Effect of ohmic dissipation on internal Alfvén-gravity waves in a conducting shear flow

Abstract

Internal Alfvén-gravity waves of small amplitude propagating in a Boussinesq, inviscid, adiabatic, finitely conducting fluid in the presence of a uniform transverse magnetic field in which the mean horizontal velocity U(z) depends on height z only are considered. We find that the governing wave equation is singular only at the Doppler-shifted frequency Ωd = 0 and not at the magnetic singularities Ωd= ± Ω λ, where Ω λ is the Alfvén frequency. Hence the effect of ohmic dissipation is to prevent the resulting wave equation from having magnetic singularities. Asymptotic solutions of the wave equation, which is a fourth-order differential equation, are obtained. They show the presence of the magnetic Stokes points Ωd = ± Ω A. The interpretation of upward and downward propagation of waves is also discussed. To study the combined effect of electrical conductivity and the magnetic field on waves at the critical level, we have used the group-velocity approach and found that the waves are transmitted across the magnetic Stokes points but are completely absorbed at the hydrodynamic critical level Ωd = 0. The general expression for the momentum flux is mathematically complicated but will be simplified under the assumption \[ \frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial y^2}\gg \frac{\partial^2h}{\partial z^2}, \] where h is the perturbation magnetic field. In this approximation we find that the momentum flux is not conserved and the waves are completely absorbed at Ωd = 0. The general theory is applied to a particular problem of flow over a sinusoidal corrugation and asymptotic solutions are obtained by applying the Laplace transformation and using the method of steepest descent.