This report describes a method of calculating the modes of small-amplitude free oscillations of a compressible, stratified, nonhydrostatic, rotating deep atmosphere, confined between two concentric spheres, including sin q and cos e Coriolis terms, where q is the latitude. Unlike the Laplace-Taylor problem for the hydrostatic primitive equations that can be solved by the separation of variables, the present problem is non-separable. In this study, normal mode solutions are obtained numerically by setting up an eigenvalueeigenfunction matrix problem by the combination of a spherical harmonics expansion in the horizontal direction and a finite-difference discretization in the radial direction. A Test of the numerical schemes is conducted for an isothermal basic state with a constant gravitational acceleration. In order to identify the species of solutions, a shallow atmosphere version of the same formulation is considered in parallel. Because the shallow nonhydrostatic normal mode problem can be solved also by the separation of variables, two different approaches to the shallow problem provide an aid to verify the numerical schemes of the deep normal mode problem. Numerical results are presented for the frequencies and eigen-structures of various kinds of normal modes in the deep model and compared with those from the shallow model.
