K. Julien, E. Knobloch and S.M. Tobias
Strongly nonlinear magnetoconvection in three dimensions
Physica D.
128 (1999) 105-129.
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 primarily 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 overstable convection with all periodic patterns having identical time-averaged Nusselt numbers and oscillation frequencies. These results are obtained via an asymptotic expansion in inverse Chandrasekhar number that determines, for each Rayleigh number, the time-averaged Nusselt number and oscillation frequency from the solution of a nonlinear eigenvalue problem for the vertical temperature profile. In the presence of variable magnetic Prandtl number zeta(z) these profiles are asymmetric, but nonetheless develop isothermal cores in the highly supercritical regime. The interesting case in which zeta > 1 near the bottom (favoring steady convection) and zeta < 1 near the top (favoring overstable convection) is discussed in detail.