Study of magnetogravity waves

We study the geometry of magnetogravity wave surfaces in an incompressible, isothermal, inviscid, nonrotating, flat, horizontally stratified perfectly conducting atmosphere. Our ultimate goal is the verification of the contention by Lighthill that such waves may contribute heavily to the heating of the upper solar atmosphere. The equations of the wave surface are determined from those of the wave normal surface using the envelope construction. For a vertical magnetic field, we find that for frequencies greater than the Vaisala-Brunt frequency N, the wave surfaces are arrow shaped, having both a point cusp and a circular cuspidal edge. The locations of these cusps are determined using the polar reciprocal relationship between the slowness (reciprocal velocity) and wave surfaces. We find that for high frequencies, these surfaces subtend small angles with respect to the origin, strongly suggesting the ability of these waves to carry energy upward. For a wave frequency equal to the Vaisala-Brunt, the surface is spherical, while for frequencies lower than the Vaisala-Brunt, the surfaces become infinite in extent. For the case of a horizontal magnetic field, we again determine the location of cuspidal edges. For oblique fields, we again find a line of self-intersection, provided the frequency ω is related to the angle zeta of the magnetic field with the vertical by ω2 > N2cos2zeta. The equation of these lines is derived: Z = cot zeta X, showing that the line is always directed along the magnetic field. The existence of the line of self-intersection suggests that magnetogravity waves, at least those of high enough frequency, can carry energy upward along the solar magnetic field, thus lending support to Lighthill's conjecture