On vertical hydromagnetic waves in a compressible atmosphere under an oblique magnetic field

Abstract

Abstract We consider vertical magneto-acoustic-gravity waves in an atmosphere under a constant magnetic field, and show that (Figure 1), in addition to the Alfvén wave, there exists a hydromagnetic-gravity wave, which is of fourth-order in the general case of an oblique magnetic field, and reduces to a second-order acoustic/magnetosonicgravity wave respectively for a vertical/horizontal magnetic field. A method is developed to solve exactly linear wave equations with certain types of variable (e.g., exponential) coefficients, describing waves of any order and arbitrary frequency in strongly stratified media. The method relies on the transformation of the wave equation into a standard type [equation (11a)], which on inspection is seen to have general properties such as: (i) cut-off frequencies and related wave components (Figure 2, Table I); (ii) the asymptotic amplitude and phase laws holding at high-altitudes (Table II); (iii) the existence and location of critical levels (Figure 3, Table III). The method also yields exact, explicit solutions, in the form of rapidly converging series, for high-order functions of the first, second and third kinds, which can be interpreted as generalizations of the second-order Bessel, Neumann and Hankel functions. These functions specify the four particular integrals of the general hydromagnetic gravity wave equation, a pair for each of the dynamic and magnetic components (Table V), which are coupled beyond the leading order. Simplified formulae are obtained in the initial and asymptotic regimes, at low- and high-altitude respectively, demonstrating (Table IV): (a) the evolution of all wave variables, viz. horizontal and vertical displacements, mass density and magnetic field perturbations, and gas and magnetic pressures; (b) the equipartition or biasing of the kinetic, compressional, potential and magnetic energy densities. The exact solutions specify the eigen-frequencies and waveforms of standing modes (Figure 4), and the amplitude and phases of propagating waves (Figures 5, 6, 7), demonstrating the dependence upon wave frequency, plasma β and magnetic field inclination.