No abstract

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.