Penetrative convection within the anelastic approximation

We present a method for studying penetrative convection within the anelastic approximation. This method is then applied to two-dimensional numerical simulations of a convective region bounded below by a stable (radiative) region. Descending plumes penetrate into the stable layer, exciting gravity waves with a spectrum of frequencies limited by the "Brunt- Väisäla" frequency. The depth of penetration depends sensitively on the "stiffness" of the boundary between the unstable and stable layers (quantified by S) and on the vigor of convection. We find scaling relations between the penetrative depth (Δ) and S, spanning 3 orders of magnitude in S. The reference state in these models is allowed to evolve; the convective region tends toward adiabaticity, and for low values of S, the stable region can be altered considerably (an extended superadiabatic region is developed). For high values of S, the stable region is left essentially unchanged.