In this paper we consider the propagation of hydromagnetic waves at altitudes below about 2000 km. This work is an extension of a previous paper in which heights below about 500 km were considered. We treat the case of plane wave propagation in the vertical direction, and assume the geomagnetic field to be vertical (polar propagation). Currently used models indicate that the ionosphere (the height region between about 80 and 500 km) may be reasonably represented by a constant Alfv•n speed and locally exponential ion-neutral collision frequency, while the lower exosphere (the region between about 500 and 2000 km) can be adequately described by an Alfv•n speed which increases exponentially with height. By using these approximate forms, we can express the solutions of the relevant forms of the electromagnetic wave equation in terms of known functions. Analytic expressions for the magnetic transmission and reflection coefi%ients are derived and analyzed, and numerical results are obtained. These results, although strictly applicable only to high magnetic latitudes, compare favorably with many geographically widespread experimental data. Of particular interest is the prediction of a prominent double transmission resonance in the daytime and a single strong resonance at night. This agrees with the measurements of various workers. Many lesser resonances are also found. The advantage of the analytic representation is the ease of interpretation of the physical results. For example, simple expressions describing the transmission resonances found by Jacobs and Watanabe for a grounded exosphere are found to be a limiting case of the equations derived here. Introduction. One of the most interesting characteristics of both natural and artificially produced geomagnetic micropulsations is the existence of narrow bands of oscillations [Maple, 1959; Tepley, 1961], or equivalently, the existence of resonant peaks in their observed energy spectrums [Ness et al., 1962; Davidson, 1964; Santirocco and Parker, 1963; Smith et al., 1961]. The micropulsation energy spectrums depend, of course, on the nature of the sources and the transmission properties of the ionized layers which surround the earth. The observed resonant frequencies are presumably charac•ristic of the transmission mediums and, in the absence of a detailed knowledgeof the sources, provide fruitful grounds for theoretical study. The transmission of.•hydromagnetic waves through regions of the ionosphere and ex(•sphere has been studied by various workers. (In what follows the term ionosphere means the altitude region between about 80 km and 500 km.)

The response of a homogeneous plasma to the sudden application of a strong, uniform electric field, E0, is studied by solving, self-consistently, the coupled equations for the one- and two-particle distribution functions. The field is assumed to be large compared to that which produces a ``runaway'' current, so that particle-particle collisions have a negligible effect on the current flow, and particle-wave interactions dominate. The principal approximations are the neglect of intrinsic three-particle correlations (i.e., linearization in fluctuations) and the use of an adiabatic ansatz for the time dependence of the fluctuations. For the case of unequal temperatures, Te ≫ Ti, two different methods of solution are employed: a moment approximation, in which the distribution functions are assumed Maxwellian, with mean velocity and width determined self-consistently; and a more exact treatment in which the distribution function for velocities parallel to the applied field is determined by direct numerical solution of the kinetic equation using on-line computational techniques. In both cases, it is found that the unstable ion acoustic waves grow from the thermal fluctuation level to a magnitude sufficient to cause a sharp and sudden decrease in the current. Examination of the kinetic equation shows this drop to be ascribable to the dynamic friction associated with the unstable waves. Although waves propagating parallel to the applied field are most important at early times, those propagating at finite angles soon become dominant; this casts some doubt on one-dimensional approximations to this problem. The limitation of the exponential growth of unstable waves resulting from the non-linear (or quasilinear) effects is clearly evident, notwithstanding the presence of the external driving force, E0.

This paper analyzes the propagation of the transverse electromagnetic (TEM) ELF mode when the earth-ionosphere waveguide is not stratified. It treats a localized disturbance by recasting the wave equation as a two-dimensional integral equation. Numerical solutions show that such a disturbance behaves like a cylindrical lens filling a narrow aperture. Lateral diffraction, focusing, and reflection can cause the TEM mode to exhibit a standing wave pattern before the disturbance, and a transverse pattern of maxima and minima beyond it. Such phenomena can contribute to the spatial fluctuations occasionally observed in ELF transmissions. The focusing and diffraction diminish when the transverse dimension of the disturbance approaches the width of the first Fresnel zone•typically, several megameters. The analysis models exceedingly widespread inhomogeneities, such as a disturbed polar cap or the day/night hemispheres, as semi-infinite regions having diffuse boundaries. It then derives full-wave analytic expressions for the lateral reflection and transmission coefficients of the TEM mode. Reflection can be important in two situations: first, when a great-circle propagation path is nearly tangential to the boundary of the disturbed polar cap and second, when the TEM mode is obliquely incident on the day/night terminator, in which case a phenomenon analogous to internal reflection can occur. INTRODUCTION Even such large inhomogeneities as sporadic-E patches, the polar cap boundary, and the day/night terminator can cause the properties of the earthionosphere waveguide to change markedly over the huge wavelength or Fresnel zone of an extremely low frequency (ELF) signal. Such inhomogeneities can therefore give rise to lateral reflection, diffraction, and focusing of ELF modes. Those phenomena are usually unimportant at higher frequencies where the waveguide can be assumed to be slowly varying in the lateral directions. The integral equation approach, which was first introduced by Wait [1964] to analyze weak VLF scattering, is one means of de•,scribing propagation when the ionosphere is not stratified. Field and Joiner [1979] solved the integral equation for ELF propagation in the presence of ionospheric disturbances that were (1) weak and localized or (2) of arbitrary strength and extent, but azimuthally sym-

This study presents a theoretical analysis of the propagation of electromagnetic waves in the frequency range 30–1000 Hz in the earth‐ionosphere waveguide. Full wave methods incorporating the vertical inhomogeneity of the ionosphere are used in conjunction with model ionospheres corresponding to ambient and disturbed (PCA) conditions. For ambient conditions, the character of the assumed ion‐density profile below 60 km substantially affects the computed attenuation rates, phase velocities, Joule heating rates, and field strengths for waves at frequencies lower than a few hundred Hz. For example, at a wave frequency of 50 Hz, 30% to 50% of the attenuation is due to losses associated with ion heating. At frequencies of a few hundred Hz or higher the losses, under ambient conditions, are related almost entirely to electron heating. Under moderate PCA conditions the losses are due mainly to ion‐heating for the entire ELF frequency range, and the attenuation rates are substantially higher than for undisturbed situations. The ionospheric height ranges which influence the ELF mode structure are quite broad being typically many tens of kilometers in thickness. The influence of the geomagnetic field upon the mode characteristics is substantial in the undisturbed nighttime, slight in the undisturbed daytime, and essentially nil under PCA conditions.

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