Planetary and Inertia-Gravity Waves on the Rotating Spherical Earth

Abstract

In the preceding chapters the physical setup included a channel and in these problems the boundary conditions that determine the eigensolutions (i.e., eigenvalues and eigenfunctions) of the eigenvalue problems are no-flow through the channel walls. For these boundary conditions it was natural to transform the set of two first-order equations, (2.4) for $\left( {V,\eta } \right)$ on a plane and (4.4) for $(V\cos \phi ,\eta )$ on a sphere, to a single second-order equations for V or $V\cos \phi$, respectively, i.e., in the channel setup, η was eliminated from the set of two first-order equations. In contrast, on the entire spherical earth there is no clear preference to one of the two variables and considerations other than those involving the wall boundary conditions should determine whether to eliminate η or $V\cos \phi$ in order to obtain the second-order eigenvalue equation.