The shallow water limit of the Zakharov Equation and consequences for (freak) wave prediction.

TitleThe shallow water limit of the Zakharov Equation and consequences for (freak) wave prediction.
Publication TypeMiscellaneous
Year of Publication2005
AuthorsJanssen, PAEM, Onorato, M
Secondary TitleTechnical Memorandum
Number464
Pagination15
Date PublishedMay
PublisherECMWF
Place PublishedShinfield Park, Reading
Abstract

Finite amplitude deep-water waves are subject to nonlinear focussing, which when the phases are right, may give rise to giant waves or freak waves. The same process is responsible for the Benjamin-Feir instability. In shallow water, finite amplitude surface-gravity waves generate a current and deviations from the mean surface elevation. This stabilizes the Benjamin-Feir instability and the process of nonlinear focussing ceases to exist when kh < 1.363. This is a well-known property of surface gravity waves (Benjamin, 1967; Whitham, 1974) and, here it is shown for the first time, that the usual starting point for wave evolution studies, namely the Zakharov equation, shares this property as well. Consequences for (freak) wave prediction are pointed out.