Parasitic Capillary Waves on Small-Amplitude Gravity Waves with a Linear Shear Current

Sunao Murashige,Wooyoung Choi

doi:10.3390/jmse9111217

Abstract

This paper describes a numerical investigation of ripples generated on the front face of deep-water gravity waves progressing on a vertically sheared current with the linearly changing horizontal velocity distribution, namely parasitic capillary waves with a linear shear current. A method of fully nonlinear computation using conformal mapping of the flow domain onto the lower half of a complex plane enables us to obtain highly accurate solutions for this phenomenon with the wide range of parameters. Numerical examples demonstrated that, in the presence of a linear shear current, the curvature of surface of underlying gravity waves depends on the shear strength, the wave energy can be transferred from gravity waves to capillary waves and parasitic capillary waves can be generated even if the wave amplitude is very small. In addition, it is shown that an approximate model valid for small-amplitude gravity waves in a linear shear current can reasonably well reproduce the generation of parasitic capillary waves.

Highlights

Eng. 2021, 9, 1217. https://doi.org/It is well known that a vertically sheared current affects the nonlinear motion of surface waves progressing on water and plays important roles in the ocean
We focus on its effects on ripples produced on the forward face of progressive gravity waves, which are referred to as “parasitic capillary waves” [2]
We numerically examine the effects of the strength of a linear shear current on the generation of parasitic capillary waves using the Unsteady Hodograph Transformation (UHT) method

Summary

Introduction

It is well known that a vertically sheared current affects the nonlinear motion of surface waves progressing on water and plays important roles in the ocean (see reviews by Peregrine [1]). The complete understanding of parasitic capillary waves helps us accurately estimate the sea surface elevation from satellite images for remote sensing of ocean surfaces. The generation of such capillary waves is intimately related to the curvature of the water surface of underlying gravity waves as well as the characteristics of the current. Few attempts have been made yet to systematically investigate the interaction between parasitic capillary waves and a vertically sheared current. The aim of this work is to numerically study this nonlinear interaction

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