Two-dimensional numerical experiments of convection
I report the results of numerical studies of 2-dimensional fully compressible convection of a fluid in which the dependence of the radiative conductivity on temperature T is taken as K, (V + aT -5)1 so that central regions of the layer are unstable to convection whereas the surrounding layers are stable. Calculations have been undertaken for Rayleigh numbers at the centre of the unstable layer Rc,, = 4.78 106 -9-56 107 and Prandtl numbers a=0.05 - 1. The main result found is that in a (statistically) stationary state the viscous dissipation decreases with decreasing Prandtl number, and that the equilibrium of the whole layer is governed by a substantial penetrative region in which the convective flux is negative. The results found here suggest that the so called "Roxburgh criterion" can be used to give a good estimate of convective penetration at small Prandtl numbers. I also report the results of three sets of numerical experiments involving the interaction of magnetic fields with 2-dimensional fully compressible convection, were the fluid has the same conductivity. For these experiments R9subscript cen)= 4.78 106 and [sigma]=1.0, typically. For one set of experiments a simple model of the evolution of a toroidal flux tube is considered. The purpose of these numerical experiments was to test the role played by magnetic buoyancy in the rise of the magnetic flux tube. It was found that magnetic buoyancy was not important, except possibly when the initial field strength was large (Chandrasekhar number Q= 10'). In another set of numerical experiments the initial magnetic field was an uniform horizontal field. One of the results found was that the magnetic field did not significantly reduce convective penetration, even when the total (integrated) magnetic energy was of the same order as the total kinetic energy. The general behaviour found was that magnetic field was expelled from the convective region, until Q- 10'. Then the initial field strength was strong enough to suppress convection completely. No oscillatory solutions were found. Finally experiments were made for initially vertical magnetic fields. An oscillatory solution is presented. Also reported are preliminary calculations of 2-dimensional penetrative convection, for a model were the effects of energy generation and self gravity are included.