The onset of thermal convection in a rapidly rotating sphere

The linear stability of convection in a rapidly rotating sphere studied her e builds on well established relationships between local and global theorie s appropriate to the small Ekman number limit. Soward (1977) showed that a disturbance marginal on local theory necessarily decays with time due to th e process of phase mixing (where the spatial gradient of the frequency is n on-zero). By implication, the local critical Rayleigh number is smaller tha n the true global value by an O(1) amount. The complementary view that the local marginal mode cannot be embedded in a consistent spatial WKBJ solutio n was expressed by Yano (1992). He explained that the criterion for the ons et of global instability is found by extending the solution onto the comple x s-plane, where s is the distance from the rotation axis, and locating the double turning point at which phase mixing occurs. He implemented the glob al criterion on a related two-parameter family of models, which includes th e spherical convection problem for particular O(1) values of his parameters . Since he used one of them as the basis of a small-parameter expansion, hi s results are necessarily approximate for our problem. Here the asymptotic theory for the sphere is developed along lines parallel to Yano and hinges on the construction of a dispersion relation. Whereas Y ano's relation is algebraic as a consequence of his approximations, ours is given by the solution of a second-order ODE, in which the axial coordinate z is the independent variable. Our main goal is the determination of the l eading-order value of the critical Rayleigh number together with its first- order correction for various values of the Prandtl number. Numerical solutions of the relevant PDEs have also been found, for values o f the Ekman number down to 10(-6); these are in good agreement with the asy mptotic theory. The results are also compared with those of Yano, which are surprisingly good in view of their approximate nature.