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Can swell increase the number of freak waves in a wind sea?

Published online by Cambridge University Press:  19 March 2010

ODIN GRAMSTAD  and
KARSTEN TRULSEN
ODIN GRAMSTAD
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway
KARSTEN TRULSEN*
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway
*
†Email address for correspondence: [email protected]
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Abstract

The effect of a swell on the statistical distribution of a directional short-wave field is investigated. Starting from Zakharov's spectral formulation, we derive a new modified nonlinear Schrödinger equation appropriate for the nonlinear evolution of a narrow-banded spectrum of short waves influenced by a swell. The swell-modified equation is solved analytically to yield an extended version of the result of Longuet-Higgins & Stewart (J. Fluid Mech., vol. 8, no. 4, 1960, pp. 565–583) for the modulation of a short wave riding on a longer wave. Numerical Monte Carlo simulations of the long-term evolution of a spectrum of short waves in the presence of a monochromatic swell are employed to extract statistical distributions of freak waves among the short waves. We find evidence that a realistic short-crested wind sea can on average experience a small increase in freak wave probability because of a swell provided the swell is not orthogonal to the wind waves. For orthogonal swell and wind waves we find evidence that there is almost no significant change in the probability of freak waves in the wind sea. If the short waves are unrealistically long crested, such that the Benjamin–Feir index serves as indicator for freak waves (Gramstad & Trulsen, J. Fluid Mech., vol. 582, 2007, pp. 463–472), it appears that the swell has much smaller relative influence on the probability of freak waves than in the short-crested case.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 650 , 10 May 2010 , pp. 57 - 79
Copyright
Copyright © Cambridge University Press 2010

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