Alfvén-gravitational waves propagating in a Boussinesq, inviscid, adiabatic, perfectly conducting fluid in the presence of a uniform aligned magnetic field in which the mean horizontal velocityU(z)depends on heightzonly are considered. The governing wave equation has three singularities, at the Doppler-shifted frequencies Ωd= 0, ± ΩA, where ΩAis the Alfvén frequency. Hence the effect of the Lorentz force is to introduce two more critical levels, called hydromagnetic critical levels, in addition to the hydrodynamic critical level. To study the influence of magnetic field on the attenuation of waves two situations, one concerning waves far away from the critical levels (i.e. Ωd[Gt ] ΩA) and the other waves at moderate distances from the critical levels (i.e. Ωd> ΩA), are investigated. In the former case, if the hydrodynamic Richardson numberJHexceeds one quarter the waves are attenuated by a factor exp{−2π(JH−¼)½} as they pass through the hydromagnetic critical levels, at which Ωd= ± ΩA, and momentum is transferred to the mean flow there. Whereas in the case of waves at moderate distances from the critical levels the ratio of momentum fluxes on either side of the hydromagnetic critical levels differ by a factor exp {−2π(J−¼)½}, whereJ(> ¼) is the algebraic sum of hydrodynamic and hydromagnetic Richardson numbers. Thus the solutions to the hydromagnetic system approach asymptotically those of the hydrodynamic system sufficiently far on either side of the magnetic critical layers, though their behaviour in the vicinity of such levels is quite dissimilar. There is no attenuation and momentum transfer to the mean flow across the hydrodynamic critical level, at which Ωd= 0. The general theory is applied to a particular problem of flow over a sinusoidal corrugation. This is significant in considering the propagation of Alfvén-gravity waves, in the presence of a geomagnetic field, from troposphere to ionosphere.
Marginal Instability?