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Modulation of the growth rate of short surface capillary–gravity wind waves by a long wave

Published online by Cambridge University Press:  26 April 2006

Yu. I. Troitskaya
Yu. I. Troitskaya
Affiliation:
Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia
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Abstract

Modulation of the growth rate of short capillary–gravity surface wind waves in the presence of a long wave with steepness much smaller than the maximum is studied theoretically. The Miles (1962) mechanism taking into account the viscous wave stresses in the air flow is considered to be the main process of short-wave generation. The short-wave growth rate is defined by the wind velocity gradient in the viscous sublayer of the logarithmic boundary layer. The long wave propagating on the wave surface induces an additional component of the wind velocity gradient oscillating with the length and time periods of the long wave, which results in modulation, with the same period, of the growth rate of the short wave riding on the long one. The growthrate modulation amplitude depends on the parameter M being of the order of the relation between the oscillating and the mean wind velocity gradients in the viscous sublayer \[M=\frac{2kac}{u^2_*}(ckv_{\alpha})^{1/2} \] (where c, k, a are the phase velocity, the wavenumber and the elevation amplitude of the long wave; va is the viscosity coefficient in the air; u* is the wind friction velocity). When M = O(1) (weak winds and long waves) the oscillating component of the shortwave growth rate is of the same order as the mean one. If M is much smaller than unity, then the relative amplitude of the growth rate is of the same order as the steepness of the long wave.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 273 , 25 August 1994 , pp. 169 - 187
Copyright
© 1994 Cambridge University Press

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References

Benjamin, T. B. 1959 Shearing flow over a wave boundary. J. Fluid Mech. 6, 513–532.Google Scholar
Forsythe, G. E. & Moler, C. B. 1967 Computer Solution of Linear Algebraic Systems. Prentice-Hall.
Landahl, M. T., Widnall, S. E. & Hultgen, L. 1978 An interactional mechanism between large and small scales for wind-generation water waves. In Proc. 12th Symp. on Naval Hydrodynamics, p. 541. National Academy of Sciences.
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535–581.Google Scholar
Longuet-Higgins, M. S. 1963 The generation of capillary waves by steep gravity waves. J. Fluid Mech. 16, 138–159.Google Scholar
Longuet-Higgins, M. S. 1987 The propagation of short surface waves on longer gravity waves. J. Fluid Mech. 177, 293–306.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1960 Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8, 565–583.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1961 The changes in the form of short gravity waves on steady, non-uniform currents. J. Fluid Mech. 10, 529–549.Google Scholar
Miles, J. W. 1962 On the generation of surface waves by shear flows. Part 4. J. Fluid Mech. 13, 433–448.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.
Phillips, O. M. 1981 The dispersion of short wavelets in the presence of a dominant long wave. J. Fluid Mech. 107, 465–485.Google Scholar
Ruvinsky, K. D., Feldstein, F. I. & Friedman, G. I. 1991 Numerical simulation of the quasistationary stage of ripple excitation by steep gravity–capillary waves. J. Fluid Mech. 230, 339–354.Google Scholar
Schlichting, H. 1955 Boundary layer theory. Pergamon.
Shyu, J.-H. & Phillips, O. M. 1990 The blockage of gravity and capillary waves by longer waves and currents. J. Fluid Mech. 217, 115–141.Google Scholar
Valenzuela, G. R. 1976 Growth of gravity–capillary waves in a shear flow. J. Fluid Mech. 76, 229–250.Google Scholar
Valenzuela, G. R. & Wright, J. W. 1979 Modulation of short gravity–capillary waves by longerscale periodic flows. – A higher-order theory. Radio Sci. 14, 1099–1110.Google Scholar