| Summary: | In this thesis I discuss a series of anelastic approximations and detail the assumptions used in the derivation. I derive an entropy and temperature formulation of the anelastic approximation along with a simplification to the entropy formulation introduced by Lantz (1992) and independently by Braginsky & Roberts (1995). I assess range of applicability of the anelastic approximation, which is often used in describing the dynamics of geophysical and astrophysical flows. I consider two linear problems: magneto convection and magnetic buoyancy and compare the fully compressible solutions with those determined by solving the anelastic problem. I further compare the Lantz-Braginsky simplification with the full anelastic formulation which I find to work well if and only if the atmosphere is nearly adiabatic. I find that for the magnetoconvection problem the anelastic approximation works well if the departure from adiabaticity is small (as expected) and determine where the approximation breaks down. When the magnetic field is large then the anelastic approximation produces results which are markedly different from the fully compressible results. I also investigate the effects of altering the boundary conditions from isothermal to isentropic and the effect of stratification on how some of the parameters scale with the Chandrasekhar number. The results for magnetic buoyancy are less straight-forward, with the accuracy of the approximation being determined by the growth rate of the instability. I argue that these results make it difficult to assess a priori whether the anelastic approximation will provide an accurate approximation to the fully compressible system for stably stratified problems. Thus, unlike the magnetoconvection problem, for magnetic buoyancy it is difficult to provide general rules as to when the anelastic approximation can be used. When the instability grows quickly or the magnetic field is large the results do not compare well with the fully compressible equations. I outline a method for a two-dimensional non-linear time-stepping computer program and explain some problems with current non-linear programs.
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