The effects of internal heating on anelastic, compressible convection have been investigated in a spherical shell, with the ratio of inner to outer radii appropriate to lower mantle convection. The internal heating and the compressiblity are characterized by two dimensionless numbers: R, the ratio of two Rayleigh numbers, and D, the dissipation number. Isothermal, stress-free boundary conditions are applied at the top and at the bottom of the shell. The equations have been solved, with a time-dependent finite difference code, in a temperature, vorticity, stream function formulation (axisymmetrical configuration) for Rayleigh numbers Rab ranging from the critical Rc up to 2000 Rc. The neutral surface computed over the (D,R) space leads to an unexpected crest shape which is correlated with a shift toward higher-degree harmonics of the most unstable modes. The phenomenon of penetrative convection is generated, for the weakly nonlinear cases, by the combination of large internal heating rates R and large dissipation numbers D. Small cells appear at the top of the shell, counterrotating with respect to the surrounding global circulation. The onset of time dependence (at Rab=Rtd) and the evolution of bottom topography have been monitored over the (D,R) space by increasing the Rayleigh numbers gradually from the critical. With no internal heating the stabilizing effect of compressibility delays the transition to time dependence. However, the combined effects of internal heating and compressibility drastically reduced the threshold of time dependence to Rtd=3 Rc, for R=16, D=0.6.

The study of the topographic deflection of the bottom interface conducted with the same set of (D, R, Rab) values shows that this deflection always decreases with the increase of the Rayleigh number. To reach situations as close as possible to mantle conditions we have focused on D=0.3 and R=16 for Rayleigh numbers up to 2000 Rc. In these highly nonlinear eigimes, penetrative convection cells are still generated at the top or at the bottom of the shell and evolved by breaking the surrounding chaotic circulations or by pairing with the closest corotating cells. The destruction of these transient cells by viscous dissipation leaves a persisting hot thermal anomaly. The internal heating combined with compressiblity reinforces the chaotic character of convection. As a consequence, the bottom topographic deflections display highly time-dependent behavior, and the thermal anomaly spectra contain energy up to degree 100. Serious assumptions are imposed on the modeling of the Earth's mantle by the mathematical and numerical difficulties of the problem, in particular by the introduction of an isothermal stress-free upper boundary. In spite of these limitations we propose that penetrative convection, due to the combination of internal heating and compressibility, could be a mechanism able to generate spontaneously layered convection and local melting in the mantle. ¿ American Geophysical Union 1989