Anisotropic Porous Penetrative Convection

A linear instability analysis and a nonlinear energy stability analysis is developed for convection in an anisotropic porous medium. The nonlinear analysis is very important since a standard energy method does not in the present situation yield unconditional stability and a weighted analysis must be employed to yield global results, and in addition the nonlinear energy results yield a valuable threshold indicating where possible subcritical instabilities may form. Significantly, we find that when a quadratic density temperature law is used in the anisotropic convection model of Tyvand & Storesletten (1991), then the growth rate σ is always complex provided we are in the anisotropic situation. Thus, the nature of bifurcation into convection is very different from the Boussinesq situation and is always via an oscillatory instability.