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Kinetic equations in a third-generation spectral wave model

Published online by Cambridge University Press:  22 January 2021

Qingxiang Liu Open the ORCID record for Qingxiang Liu [Opens in a new window] ,
Odin Gramstad Open the ORCID record for Odin Gramstad [Opens in a new window]  and
Alexander Babanin Open the ORCID record for Alexander Babanin [Opens in a new window]
Qingxiang Liu*
Affiliation:
Department of Infrastructure Engineering, University of Melbourne, VIC 3010, Australia
Odin Gramstad
Affiliation:
Group Technology and Research, DNV GL, 1363 Høvik, Norway
Alexander Babanin
Affiliation:
Department of Infrastructure Engineering, University of Melbourne, VIC 3010, Australia Laboratory for Regional Oceanography and Numerical Modeling, National Laboratory for Marine Science and Technology, 266237 Qingdao, PR China
*
†Email addresses for correspondence: qingxiang.liu@unimelb.edu.au, qxiangliu@gmail.com
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Abstract

The Hasselmann kinetic equation (HKE) forms the cornerstone of present-day spectral wave models. It describes the redistribution of energy over the wave spectrum as a result of resonant four-wave interactions, and theoretically prescribes wave evolution on a slow $O(1/\varepsilon ^4\omega _0)$ time scale, where $\varepsilon$ and $\omega _0$ are typical wave steepness and frequency. Alternatives to the HKE (e.g. the generalized kinetic equation (GKE)), including the effects of non-resonant four-wave interactions, are believed capable of evolving wave fields on a fast $O(1/\varepsilon ^2\omega _0)$ time scale. It is beyond doubt that these alternatives could reasonably predict changes of unidirectional waves whereas the HKE cannot. For angular spread wave fields, however, it is still ambiguous whether the GKE behaves remarkably differently from the HKE because previous research in this direction was not fully consistent. In this study, we revised the GKE algorithm implemented in the spectral wave model WAVEWATCH III (WW3) by correcting two numerical aspects related to the discretization of the GKE and to the source term integration. It is proved that once updated, the GKE in WW3 does not give rise to significant deviation from the HKE-based results, provided that the wave spectra are fairly smooth and the directionality is sufficiently broad. These results, although unexpected, are in good agreement with findings reported by Annenkov & Shrira. More strikingly, the HKE and GKE are observed to operate at the same fast $O(1/\varepsilon ^2\omega _0)$ time scale for the spectral peak downshift and angular broadening, indicating that the HKE, solved by the well-established Webb–Resio–Tracy algorithm, seems more robust than usually expected.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A50
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© The Author(s), 2021. Published by Cambridge University Press

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References

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