Strongly nonlinear magnetoconvection in three dimensions

Fully nonlinear three-dimensional convection in a strong vertical magnetic field is studied. In this regime, the convective velocities are not strong enough to distort the magnetic field substantially and the field remains pr imarily vertical. Consequently, the leading order nonlinearity arises from the distortion of the horizontally averaged temperature profile only. As a result all steady spatially periodic patterns have the same Nusselt numbers and mean temperature profile, A similar degeneracy is present in overstabl e convection with all periodic patterns having identical time-averaged Nuss elt numbers and oscillation frequencies. These results are obtained via an asymptotic expansion in inverse Chandrasekhar number that determines, for e ach Rayleigh number, the time-averaged Nusselt number and oscillation frequ ency from the solution of a nonlinear eigenvalue problem for the vertical t emperature profile. In the presence of variable magnetic Prandtl number zet a(z) these profiles are asymmetric, but nonetheless develop isothermal core s in the highly supercritical regime. The interesting case in which zeta > 1 near the bottom (favoring steady convection) and zeta < 1 near the top (f avoring overstable convection) is discussed in detail. (C) 1999 Elsevier Sc ience B.V. All rights reserved.