Penetrative convection in a fluid layer with throughflow

Linear and nonlinear stability analises of vertical throughflow in a fluid layer, where the density is quadratic in temperature, are studied. To avoid the loss of key terms a weighted functional is used in the energy analysis. Both conditional and unconditional thresholds are derived. When the throughflow is ascending the linear and nonlinear boundaries show substantially agreement. The linear boundary remains close to the conditional nonlinear bioundary for descending throughflow, whilst the unconditional threshold begins to diverge.