Normal modes of deep atmospheres.: II:: f-F-plane geometry

Numerical results have shown that, for a rotating spherical atmosphere with terrestrial parameters, making the shallow-atmosphere approximation has only a very small impact on the frequency of linear, unforced normal modes, compared with those obtained from the full, unapproximated equations. In nearly all cases the normal-mode frequencies are smaller in magnitude when the shallow-atmosphere approximation is relaxed. Relaxing the approximation was also found to have only a small impact on the structures of normal modes, with the exception of long-zonal-wavelength internal acoustic modes. These results are particularly surprising in the tropics where the inclusion of the F = 2Omega cos phi Coriolis terms (which are dropped in the shallow-atmosphere approximation) might be expected to dominate the usual f = 2Omega sin phi Coriolis terms. The complexity of the full equations, however, prevents analysis of why this insensitivity to the extra terms arises. In this paper normal modes are examined under the f-F-plane approximation and compared with those on the more usual f-plane. The resulting equations are more amenable to analysis than the full equation set, and analytic expressions for the dispersion relation and for the normal-mode structures are obtained for the particular case of an isothermal reference profile. This simplified geometry allows the effects of the F Coriolis terms to be examined while eliminating the geometrical effects of relaxing the shallow-atmosphere approximation, giving some insight into the relative importance of the two types of effect as well as the physical mechanisms at work. The F Coriolis terms are found to be responsible for the structural changes to Ion g-zonal-wavelength internal acoustic modes, and can also affect extremely shallow and extremely deep gravity modes. However, these terns are found to have only a small effect on normal-mode frequencies, and geometrical effects, rather than these Coriolis terms, are responsible for the systematic reduction in the magnitude of normal-mode frequencies in a deep spherical atmosphere. In planar geometry the inclusion of the F terms gives rise to a new kind of normal mode in addition to the usual Rossby, gravity, and acoustic modes. The new modes are inertial in character, have frequency very close to f, and have extremely strong vertical tilt.