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 and 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 correspondingly appears in the boundary conditions. In certain cases, the eigenvalue problem can be formulated as an eigenvalue problem in the Hilbert space L2⊕C and this theory is well-developed. Less explored is the case when the eigenvalue problem takes place in a Pontryagin space, as in the Rossby wave problem over sloping topography. This article develops the theory of such problems and explores the properties of wave problems with dynamically active boundaries. The theory allows us to solve the initial value problem for quasigeostrophic Rossby waves in a region with sloping bottom (we also apply the theory to two Boussinesq problems with a free-surface). For a step-function perturbation at a dynamically active boundary, we find that the resulting time-evolution consists of waves present in proportion to their projection onto the dynamically active boundary.
Highlights
An important tool in the study of wave motion near a stable equilibrium is the separation of variables
We have developed a mathematical framework for the analysis of three-dimensional wave problems with dynamically active boundaries
The resulting waves have vertical structures that depend on the wavevector k: For Boussinesq gravity waves, the dependence is only through the wavenumber k, whereas the dependence for quasigeostrophic Rossby waves is on both the wavenumber k and the propagation direction k/k
Summary
An important tool in the study of wave motion near a stable equilibrium is the separation of variables. Examples of boundary-confined restoring forces include the gravitational force at a free-surface (i.e., at a jump discontinuity in the background density), forces arising from gradients in surface potential vorticity (Schneider et al, 2003), and the molecular forces giving rise to surface tension These restoring forces, respectively, result in surface gravity waves (Sutherland, 2010), topographic/thermal waves (Hoskins et al, 1985), and capillary waves (Lamb, 1975). In the absence of boundary-confined restoring forces, we can often apply Sturm–Liouville theory [e.g., Hillen et al (2012); Zettl (2010)] to the resulting eigenvalue problem. This article shows that by including boundary-confined restoring forces, we obtain a set of modes with additional degreesof-freedom. These degrees-of-freedom manifest in the behavior of eigenfunction expansions at the boundaries.
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