Linearized Plasma Oscillations in Arbitrary Electron Velocity Distributions

Abstract

This paper is a mathematical examination of the linearized small disturbances in the steady distribution f0(q) of the velocities q of the electrons in an electrostatic, collisionless plasma with motionless protons. It is assumed that g0(u)=∫∫f0(u,v,w)dvdw has an integrable derivative with respect to u for all axis orientations. An existence and uniqueness theorem for the initial value problem is given, and it is shown that no disturbance can grow faster than expωpt, where ωp is the electron plasma frequency. Consequently, one can base a stability theory on Laplace transforms with respect to time, as Landau has done. The limits of validity of Landau's stability criterion are explored: that g0(u) is stable if there are no wave numbers k for which L(s)=k2ωp−2−∫−∞∞g0′(u)(u−s)−1du has zeros in the upper complex s half-plane. To ensure instability, the zeros must have positive imaginary parts or a multiplicity of 2 or greater. To insure stability, the initial disturbance must be not only integrable, but square integrable with respect to u. The Maxwell distribution is unstable to certain integrable disturbances. All isotropic, three-dimensional distributions f0(q) = h(q2) for which x¼h(x) is absolutely continuous and square integrable, and h(x)+2xh′(x) is bounded, are stable to integrable, square integrable disturbances. This explains Van Kampen's ability to solve the initial value problem by superposing normal modes (solutions with complex, exponential time dependence) with real frequencies; he implicitly introduced stability by considering only isotropic distributions f0(q). His method is extended to unstable f0 as a technique independent of Landau's for solving the initial value problem. If f0 is unstable, the normal modes are not complete, and a normal mode analysis can lead to erroneous positive conclusions about stability. Finally, the linear theory predicts that in stable plasmas the neglected term will grow linearly with time at a rate proportional to the initial disturbance amplitude, destroying the validity of the linear theory, and vitiating positive conclusions about stability based on it. In a thermonuclear plasma with T = 108 °K and N = 1015 electrons/cm3, a disturbance of wavelength 1 cm and initial amplitude 1 v can no longer be treated by the linear theory after 220 μsec.