Magnetoconvection in a rapidly rotating sphere: the weak–field case

Abstract

The linear stability of magnetoconvection in a rapidly rotating sphere is investigated. The weak–field regime is studied where the Elsasser number Λ is ;O(;E1/3) 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.