Eigen solutions of internal waves over subcritical topography

Diapycnal mixing plays an important role in the ocean circulation. Internal waves are a kind ofbridge relating the diapycnal mixing to external sources of mechanical energy. Difficulty in obtainingeigen solutions of internal waves over curved topography is a limitation for further theoretical studyon the generation problem and scattering process. In this study, a kind of transform methodis 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 spaceto flat bottom in transform space while the governing equation of internal waves is still hyperbolicif proper transform function is selected. Thus, one can obtain eigen solutions of internal wavesin 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 areillustrated. A method, using a polynomial to approximate the transform function and least squaresmethod to estimate the undetermined coefficients in the polynomial, is introduced to calculate theapproximate expression of the transform function for the given subcritical topography.