Wave action and critical surfaces for hydromagnetic-inertial-gravity waves

The propagation properties of hydromagnetic-inertial-gravity waves riding a basic state which varies slowly in two independent coordinates are examined in the Boussinesq approximation. The amplitudes of the waves are governed by an equation representing conservation of wave action. A study of the dispersion relation shows that the existence of critical surfaces (i.e. the analogue of critical levels in two-dimensions) is governed by nonlinear partial differential equations for the phase function of the waves. Although a solution of these equations is not readily obtainable, the geometric representation of the dispersion relation indicates the existence of critical surfaces for certain types of basic state. These are composed of magnetic field lines and, in contrast to the non-magnetic case, they are associated with energy propagation.