ABSTRACT
This chapter summarizes some of the fully nonlinear results that can be obtained by reformulating the problem of convection in an imposed magnetic field as a nonlinear eigenvalue problem. Such a reformulation is possible when the field strength is large and the distortion of the field by the flow remains small. Motivated by convection in sunspots both vertical and inclined imposed fields are considered. For overstable convection this profile is determined from the solution of a nonlinear eigenvalue problem for the Nusselt number and oscillation frequency, and evolves towards an isothermal profile with increasing Rayleigh number. The study of convection in an imposed magnetic field is motivated primarily by astrophysical applications, particularly by the observed magnetic field dynamics in the solar convection zone. For small tilt angles the magnetic field plays a minor role in inhibiting convection and the Nusselt number is an increasing function of the Rayleigh number.
