Highly supercritical convection in strong magnetic fields

Fully nonlinear convection in a strong imposed magnetic field is studied in the regime in which the convective velocities are not strong enough to distort the magnetic field substantially. Motivated by convection in sunspots both vertical and inclined imposed fields are considered. In this regime the leading order nonlinearity is provided by the distortion of the horizontally averaged temperature profile. For overstable convection this profile is determined from the solution of a nonlinear eigenvalue problem for the (time-averaged) Nusselt number and oscillation frequency, and evolves towards an isothermal profile with increasing Rayleigh number. In the presence of variable magnetic Prandtl number ζ(z) the profile is asymmetric with respect to midlevel, but nonetheless develops an isothermal core in the highly supercritical regime. A hysteretic transition between two distinct convection regimes is identified in the inclined case, and used to suggest an explanation for the sharp boundary between the sunspot umbra and penumbra. These results are obtained via an asymptotic expansion in inverse powers of the Chandrasekhar number, and generalize readily to a polytropic atmosphere.