Theory of hydromagnetic propagation in the ionospheric waveguide

Abstract

In this paper, we consider the propagation of low-frequency (Pc 1) hydromagnetic waves in the ionospheric duct, which results from the minimum in the Alfven speed near the F2 ionization peak. We consider an inhomogeneous (in the vertical direction) waveguide in the presence of a uniform static magnetic field and treat the case of horizontal propagation in the plane of the magnetic meridian. Propagation in the waveguide is governed by two coupled equations for the horizontal components of the electric field. For nighttime conditions, where the ionized region of interest starts at sufficiently high altitude that the collision frequency of ions with neutrals is small compared with the ion cyclotron frequency, the two components become uncoupled, and only the isotropic, or fast wave, need be considered. For daytime conditions, however, coupling between the fast and slow waves must be taken into account; to simplify the analysis, the static field is assumed uniform in this case. To obtain analytic solution of the waveguide equations, it is necessary to have analytic approximations to the height dependence of the relevant ionospheric parameters; such approximations are constructed to fit some representative tabulated ionospheric profiles for a variety of conditions of time of day and sunspot activity. Solution of the waveguide equations subject to the appropriate boundary conditions results in a rather complicated dispersion relation between complex (horizontal) wave number and angular frequency. The solutions of the dispersion relation prescribe the allowed bands of propagation in the guide. It is shown that each band, including the lowest, has a low-frequency cutoff, and that the existence of a low-frequency cutoff is a consequence of the boundary conditions and not of attenuation, as assumed in some theories. Numerical solutions of the dispersion equation are obtained for the two lowest bands for each of the conditions considered. From the dependence of the real part of the wave number on frequency, the cutoff frequencies and the phase and group velocities are determined, while the imaginary part provides the attenuation length. For nighttime conditions, attenuation is not large, and propagation can take place over distances of thousands of km, whereas daytime propagation is restricted by attenuation to distances of the order of hundreds of km. For nighttime minimum conditions, the calculated waveguide cutoff is about 0.4 cps, and the group velocity for the lowest band is about 720 km/sec, both in reasonable agreement with experiment. Finally, it is pointed out that there is an effective high-frequency ‘cutoff’ (in any band) as far as ground-level signals are concerned due to an exponential decrease of transmission coefficient with frequency at high frequencies.