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A similarity relation for the nonlinear energy transfer in a finite-depth gravity-wave spectrum

Published online by Cambridge University Press:  19 April 2006

K. Herterich  and
K. Hasselmann
K. Herterich
Affiliation:
Max-Planck-Institut für Meteorologie, Hamburg
K. Hasselmann
Affiliation:
Max-Planck-Institut für Meteorologie, Hamburg
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Abstract

The energy transfer in a finite-depth gravity-wave spectrum is investigated in the approximation of a narrow spectrum. It is shown that for ocean depths larger than approximately one tenth of the wavelength (kh [ges ] 0·7) the finite-depth case can be reduced to Longuet-Higgins’ (1976) result for an infinitely deep ocean by a similarity transformation involving changes in scale of the angular spreading function and the transfer rate. For shallower water (kh < 0·7) Longuet-Higgins’ expansion technique is no longer applicable without modification, as the nonlinear coupling coefficient develops a discontinuity at the origin of the expansion. In the range kh [ges ] 0·7 both the magnitude and the two-dimensional frequency-directional distribution of the energy transfer are found not to differ significantly (to within variations by a factor of 2) from the case of an infinitely deep ocean. The transformation rules relating the infinite-depth and finite-depth cases may provide a useful guide for constructing parametrizations of the nonlinear transfer for finite-depth wave prediction models.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 97 , Issue 1 , 11 March 1980 , pp. 215 - 224
Copyright
© 1980 Cambridge University Press

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References

Bouws, E. 1978 Wind and wave climate in the Netherlands sector of the North Sea between 53° and 54° north latitude. KNMI, Scientific Rep. W.R. 78-9.
Fox, M. J. H. 1976 On the nonlinear transfer of energy in the peak of a gravity-wave spectrum. II. Proc. Roy. Soc. A 348, 467–83.Google Scholar
Hasselmann, K. 1961 On the nonlinear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech. 12, 481–500.Google Scholar
Hasselmann, K. 1963a On the nonlinear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility. J. Fluid Mech. 15, 273–281.Google Scholar
Hasselmann, K. 1963b On the nonlinear energy transfer in a gravity-wave spectrum. Part 3. Evaluation of the energy flux and swell-sea interaction for a Neumann spectrum. J. Fluid Mech. 15, 385–398.Google Scholar
Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright, D. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E., Kruseman, P., Meerburg, A., Müller, P., Olbers, D. J., Richter, K., Sell, W., Walden, H. 1973 Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Deutsche Hydrograph. Zeit., Ergänzungsheft Reihe A (8°), Nr. 12.
Hasselmann, K., Ross, D. B., Müller, P. & Sell, W. 1976 A parametric wave prediction model. J. Phys. Oceanogr. 6, 200–228.
Kitaigorodskii, S. A., Krasitskii, V. P. & Zaslavskii, M. M. 1975 On Phillips’ theory of equilibrium range in the spectra of wind generated gravity waves. J. Phys. Oceanogr. 5, 410–420.Google Scholar
Longuet-Higgins, M. S. 1976 On the nonlinear transfer of energy in the peak of a gravity-wave spectrum: a simplified model. Proc. Roy. Soc. A 347, 311–328.Google Scholar
Mitsuyasu, H. 1968 On the growth of the spectrum of wind-generated waves. 1. Rep. Res. Inst. Appl. Mech., Kyushu Univ. 16, 459–482.Google Scholar
Mitsuyasu, H. 1969 On the growth of the spectrum of wind-generated waves. 2. Rep. Res. Inst. Appl. Mech., Kyushu Univ. 17, 235–248.Google Scholar
Sell, W. & Hasselmann, K. 1973 Computations of nonlinear energy transfer for JONSWAP and empirical wind-wave spectra. Rep. Inst. Geophys., Univ. Hamburg.
Webb, D. J. 1978 Nonlinear transfer between sea waves. Deep-Sea Res. 25, 279–298.Google Scholar