On the Propagation and Stability of Wave Motions in Rapidly Rotating Spherical Shells:I. The Non-Magnetic Case

The linear propagation properties of wave motions in a rapidly rotating stratified Boussinesq spherical shell, of outer radius 1 and inner radius 0. are studied in the small Prandtl number limit. When n=O, the various possible motions can be accommodated in two classes: F and D. The class F is closely related to the class of free oscillations of the inviscid unstratified fluid shell while D corresponds to diffusive inertial-gravity waves. Both F and D are subdivided into two infinite sets of modes one of which (E) is symmetric and the other (0) is anti-symmetric about the equatorial plane. The sets E and 0 for class F are further subdivided into two infinite subsets one of which propagates (in phase) eastward and the other westward. Waves of class D propagate eastward. The two classes F and D are decoupled except for one mode which belongs to D on and in the immediate neighbourhood of the axis of rotation and transforms into F away from the axis. This mode provided the already known mode of convection of the linear stability of a fluid sphere containing a uniform distribution of heat sources. When an inner solid core is present (and q is nonzero) all the modes of classes F and D persist outside C, (where C, is the coaxial cylinder whose generators touch the inner core at its equator) but the set E of modes (of both F and D) is suppressed within C, The stability of the elements of F and D is examined for various forms of the temperature gradient, 8, providing the stratification. If the shell is unstably stratified everywhere then every wave can be destabilized if is large enough. If, however, the shell is composed of an inner region of unstably stratified fluid surrounded by a stably stratified region then only a finite number of modes of F can be made unstable whatever the value of j. The critical mode of convection almost always belongs to the same mode but the location of the preferred cell of convection depends on b. For example, in the case of a differentially heated shell it is situated on C, for all values of n.