On three-dimensional magnetosonic waves in an isothermal atmosphere with a horizontal magnetic field

Abstract

Magnetosonic–gravity waves in an isothermal non-dissipative atmosphere, with a uniform horizontal external magnetic field have been considered in the literature in two cases: (i) ‘one-dimensional’ magnetosonic–gravity waves, in the case of zero horizontal wavenumber and (ii) ‘two-dimensional’ magnetosonic–gravity waves, in which the horizontal wave vector lies in the plane of gravity and the external magnetic field. In the present paper, an extension of case (i) is considered that is distinct from case (ii). This case (iii) is that of magnetosonic–gravity waves with a horizontal wave vector orthogonal to the plane of gravity and the external magnetic field. Since the wave fields depend only on two spatial coordinates and time, the problem could be called ‘two-and-half’-dimensional. The three-dimensional magnetosonic–gravity wave propagates a magnetic field perturbation parallel to the external magnetic field, and velocity perturbations transverse to it. Elimination for the vertical velocity perturbation leads to a second-order wave equation, with four regular singularities. Three regular singularities specify (a) the wave fields at high altitude, where there are two cut-off frequencies involving the acoustic cut-off frequency; (b) the wave fields in the deep layers, where another two cut-off frequencies appear, involving both the acoustic and gravity cut-off frequencies; and (c) the transition between the two regimes, occurring across a critical layer, where one solution of the wave equation vanishes and the other has a logarithmic singularity in the amplitude and also a phase jump. The whole altitude range can be covered using the three pairs of solutions of the wave equation, obtained by expanding in Frobenius–Fuchs series about each regular singularity. The power series solutions are used to plot the wave fields, for several values of the three dimensionless parameters of the problem, namely the plasma $\beta$, frequency and wavenumber. It is shown that the presence of a horizontal wave vector transverse to the plane of gravity and the external magnetic field, can change the properties of the waves significantly: first, the two cut-off frequencies may cease to exist, in which case the full wave frequency spectrum can propagate; secondly, the critical layer occurs at different altitudes for different frequencies, allowing gradual absorption of the waves (e.g. in the solar transition region).