Convective Instability in a Compressible Atmosphere.

The linear equations for time4ndependent convection in a plane-parallel layer of perfect gas are studied for the case of constant viscosity and conductivity. These equations determine the condition for the onset of steady convection. The equations are treated from three points of view. First a perturbation expansion in terms of layer thickness is carried out to first order. The zeroth-order terms give the Boussinesq equations studied by Lord Rayleigh. The first-order terms show that if the Rayleigh number is evaluated at the mid-height of the layer, its eigenvalues are stationary with respect to variations in layer thickness. The first-order eigenfunctions are numerically calculated for a particular polytropic index. Next, two variational statements are written and some sample numerical results are presented. It is found that variational techniques are not effective in the present problem. The WKB approach is then explored and extended to the determination of uniformly valid asymptotic solutions. These provide not only stability criteria (from which explicit numerical results are derived), but also produce analytic approximations for all the eigenmodes. This method appears to be generally useful for problems of this kind.