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Experimental study of dispersion and modulational instability of surface gravity waves on constant vorticity currents

Published online by Cambridge University Press:  17 December 2019

James N. Steer Open the ORCID record for James N. Steer [Opens in a new window] ,
Alistair G. L. Borthwick Open the ORCID record for Alistair G. L. Borthwick [Opens in a new window] ,
Dimitris Stagonas ,
Eugeny Buldakov Open the ORCID record for Eugeny Buldakov [Opens in a new window]  and
Ton S. van den Bremer Open the ORCID record for Ton S. van den Bremer [Opens in a new window]
James N. Steer*
Affiliation:
School of Engineering, University of Edinburgh, EdinburghEH9 3FB, UK Wind and Marine Energy Systems CDT, University of Strathclyde, GlasgowG1 1XW, UK
Alistair G. L. Borthwick
Affiliation:
School of Engineering, University of Edinburgh, EdinburghEH9 3FB, UK
Dimitris Stagonas
Affiliation:
School of Water, Energy and Environment, Cranfield University, CranfieldMK43 0AL, UK
Eugeny Buldakov
Affiliation:
Department of Civil, Environmental and Geomatic Engineering, University College London, Chadwick Building, LondonWC1 6BT, UK
Ton S. van den Bremer
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
*
†Email address for correspondence: james.steer@ucd.ie
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Abstract

This paper examines experimentally the dispersion and stability of weakly nonlinear waves on opposing linearly vertically sheared current profiles (with constant vorticity). Measurements are compared against predictions from the unidirectional $(1\text{D}+1)$ constant vorticity nonlinear Schrödinger equation (the vor-NLSE) derived by Thomas et al. (Phys. Fluids, vol. 24, no. 12, 2012, 127102). The shear rate is negative in opposing currents when the magnitude of the current in the laboratory reference frame is negative (i.e. opposing the direction of wave propagation) and reduces with depth, as is most commonly encountered in nature. Compared to a uniform current with the same surface velocity, negative shear has the effect of increasing wavelength and enhancing stability. In experiments with a regular low-steepness wave, the dispersion relationship between wavelength and frequency is examined on five opposing current profiles with shear rates from $0$ to $-0.87~\text{s}^{-1}$. For all current profiles, the linear constant vorticity dispersion relation predicts the wavenumber to within the $95\,\%$ confidence bounds associated with estimates of shear rate and surface current velocity. The effect of shear on modulational instability was determined by the spectral evolution of a carrier wave seeded with spectral sidebands on opposing current profiles with shear rates between $0$ and $-0.48~\text{s}^{-1}$. Numerical solutions of the vor-NLSE are consistently found to predict sideband growth to within two standard deviations across repeated experiments, performing considerably better than its uniform-current NLSE counterpart. Similarly, the amplification of experimental wave envelopes is predicted well by numerical solutions of the vor-NLSE, and significantly over-predicted by the uniform-current NLSE.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 884 , 10 February 2020 , A40
Copyright
© 2019 Cambridge University Press

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