Linear and nonlinear evolution of isolated disturbances in a growing thermal boundary layer in porous media

We consider the onset and development of convection in a saturated porous half-space which is initially cold, but where the lower boundary has its temperature raised suddenly to a new uniform level. The resulting thermal boundary layer diffuses upwards and eventually becomes thermoconvectively unstable. Previous works by the present authors have considered in turn linearised theory, the nonlinear development of cells, and the destabilisation of such cells due to subharmonic disturbances. In all three papers it was assumed that the convection pattern is horizontally periodic. In the present paper we relax this restriction by considering how an isolated disturbance develops in time, and this is compared with the horizontally periodic flows. New cells are generated outboard of existing ones so that the convecting region spreads horizontally with time. The effective wavelengths are also found to increase with time, newer cells having larger wavelengths than older ones. We also consider the nonlinear development of these disturbances.