Surface waves on arbitrary vertically-sheared currents

TL;DR

This paper develops a numerical method to analyze surface gravity waves on arbitrary vertically-sheared currents, extending previous models to handle any wave-current angle and broad wavelength ranges, with applications to ship wakes and wave dynamics.

Contribution

The authors extend a piecewise linear approximation method to arbitrary wave-current angles, enabling accurate analysis of wave dispersion on depth-varying currents for all wavelengths.

Findings

The method accurately computes group and phase velocities across different regimes.

Qualitative differences in wake patterns are observed between concave down and up shear profiles.

Additional vorticity wave solutions are identified as artifacts of the piecewise model.

Abstract

We study dispersion properties of linear surface gravity waves propagating in an arbitrary direction atop a current profile of depth-varying magnitude using a piecewise linear approximation, and develop a robust numerical framework for practical calculation. The method has been much used in the past for the case of waves propagating along the same axis as the background current, and we herein extend and apply it to problems with an arbitrary angle between the wave propagation and current directions. Being valid for all wavelengths without loss of accuracy, the scheme is particularly well suited to solve problems involving a broad range of wave vectors, such as ship waves and Cauchy-Poisson initial value problems for example. We examine the group and phase velocities over different wavelength regimes and current profiles, highlighting characteristics due to the depth-variable vorticity. We show an example application to ship waves on an arbitrary current profile, and demonstrate qualitative differences in the wake patterns between concave down and concave up profiles when compared to a constant shear profile with equal depth-averaged vorticity. We also discuss the nature of additional solutions to the dispersion relation when using the piecewise-linear model. These are vorticity waves, drifting vortical structures which are artifacts of the piecewise model. They are absent for a smooth profile and are spurious in the present context.