Linear waves on the spheroidal Earth

The Linearized Shallow Water Equations (LSWE) are formulated on an oblate spheroid (ellipsoid of revolution) that approximates Earth's geopotential surface more accurately than a sphere. The application of a previously developed invariant theory (i.e. applied to an arbitrary smooth surface) to oblate spheroid yields exact equations for the meridional structure function of zonally propagating wave solutions such as Planetary (Rossby) waves and Inertia-Gravity (Poinacré) waves. Approximate equations (that are accurate to first order only of the spheroid's eccentricity) are derived for the meridional structure of Poincaré (Inertia-Gravity) and Rossby (Planetary) and the solutions of these equations yield expressions in terms of prolate spheroidal wave functions. The eigenvalues of the approximate equations provide explicit expressions for the dispersion relations of these waves. Comparing our expressions for the dispersion relations on a spheroid to the known solutions of the same problem on a sphere shows that the relative error in the dispersion relations on a sphere is of the order of the square of spheroid's eccentricity (i.e. about 0.006 for Earth) for both Poincaré and Rossby waves.