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# On the generation of waves by turbulent wind

Abstract

A theory is initiated for the generation of waves upon a water surface, originally at rest, by a random distribution of normal pressure associated with the onset of a turbulent wind. Corrlations between air and water motions are neglected and the water is assumed to be inviscid, so that the motion of the water, starting from rest, is irrotational. It is found that waves develop most rapidly by means of a resonance mechanism which occurs when a component of the surface pressure distribution moves at the same speed as the free surface wave with the same wave-number.

The development of the waves is conveniently considered in two stages, in which the time elapsed is respectively less or greater than the time of development of the pressure fluctuations. An expression is given for the wave spectrum in the initial stage of development (§ 3.2), and it is shown that the most prominent waves are ripples of wavelength λcr = 1·7 cm, corresponding to the minimum phase velocity c = (4gT/ρ)1/4 and moving in directions cos-1(c/Uc) to that of the mean wind, where Uc is the ‘convection velocity’ of the surface pressure fluctuations of length scale λcr or approximately the mean wind speed at a height λcr above the surface. Observations by Roll (1951) have shown the existence under appropriate conditions, of waves qualitatively similar to those predicted by the theory.

Most of the growth of gravity waves occurs in the second, or principal stage of development, which continues until the waves grow so high that non-linear effects become important. An expression for the wave spectrum is derived (§ 4.1), from which follows the result $\overline {\xi ^2} \sim \frac {\overline{p^2}t} {2 \surd 2\rho ^2 U_c g},$ where $\overline {\xi ^2}$ is the mean square surface displacement, $\overline {p ^2}$ the mean square turbulent pressure on the water surface, t the elapsed time, Uc the convection speed of the surface pressure fluctuations, and ρ the water density. This prediction is consistent with published oceanographic measurements (§ 4.3).

It is suggested that this resonance mechanism is more effective than those suggested by Jeffreys (1924, 1925) and Eckart (1953), and may provide the principal means whereby energy is transferred from the wind to the waves.

References
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Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Danel, P. 1956 Proc. IV Journées de l'Hydraulique (Paris).
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Ellison, T. H. 1956 Surveys in Mechanics, p. 400. Cambridge University Press.
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Jeffreys, H. 1925 Proc. Roy. Soc. A, 110, 341.
Keulegan, G. H. 1951 J. Res. Nat. Bur. Stand., Wash., 50, 99.
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MacCready, P. B. 1953a J. Met. 10, 325.
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Roll, H. U. 1951 Ann. Met. (Hamburg) 4, 269.
Sverdrup, H. U. & Munk, W. 1947 Wind, Sea and Swell. Theory of Relations for Forecasting. Publ. Hydrog. Off., Wash., no. 601.
Taylor, R. J. 1952 Quart. J. Roy. Met. Soc. 78, 179.
Taylor, R. J. 1955 Aust. J. Phys. 8, 535.
Ursell, F. 1956 Surveys in Mechanics, p. 216. Cambridge University Press.
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Journal of Fluid Mechanics
• ISSN: 0022-1120
• EISSN: 1469-7645
• URL: /core/journals/journal-of-fluid-mechanics
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