Approximate dispersion relations for waves on arbitrary shear flows

TL;DR

This paper derives an approximate dispersion relation for surface waves on shear flows with variable depth profiles, improving accuracy and applicability over previous models, and provides criteria for their validity.

Contribution

It introduces a new, more robust first and second order approximation for wave dispersion on shear flows, extending and refining prior models like Kirby & Chen.

Findings

The new model accurately approximates wave dispersion across various shear flows.

It extends the applicability of existing models to more complex flow conditions.

The second order approximation significantly improves accuracy in difficult cases.

Abstract

An approximate dispersion relation is derived and presented for linear surface waves atop a shear current whose magnitude and direction can vary arbitrarily with depth. The approximation, derived to first order of deviation from potential flow, is shown to produce good approximations at all wavelengths for a wide range of naturally occuring shear flows as well as widely used model flows. The relation reduces in many cases to a 3D generalization of the much used approximation by Skop [1987], developed further by Kirby & Chen [1989], but is shown to be more robust, succeeding in situations where the Kirby & Chen model fails. The two approximations incur the same numerical cost and difficulty. While the Kirby & Chen approximation is excellent for a wide range of currents, the exact criteria for its applicability have not been known. We explain the apparently serendipitous success of the latter and derive proper conditions of applicability for both approximate dispersion relations. Our new model has a greater range of applicability. A second order approximation is also derived. It greatly improves accuracy, which is shown to be important in difficult cases. It has an advantage over the corresponding 2nd order expression proposed by Kirby \& Chen that its criterion of accuracy is explicitly known, which is not currently the case for the latter to our knowledge. Our 2nd order term is also arguably significantly simpler to implement, and more physically transparent, than its sibling due to Kirby & Chen.