By investigating the effects of compressibility on laminar convective flow, I have identified the specific conditions under which a compressible code can be used to model a Boussinesq fluid. As well as requiring the Mach number to be small, the density contrast of the initial distribution needs to be unity. Under such conditions, the Nusselt numbers of steady convective flow are independent of the ratio of specific heats, and the Hopf-bifurcation from steady to oscillatory convection, oscillation frequencies and period doubling bifurcations, are within a few percent of benchmark laboratory experiments with liquids. By measuring the r.m.s. density fluctuation, the oscillation frequency of the rolls was found to scale with the convective growth rate, [square root]gp[prime]/dp_o where g, p[...[ Read more ]

By investigating the effects of compressibility on laminar convective flow, I have identified the specific conditions under which a compressible code can be used to model a Boussinesq fluid. As well as requiring the Mach number to be small, the density contrast of the initial distribution needs to be unity. Under such conditions, the Nusselt numbers of steady convective flow are independent of the ratio of specific heats, and the Hopf-bifurcation from steady to oscillatory convection, oscillation frequencies and period doubling bifurcations, are within a few percent of benchmark laboratory experiments with liquids. By measuring the r.m.s. density fluctuation, the oscillation frequency of the rolls was found to scale with the convective growth rate, [square root]gp[prime]/dp_o where g, p[prime], p_o and d are the gravitational acceleration, density fluctuation, average density and depth of the layer.

In the second part of my PhD, the large eddy simulation approach is used to solve the full Navier Stokes equations in spherical coordinates. The purpose is to explain the observed differential rotation in the solar convection zone. Using helioseismology, the isorotation surfaces in the convection zone were found to be 'cone-like' (aligned radially), in disagreement with most previous numerical simulations, which tended to produce isorotation contours parallel to the rotation axis.

My latest results indicate that the Reynolds stress plays only a minor role, compared to the latitudinal pressure gradient and Coriolis force, in producing the desired differential rotation. By using an implicit code, in conjunction with an explicit code when gathering statistics, a stronger emphasis is placed on complete thermal relaxation. With a small Rossby number and fully relaxed thermal structure, I have obtained isorotation surfaces in approximate agreement with observational results.