Two-dimensional Cartesian finite difference models of compressible convection with constant and variable viscosity and fixed bottom temperature are presented. Density variations according to the Adams Williamson equation of state are included. In the case of constant viscosity convection, viscous and adiabatic heating damp the flow. Compressible density variations hardly influence geoid undulations; however, the lower thermal boundary layer becomes thinner, and the mean cell temperature increases. In the case of variable viscosity, the nonlinear coupling between compression, adiabatic and viscous heating, and a temperature-, pressure-, and stress-dependent rheology leads to important consequences: The upwelling flow broadens and plumes are retarded. The flow strongly concentrates toward the downwelling region and becomes mechanically decoupled from the interior of the cell by a low-viscosity region. This mechanism seems to be important for the formation of subducting slabs. Extrapolated to mantle conditions, two low-viscosity regions are predicted flanking the slab on either side and inhibiting an early dispersal and mixing of slab material into the mantle. This process might be aided by an increase of negative buoyancy forces with depth as observed in the models. Further results are as follows: Increasing the dissipation number in variable viscosity convection may either damp or speed up convection, depending on the rheology. Models with internal heating and a fixed bottom temperature show that the threshold to time-dependent variable viscosity convection is drastically reduced if the anelastic liquid approximation is applied instead of the extended Boussinesq approximation. ¿ American Geophysical Union 1989