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- W3123030852 abstract "Three-dimensional geophysical fluids support both internal and boundary-trapped waves. To obtain the normal modes in such fluids we must solve a differential eigenvalue problem for the vertical structure (for simplicity, we only consider horizontally periodic domains). If the boundaries are dynamically inert (e.g., rigid boundaries in the Boussinesq internal wave problem, flat boundaries in the quasigeostrophic Rossby wave problem) the resulting eigenvalue problem typically has a Sturm-Liouville form and the properties of such problems are well-known. However, when restoring forces are also present at the boundaries, then the equations of motion contain a time-derivative in the boundary conditions and this leads to an eigenvalue problem where the eigenvalue appears in the boundary conditions. This article develops the mathematical theory of such problems and explores the novel properties of wave problems with dynamically active boundaries. Physically, the presence of a restoring force at the boundary leads to an additional wave (e.g., surface gravity wave, topographic Rossby wave) and this boundary-trapped wave allows us to expand a larger collection of physical states in terms of these modes. The vertical structure of these waves depends on the horizontal wavevector, they satisfy an indefinite orthogonality relation containing Dirac delta contributions, and they may have jump discontinuities at the boundaries. Additionally, such problems include a $delta$-sheet formulation analogous to the Bretherton (1966) interpretation of boundary buoyancy gradients as infinitely thin sheets of potential vorticity. We apply the theory to four examples: two Boussinesq problems with a free-surface and two quasigeostrophic problems with a sloping lower boundary." @default.
- W3123030852 created "2021-02-01" @default.
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- W3123030852 date "2021-01-18" @default.
- W3123030852 modified "2023-09-27" @default.
- W3123030852 title "Normal modes with boundary dynamics" @default.
- W3123030852 hasPublicationYear "2021" @default.
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