Elliptica topographic waves

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Abstract
Simple analytical solutions are presented for nondivergent topographic waves (rotational modes) in a certain type of elliptical basin. Under the assumption that the basin's depth contours form a family of confocal ellipses, the governing potential vorticity equation in elliptic cylindrical coordinates reduces to a Cartesian form, independent of the coordinate scale factors. As a consequence, for the exponential depth profile h=e −bξ, where ξ is the radial coordinate, the radial eigenfunctions for elliptically travelling waves in a basin with a partial vertical barrier along the centerline can be expressed in terms of elementary functions. For a lake without a barrier, approximate analytical solutions are obtained by the Rayleigh-Ritz (variational) method. The periods and streamline patterns of the first few modes of the variational solutions are compared with those due to Ball (1965) for an elliptic paraboloid. The gravest mode period of one of the variational solutions has also been computed for three Swiss lakes and two Laurentian Great Lakes, and the results are shown to agree favorably with observations.

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