A Lorenz model for an anelastic Oberbeck-Boussinesq system

In an Oberbeck–Boussinesq model, rigorously derived, which includes compressibility, one could expect that the onset of convection for Bénard’s problem occurs at a higher critical Rayleigh number with respect to the classic O–B solutions. The new partial differential equations exhibit non constant coefficients and the unknown velocity field is not divergence-free. By considering these equations, the critical Rayleigh number for the instability of the rest state in Lorenz approximation system is shown to be higher than the classical value, so proving increased stability of the rest state as expected.