Flux Expulsion in Shallow Water Magnetohydrodynamics

Abstract

The process of flux expulsion, pioneered by Weiss (1966) involves imposing an electrically conducting fluid onto an initially uniform magnetic field. This interaction results in the expulsion of magnetic flux from a region of closed streamlines, effectively transporting and creating concentrated regions of magnetic field. Flux expulsion has been proposed as a mechanism through which electrically conducting eddies in the Sun's convective zone can give rise to concentrated magnetic fields. Weiss (1966) found that the time associated with the expulsion of magnetic flux from a region of closed streamlines scales as t∼ϵ^(-1/3), where ϵ is the magnetic diffusivity. This thesis explores kinematic flux expulsion by simulating the nonlinear 2D MHD equations with periodic boundary conditions. We find that initialising the system with a sinusoidal flow confirms the theoretical scaling law of t∼ϵ^(-1/3), however, initialising the system with a Roberts flow results in a shallower scaling of t∼ϵ^(-0.13) due to magnetic flux ropes forming along the flow's separatrices. By creating a new method to identify the magnetic field lines contributing to flux expulsion, we find that these magnetic field lines expel flux on a time scale of t∼ϵ^(-1/3). Recent studies incorporate dynamical effects into flux expulsion, considering regimes where the Lorentz force becomes significant. In Gilbert, Mason and Tobias (2016), a 1D quasi-linear model demonstrates that the minimum magnetic field strength required to suppress flux expulsion scales as M∼ϵ^(1/3). This thesis simulates the full nonlinear 2D MHD and SWMHD equations with periodic boundary conditions, and by defining new criteria to identify the threshold of when flux expulsion occurs, we find scaling laws of M∼ϵ^(1/3), for both systems, when they are initialised with a sinusoidal flow. Additionally, when comparing a 2D MHD and a SWMHD system in the dynamical regime for a Roberts flow, we identify a new criterion for the minimum magnetic field strength required to suppress flux expulsion, revealing a shallow scaling law of M∼ϵ^0.15.