A model for penetrative convection in which a stably stratified layer of fluid is bounded by two unstable layers is considered. This configuration is obtained in a horizontal water layer around its density extremum when a quadratic temperature profile is maintained by internal heating. A linear stability analysis shows that either stationary or oscillatory modes set in at the onset of instability depending on the values of the control parameter. Moreover, two types of stationary modes have been identified that differ by the value of their critical wave number and the number of vertical cells. These results are discussed in the light of recent studies on related topics.

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