Magnetoconvection in sunspot umbrae :steady and oscillatory localised states

Abstract

Astrophysical observations of the solar photosphere uncover a wealth of detailed structures that arise from the interaction of vigorously convecting plasma and the internally generated magnetic elds. The most prominent features are sunspots, which exhibit sub-structures on a range of scales. Speci cally within the umbra is an intensity pattern consisting of individual small bright points, referred to as umbral dots. These states are thought to indicate the presence of localised magnetoconvective motions. This thesis discusses the applications of magnetoconvection to the umbra, with the aim of investigating the occurrence of steady and oscillatory localised states known as convectons. Convectons are isolated convective plumes from which magnetic ux is at least partially expelled. In two-dimensional Boussinesq magnetoconvection we examine both a simpli ed model, in which the vertical structure has been reduced, and a fully-resolved model. In performing parametric surveys of the steady modes we attempt to understand how localised states di er between the two models. Examining the oscillatory localised cells we locate, for the rst time, these states in the fully-resolved system. Both of these models are horizontally periodic. We nd that by altering these horizontal boundaries so that they are impermeable to uid motions does not impede the existence of these states but leads to the additional existence of a new set of solutions that are localised at the boundaries. To examine the bifurcation structure of these states we develop a numerical continuation model. However, due to the limitations of the continuation program, AUTO-07p, this model has restricted symmetries and impermeable horizontal boundaries. Despite these simpli cations the symmetries of the model ensure that convectons can still be found and in addition allows the examination of the wall states. The remainder of this thesis focuses on compressible magnetoconvection. In studying oscillatory convectons in two-dimensions we nd a new type of oscillation not found in the Boussinesq models. This state no longer retains Boussinesq point symmetry but has more gentle extended up ows characteristic of a three-dimensional cylindrical plume. In three dimensions a new type of steady convecton is found with a broken symmetry such that the cross-section corresponds to a single overturning roll.