crit> with­out instability occurring. This expectation has been confirmed by an explicit numerical solution of the LB equation. For several values of 6e/8j, the velocity distribution has been determined as a function of E0. For E0«Ecrit, the distribution is nearly identical to that of SH. But as E0 approaches £Crit> t n e enhanced wave exchange causes the electron distribution to become more isotropic in the ion frame, thereby preserving the plasma stability. In fact, as E0 increases further, the distribution keeps adjusting itself so as to remain stable.

This effect is illustrated in Fig. 1, which shows the damping rate of that ion wave which would be the first to go unstable on the basis

STABILIZATION OF A CURRENT-CARRYING PLASMA BY WAVE-PARTICLE INTERACTION*

Gary A. Pearsont

Lawrence Radiation Labora tory , Universi ty of California, Berkeley , California

and

Allan N. Kaufman

Depar tment of Phys i c s , Universi ty of California, Los Angeles , California (Received 1 April 1965)

735

Page 2: Stabilization of a Current-Carrying Plasma by Wave-Particle Interaction

V O L U M E 14, N U M B E R 18 P H Y S I C A L R E V I E W L E T T E R S 3 MAY 1965

FIG. 1. Landau damping r a t e y(E0) of that ion wave which would be the f i rs t to go unstable at £ c r i t if the SH dis t r ibut ion w e r e used. The dashed curve r e p r e ­sen ts the SH r e s u l t . y0 = y ( £ 0 = O).

of the SH distribution. The dashed curve is the SH result, while the solid curves are the LB results. It is seen that the wave never be­gins to grow, for all values of E0 considered. (Further increase of E0 leads to prohibitive numerical computing time.) Whether instabil­ity would ever occur on the basis of the LB equation is an academic question, since with

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at ^cr it/^run-increasing fluctuation energy and decreasing damping rate, the LB equation eventually be­comes invalid, for it does not include the ef­fects that wave-wave interactions (mode cou­pling) and collisions have on the waves.

In Fig. 2, the fluctuation energy is illustra­ted as a function of wave speed and direction relative to E0 for a value of EQ well beyond £cr i t (0 = n is the direction of the electron flow). These enhanced fluctuations could presumably be observed by scattering of electromagnetic waves.5

The decreased anisotropy caused by wave exchange reduces the electrical conductivity, o=j/E0. This is shown in Fig. 3.

A detailed account of this work is in prepa­ration. It will discuss the approximations we made to enable us to solve the LB equation numerically, and will also present the results in more detail.

Phys . Fluids 4, 1037

*This work done under the auspices of the U. S. Atomic Energy Commiss ion .

f This work was supported by a National Science Foundation Graduate Fel lowship.

1 L. Spitzer and R. H a r m , Phys . Rev. 89, 977 (1953). 2I . Berns te in and R. Kulsrud,

(1961). 3See, e.g., N. Ros toker , Nucl. Fusion 1, 101(1961). 4A. Lenard, Ann. Phys . (N.Y.) 10., 390 (1960); R. Ba-

lescu , Phys . Fluids 3_, 52 (1960); R. Guernsey , t h e s i s , Universi ty of Michigan, 1960 (unpublished); and J . Hub­bard , P r o c . Roy. Soc. (London) A260, 114 (1961).

5M. N. Rosenbluth and N. Ros toker , Phys . Fluids £, 776 (1962).

736