A MATHEMATICAL MODEL FOR ANALYSIS OF WAVE PROPAGATION IN A LINEARIZED VERTICALLY NONUNIFORM PARTIALLY IONIZED GAS

Abstract

Wave propagation in fully and partially ionized gases, with and without magnetic fields, has been treated by several workers; e.g., Tanenbaum and Mintzer (1962) obtained dispersion relations for a linearized and spatially uniform gas of electrons, positive ions, and neutrals. The present paper discusses the basic formulation and mathematical treatment of wave propagation in a linearized electron – ion – neutral gas, with static magnetic field, in which ambient-gas parameters vary arbitrarily vertically and are uniform horizontally.A standard formulation of the general problem is discussed via Boltzmann and Maxwell equations. By momentum-space averaging, the Boltzmann equation yields motion, continuity, and dynamic adiabatic state equations. These are combined to yield neutral and plasma equations of motion, continuity, and adiabatic state and a generalized Ohm's Law. Steady-state plane-wave solutions of the form exp[−i(ωt – kxx)] are assumed, reducing the x, y, and t dependence to algebraic relations, but the equations remain differential in z. The system consists of 10 simultaneous coupled ordinary first-order complex differential equations and 11 simultaneous complex algebraic equations in 21 complex unknowns.The second part of the paper is a discussion of the solution of this coupled algebraic differential equation system, equivalent to the system arising in the analysis of coupled linear electrical networks. Referring to the literature of differential equations and modern automatic control systems, various purely analytical approaches are discussed with emphasis on their deficiencies in obtaining practical numerical results with an arbitrary z variation. The Runge–Kutta step-by-step-procedure was invoked eventually and a Fortran program based on this technique was written. The program can be used to obtain accurate numerical solutions to many problems involving wave propagation in a linearized, vertically nonuniform electron – ion – neutral gas without requiring drastic simplifying assumptions for the vertical nonuniformity. This program can be used, by changing input parameter values, to treat such diverse problems as the perturbing effect of acoustic–gravity waves on ionospheric electron density, electromagnetic wave propagation in the vertically inhomogeneous ionosphere, MHD waves high in the ionosphere, or various kinds of wave propagation in prepared plasmas with a one-dimensional inhomogeneity. Numerical solutions for the acoustic – gravity wave – plasma interaction problem and their interpretation will be reported in a later paper.