The completeness theorem for Rossby normal modes of a stably stratified flat ocean with an arbitrary form of side boundary

A mid-latitude flat ocean on a β \beta -plane has characteristic oscillations called Rossby normal modes, where the motion is governed by the quasigeostrophic vorticity equation. Although the relevant eigenvalue problem differs from the usual one of Hilbert-Schmidt type, a variational proof is obtained that the Rossby normal modes constitute a complete orthonormal set for a basin with an arbitrary profile of stable density stratification and an arbitrary form of side boundary. In particular, for each fixed vertical mode, the set of the horizontal modes is complete and orthonormal in a two-dimensional Hilbert space. General solutions are expressed in terms of Rossby normal modes, not only to the initial-value problem, but also to the response problem of the closed basin.