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 sigma 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.