Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T14:47:08.044Z Has data issue: false hasContentIssue false
  • English
  • Français

Energy transport in a nonlinear and inhomogeneous random gravity wave field

Published online by Cambridge University Press:  29 March 2006

Jürgen Willebrand
Jürgen Willebrand
Affiliation:
Institut für Meereskunde an der Universität Kiel, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Certain tertiary resonant interactions of gravity waves which have been found previously can be obtained more easily by using a simple extension of Whitham's formalism. The contribution of these interactions to the total energy transfer in an inhomogeneous random field of gravity waves is calculated. It is found to be small for open-ocean waves, but to be of some importance for shallow-water waves, where topography or mean shear currents may produce strong inhomogeneities. The nonlinear splitting of the group velocity is found to be unimportant in wave fields with sufficiently smooth spectra.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 70 , Issue 1 , 15 July 1975 , pp. 113 - 126
Copyright
© 1975 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Shortened version of the author's Ph.D. thesis, hereafter cited as I.

References

Bisshop, F. E. & Wilson, R. B. 1968 Averaged stationarity principles. Brown University, Tech. Rep. no. 7.Google Scholar
Bretherton, F. P. & Garrett, C. J. R. 1968 Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. A 302, 529–554.Google Scholar
Dewar, R. L. 1970 Interaction between hydromagnetic waves and a time-dependent inhomogeneous medium. Phys. Fluids, 13, 2710–2720.Google Scholar
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity wave spectrum, Part 1. General theory. J. Fluid Mech. 12, 481–500.Google Scholar
Hasselmann, K. 1968 Weak interaction theory of ocean waves. Basic Developments in Fluid Dynamics, vol. 2, pp. 117–182. Academic.
Hasselmann, K. 1970 Wave refraction in the presence of time-dependent currents and depths. Unpublished manuscript.
Hasselmann, K. 1973 On the spectral dissipation of ocean waves due to white capping. Boundary-Layer Met. 6, 107–127.Google Scholar
Hasselmann, K. & Collins, J. I. 1968 Spectral dissipation of finite depth gravity waves due to turbulent bottom friction. J. Mar. Res. 26, 1–12.Google Scholar
Hasselmann, K. et al. 1973 Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (Jonswap). Dtsch. Hydrogr. Z., Erg. H. Reihe, A12, 1–95.Google Scholar
Lighthill, M. J. 1965 Group velocity. J. Inst. Math. Appl. 1, 1–28.Google Scholar
Longuet-Higgins, M. S. & Phillips, O. M. 1962 Phase velocity effects in tertiary wave interactions. J. Fluid Mech. 12, 333–336.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1961 The changes in amplitude of short gravity waves on steady non-uniform currents. J. Fluid Mech. 10, 529–549.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1962 Radiation stress and mass transport in gravity waves, with application to surf beats. J. Fluid Mech. 13, 481–504.Google Scholar
Longuet-Higgins, M. S. & Turner, J. S. 1974 An ‘entraining plume' model of a spilling breaker. J. Fluid Mech. 63, 1–20.Google Scholar
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204.Google Scholar
Phillips, O. M. 1957 On the generation of waves by turbulent wind. J. Fluid Mech. 2, 417–445.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. J. Fluid Mech. 9, 193–217.Google Scholar
Pierson, W. J. & Moskowitz, L. 1964 A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskij. J. Geophys. Res. 69, 5181–5190.Google Scholar
Schrader, J. P. 1968 Kennzeichnende Seegangsgrößen für drei Meßpunkte in der Elbmündung. Hamburger Küstenforschung, 4, 1–76.Google Scholar
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. Roy. Soc. A 309, 551–575.Google Scholar
Tick, L. J. 1961 Nonlinear probability models of ocean waves. In Ocean Wave Spectra, pp. 163–169. Prentice Hall Inc.
Whitham, G. B. 1965a A general approach to linear and nonlinear dispersive waves using a Lagrangian. J. Fluid Mech. 22, 273–283.Google Scholar
Whitham, G. B. 1965b Nonlinear dispersive waves. Proc. Roy. Soc. A 283, 238–261.Google Scholar
Whitham, G. B. 1967 Nonlinear dispersion of water waves. J. Fluid Mech. 27, 399–412.Google Scholar
Willebrand, J. 1973 Zum Energietransport in einem nichtlinearen raumzeitlich inhomogenen Seegangsfeld. Ph.D. dissertation, Universität Kiel.