Resonant penetrative convection with an internal heat source/sink.

We develop a detailed linear instability and nonlinear stability analysis for the situation of convection in a horizontal plane layer of fluid when there is a heat sink/source which is linear in the vertical coordinate which is in the opposite direction to gravity. This can give rise to a scenario where the layer effectively splits into three sublayers. In the lowest one the fluid has a tendency to be convectively unstable while in the intermediate layer it will be gravitationally stable. In the top layer there is again the possibility for the layer to be unstable. This results in a problem where convection may initiate in either the lowest layer, the upmost layer, or perhaps in both sublayers simultaneously. In the last case there is the possibility of resonance between the upmost and lowest layers. In all cases penetrative convection may occur where convective movement in one layer induces motion in an adjacent sublayer. In certain cases the critical Rayleigh number for thermal convection may display a very rapid increase which is much greater than normal. Such behaviour may have application in energy research such as in thermal insulation