Reduced partial differential equations valid for convection in a strong imposed magnetic field (vertical or oblique) are derived and discussed. These equations filter out fast, small-scale Alfvén waves, and are valid outside of passive horizontal boundary layers. In the regime in which the convective velocities are not strong enough to distort substantially the field, exact, fully nonlinear, single-mode solutions exist. These are determined from the reduced PDEs reformulated as a nonlinear eigenvalue problem whose solution also gives, for each Rayleigh number, the time-averaged Nusselt number and oscillation frequency together with the mean vertical temperature profile. In the oblique case a hysteretic transition between two distinct convection regimes is identified. Possible applications to sunspots are discussed.

Introduction

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 (Hughes & Proctor 1988). Applications to sunspots (Thomas & Weiss 1992) have leds everal authors to investigate the suppression of convection by strong “vertical” or “horizontal” magnetic fields. However, the magnetic field in sunspots is neither vertical nor horizontal, and this has led to recent nonlinear investigation of convection in an oblique magnetic field (Matthews et al. 1992, Julien et al. 2000). Numerical simulations of magnetoconvection are unable to reach the parameter values, both in terms of field strengths and Reynolds number (Re), characteristic of convection in sunspots.