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Evolution of wind-induced wave groups in water of finite depth

Published online by Cambridge University Press:  12 April 2024

Montri Maleewong Open the ORCID record for Montri Maleewong [Opens in a new window]  and
Roger Grimshaw Open the ORCID record for Roger Grimshaw [Opens in a new window]
Montri Maleewong*
Affiliation:
Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand
Roger Grimshaw
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
*
†Email address for correspondence: montri.m@ku.th
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Abstract

The generation of water waves by wind is a fundamental and much-studied problem of scientific and operational concern. One mechanism that has obtained a wide degree of acceptance and use is the Miles air shear flow instability theory in which water waves grow due to energy transfer from the air at a critical level where the wave phase speed matches the wind speed. In this paper we revisit and extend this theory by examining how the predicted wave growth rate depends on the water depth and the wind shear parameters. At the same time we include frictional effects due to water bottom stress and at the water surface, and add an additional driving term due to wave stress in the air near the sea surface, using a turbulent parametrisation. The theory is developed using a frictional modification of the usual potential flow theory for water waves, the modified Euler equations, coupled to linearised air-flow equations. To assist our analysis we develop a long-wave approximation for the key Miles growth parameter, which explicitly shows how this depends on the water depth and the wind shear parameters. Since our focus is on the development of wave groups, we present a forced nonlinear Schrödinger equation in which the forcing term describes wave growth or decay. For very shallow water the modified Euler equations are reduced to their shallow-water counterpart, which are briefly discussed using Riemann invariants.

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 985 , 25 April 2024 , A2
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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