FROM THE GENERALIZED BOUSSINESQ APPROXIMATION TO THE MARGINALLY SUPER-ADIABATIC LIMIT

The prevailing view of the Rayleigh-Bénard problem in compressible fluids is that for small temperature differences the Boussinesq approximation holds, provided that it is based on a modified Rayleigh number incorporating the potential-temperature gradient. However, for small values of the latter, the onset of convection is characterized by distinct non-Boussinesq features. We consider the linear temporal stability problem and identify the origin of the nonuniformity in the convection and compression-work terms of the perturbation energy balance. We thereby regularize the transition with diminishing potential-temperature gradient from the generalized Boussinesq approximation to the limit when the temperature gradient is only marginally super-adiabatic. It is demonstrated that this transition is accomplished in two phases. Initially, the critical Rayleigh number rapidly increases, which is accompanied by only slight variations of the corresponding wave number. Subsequently, with further diminishing potential-temperature gradients, the critical wave number rapidly increases as well, and the resulting convection becomes effectively confined to a narrowing fluid layer adjacent to the upper wall.

Keywords

• Compressibility,
• Generalized Boussinesq approximation,
• Rayleigh-Bénard instability,
• Super-adiabatic limit