An atmospheric Bénard problem

In a reformulation for the atmosphere of the classical Bénard problem for the atmosphere, the characterizing elements are the height of the domain and the use of potential temperature in the thermodynamic equation. The constant-temperature boundary conditions induce a coupling of the odd and even modes that are separate in the "laboratory" version of the problem. A non-dimensional parameter (?) controls the coupling, and takes values of O( ) in the Earth's atmosphere. The changes to the critical Rayleigh number for instability are correspondingly small, i.e. an increase of a few percent. Larger corrections can be obtained for larger values of ?, appropriate for different geo- or astrophysical systems, but the approximations applied here limit the allowed range.