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On waves in a thin shell of stratified rotating fluid

The periods of free oscillation of an incompressible inviscid fluid in a spherical annulus rotating as if rigid with angular velocity, Ω, have been studied by Haurwitz (1940) and extended by Stewartson & Rickard (1969). In this paper we discuss the effect of the addition of a slight radial temperature gradient, β. It is found that when N2 ≪ Ω2 where N is the Brunt—Väisälä frequency the governing equation is hyperbolic everywhere but for N2 ≫ Ω2 wave motions are confined between the critical latitudes. Buoyancy forces do not affect the first term in a formal expansion in powers of ε = (a - b)/(a + b) but they cause a singularity in the second term at the critical latitudes. Closer examination of these singularities leads to the forcing of a rapidly oscillating inertial wave when N2 ≪ Ω2 which is singular on the characteristics. For N2≫Ω2 a rapidly oscillating internal wave is forced which is regular everywhere. These waves are then continued all round the shell and periods of free oscillations of this new kind are found. The results are discussed with reference to the Earth’s atmosphere, oceans and liquid core.