Jones, C.A., Mussa, A.I. and Worland, S.J., 2003.
Proceedings of the Royal Society of London, Series A, 459, 773-797.
The linear stability of magnetoconvection in a rapidly rotating sphere is investigated. The weak-field regime is studied where the Elsasser number Lambda is O(E-1/3) and E is the Ekman number, assumed to be small. In this regime, the magnetic field is strong enough to affect 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 differs 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 first-order correction terms, including those from Hartmann boundary layers. Numerical solutions of the relevant partial differential 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.