A unified linear wave theory of the shallow water equations on a rotating plane

The linearized Shallow Water Equations (LSWE) on a tangent (x, y) plane to the rotating spherical Earth with Coriolis parameter f(y) that depends arbitrarily on the northward coordinate y is considered as a spectral problem of a self-adjoint operator. This operator is associated with a linear second-order equation in x - y plane that yields all the known exact and approximate solutions of the LSWE including those that arise from different boundary conditions, vanishing of some small terms (e.g. the beta-term and frequency) and certain forms of the Coriolis parameter f (y) on the equator or in mid-latitudes. The operator formulation is used to show that all solutions of of the LSWE are stable. In some limiting cases these solutions reduce to the well-known plane waves of geophysical fluid dynamics: Inertia-gravity (Poincare) waves, Planetary (Rossby) waves and Kelvin waves. In addition, the unified theory yields the non-harmonic analogs of these waves as well as the more general propagating solutions and solutions in closed basins.