Eigen solutions of internal waves over subcritical topography

Abstract

Diapycnal mixing plays an important role in the ocean circulation. Internal waves are a kind of bridge relating the diapycnal mixing to external sources of mechanical energy. Difficulty in obtaining eigen solutions of internal waves over curved topography is a limitation for further theoretical study on the generation problem and scattering process. In this study, a kind of transform method is put forward to derive the eigen solutions of internal waves over subcritical topography in two-dimensional and linear framework. The transform converts the curved topography in physical space to flat bottom in transform space while the governing equation of internal waves is still hyperbolic if proper transform function is selected. Thus, one can obtain eigen solutions of internal waves in the transform space. Several examples of transform functions, which convert the linear slope, the convex slope, and the concave slope to flat bottom, and the corresponding eigen solutions are illustrated. A method, using a polynomial to approximate the transform function and least squares method to estimate the undetermined coefficients in the polynomial, is introduced to calculate the approximate expression of the transform function for the given subcritical topography.