Asymptotically Reduced Models in Strong Field Magnetoconvection

Abstract

The magnetoconvection problem represents a doubly-diffusive system where the velocity, temperature and magnetic fields of a conducting fluid interact in a nontrival manner. This thesis uses asymptotic and computational methods in an attempt to understand the dynamics of the incompressible magnetoconvection problem, when the background magnetic field is strong and the primary instability is direct (stationary) or oscillatory (Hopf). By asymptotically solving the magnetoconvection system and projecting out the coordinate parallel to the direction of convection, we derive reduced models in the plane that describe the system from the onset of convection far into the nonlinear regime. In the first part of this thesis we develop the reduced models. Two models are derived for the stationary bifurcation, one where the Prandtl number is asymptotically small and another where the Prandtl number is order one in magnitude. For the Hopf bifurcation, a single model is derived for travelling wave solutions with both Prandtl numbers order one in magnitude. Remarkably, the derivation automatically chooses the supercritical branch of the bifurcation, while the leading order perturbation is shown to be independent of the growth rate. The second part of this thesis examines the numerical solution of the reduced models. We use the Dedalus code to compute high resolution solutions in the direct problem. We find that the poloidal component in the large Prandtl number model converges to a random Gaussian field, regardless of the magnitudes of the magnetic Prandtl number and bifurcation parameter. The asymptotically small Prandtl number problem displays a much wider range of characteristics, from rolls that are frozen in to strongly localised, rapidly rotating vortices, depending on the relative sizes of the Prandtl numbers and the strength of the bifurcation parameter. We also find that heat transport is optimal when rolls are the preferred planform and the vorticity is small.