The critical level for hydromagnetic waves in a rotating fluid

Abstract

The propagation of plane hydromagnetic waves in a fluid rotating with angular velocity ω and permeated by a magnetic field B = {Bx(z), By(z), 0} varying in both magnitude and direction with z is studied by techniques recently applied to the propagation of internal gravity waves in a shear flow (Bretherton 1966; Booker & Bretherton 1967). Particular attention is paid to a class of 'slow’ hydromagnetic waves of interest in connexion with the dynamics of the earth's liquid core. While, in general, rotation permits propagation across the lines of force, there is associated with each wave a ‘critical level’ z = zc, which acts as a valve by effectively permitting the wave to penetrate it from one side only. A slow hydromagnetic wave with frequency ω and wavenumber components k,l normal to the magnetic field gradient can only effectively penetrate its critical level if its propagation speed across field lines W is such that Wωz,(ωx,k+ ωy,l)ω < 0. The phenomenon of ‘critical-layer absorption’ evidently does not in general require the presence of a mean shear flow; a non-uniform magnetic field gives rise to similar effects provided that some other restoring mechanism (in this case the Coriolis force) is available to permit hydromagnetic waves to propagate across field lines.