The linear stability of convection in a rapidly rotating sphere studied here builds on well established relationships between local and global theories appropriate to the small Ekman number limit. Soward (1977) showed that a disturbance marginal on local theory necessarily decays with time due to the process of phase mixing (where the spatial gradient of the frequency is non-zero). By implication, the local critical Rayleigh number is smaller than 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 solution was expressed by Yano (1992). He explained that the criterion for the onset of global instability is found by extending the solution onto the complex 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 global criterion on a related two-parameter family of models, which includes the 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, his 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 Yano'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 leading-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 of the Ekman number down to 10−6; these are in good agreement with the asymptotic theory. The results are also compared with those of Yano, which are surprisingly good in view of their approximate nature.

The linear stability of magnetoconvection in a rapidly rotating sphere is investigated. The weak-eld regime is studied where the Elsasser number ¤ is O(E 1=3 ) and E is the Ekman number, assumed to be small. In this regime, the magnetic eld is strong enough to a¬ect the critical Rayleigh number, frequency and preferred azimuthal wavenumber by order-one amounts, but is weak enough that the convection still has the form of columnar rolls. A global asymptotic theory is constructed that di¬ers from previous local theories of the onset of convection at asymptotically small Ekman number, and it provides a consistent Wentzel{Kramers{Brillouin solution which takes account of the phase-mixing phenomenon.The asymptotic theory is developed to give the leading-order and rst-order correction terms, including those from Hartmann boundary layers. Numerical solutions of the relevant partial di¬erential equations have also been found, for values of the Ekman number down to 10 ¡6:5 , and these are compared with the asymptotic theory. Good agreement with the asymptotic theory is found.