Analytical and Numerical Linear Analyses of Convection Revisited

We conduct linear analyses of convection in domains larger than the temperature scale height. We employ both analytical and numerical methods in these analyses. In the case excluding all dissipation, the typical time scale of convection is determined by the free fall time over the temperature scale height. We quantitatively show the condition for the Boussinesq and Wentzel-Kramers-Brillouin (WKB) approximations to be applicable. We provide a reassessment of the critical Rayleigh number, a key indicator of convection, and show that WKB approximation tends to underestimate the critical Rayleigh number, particularly when the temperature scale height is comparable to or smaller than the domain height. We show clear explanation why both thermal conduction and viscosity are required for stabilizing negative entropy gradient medium.