Electrostatic oscillations in inhomogeneous plasmas

Abstract

The problem of longitudinal electrostatic oscillations in inhomogeneous, one-dimensional plasmas is examined in several areas. First the question is considered regarding whether the Vlasov-Maxwell equations will allow a plasma system to evolve naturally toward a preassigned stationary, inhomogeneous state such as one of the Bernstein-Greene-Kruskal modes. For the particular case in which the stationary state is characterized by an energy distribution function which is a constant for all energies up to a given maximum and zero for energies beyond, an extremization principle is developed which indicates that the total system energy is a minimum for the plasma. This stationary state then is inaccessible in general for a plasma prepared initially in a less restrictive manner. Further, such an inhomogeneous, stationary state is necessarily stable, a fact which is demonstrated directly by treating this case through an integral equation method. This integral equation is shown to degenerate into a Schrödinger-type differential equation, which is formally similar to the differential equation obtained in the hydrodynamic approximation for Maxwellian velocity distribution functions.

Electrostatic oscillations in more general cases are treated through the Fourier-Hermite transformation of the coordinate and velocity variables. A new recursion technique utilizing an electronic computer is used to reduce the resulting algebraic equations to a homogeneous matrix equation whose column vector is proportional to the Fourier components of the electric field. The addition of a small collision term to the Vlasov equation facilitates the calculation of the landau damping in the inhomogeneous plasma and allows the determination of the effects of collisions on the growth rate of the two-stream instability. The method is capable of treating arbitrary wavelengths, but works best for small degrees of inhomogeneity.

Results of the work show that the usual practice in the long-wavelength regime of dropping the stationary electric field term from the Vlasov equation has the effect of overestimating the landau damping when applied to the intermediate wavelength range. The effect of inhomogeneity upon the growth rates of the two-stream instability is shown to be a relative increase in the growth rates for the smaller wavenumbers and a more pronounced relative decrease for the higher wavenumbers. Collisions are shown to depress the growth rate of this instability, in contrast to some other work reported in the literature.