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The Zakharov equation with separate mean flow and mean surface

Published online by Cambridge University Press:  05 January 2014

Odin Gramstad
Odin Gramstad*
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway Centre for Ocean Engineering Science and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
*
†Email address for correspondence: oding@math.uio.no
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Abstract

Using the Hamiltonian approach of Krasitskii (J. Fluid Mech., vol. 272, 1994, pp. 1–20), we derive a variant of the Zakharov equation in which the wave-induced mean surface elevation and the surface potential of the wave-induced mean flow are represented as separate variables governed by separate evolution equations. The kernel function of this new variant is simpler, and in particular also well defined in the uniform-wave-train limit for waves on finite depth. This form of the Zakharov equation may be advantageous in some applications. One example is the derivation of nonlinear Schrödinger equations in the narrow-band limit, where the handling of the mean flow and mean surface is significantly simpler than when starting from the original Zakharov equation. In this paper we have used the alternative form of the Zakharov equation to derive a Hamiltonian nonlinear Schrödinger equation for directional waves on arbitrary depth, valid to one order higher in bandwidth than the Hamiltonian equation recently presented by Craig, Guyenne and Sulem (Wave Motion, vol. 47, 2010, pp. 552–563).

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 740 , 10 February 2014 , pp. 254 - 277
Copyright
© 2014 Cambridge University Press 

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