Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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


Author: Akira Masuda
Journal: Quart. Appl. Math. 51 (1993), 425-439
MSC: Primary 76C20; Secondary 35Q35, 76V05, 86A05
DOI: https://doi.org/10.1090/qam/1233523
MathSciNet review: MR1233523
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Abstract | References | Similar Articles | Additional Information

Abstract: A mid-latitude flat ocean on a $ \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.


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DOI: https://doi.org/10.1090/qam/1233523
Article copyright: © Copyright 1993 American Mathematical Society


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