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Spectral and statistical properties of the equilibrium range in wind-generated gravity waves

  • O. M. Phillips (a1)

Recent measurements of wave spectra and observations by remote sensing of the sea surface indicate that the author's (1958) conception of an upper-limit asymptote to the spectrum, independent of wind stress, is no longer tenable. The nature of the equilibrium range is reexamined, using the dynamical insights into wave–wave interactions, energy input from the wind and wave-breaking that have been developed since 1960. With the assumption that all three of these processes are important in the equilibrium range, the wavenumber spectrum is found to be of the form $(\cos\theta)^p u_{*} g^{-\frac{1}{2}}k^{-\frac{7}{2}}$, where p ∼ ½ and the frequency spectrum is proportional to u*gσ−4. These forms have been found by Kitaigorodskii (1983) on a quite different dynamical basis; the latter is consistent with the form found empirically by Toba (1973) and later workers. Various derived spectra, such as those of the sea-surface slope and of an instantaneous line traverse of the surface, are also given, as well as directional frequency spectra and frequency spectra of slope.

The theory also provides expressions for the spectral rates of action, energy and momentum loss from the equilibrium range by wave-breaking and for the spectrally integrated rates across the whole range. These indicate that, as a wave field develops with increasing fetch or duration, the momentum flux to the underlying water by wave-breaking increases asymptotically to a large fraction of the total wind stress and that the energy flux to turbulence in the water, occurring over a wide range of scales, increases logarithmically as the extent of the equilibrium range increases. Interrelationships are pointed out among different sets of measurements such as the various spectral levels, the directional distributions, the total mean-square slope and the ratio of downwind to crosswind mean-square slopes.

Finally, some statistical characteristics of the breaking events are deduced, including the expected length of breaking fronts (per unit surface area) with speeds of advance between c and c+dc and the number of such breaking events passing a given point per unit time. These then lead to simple expressions for the density of whitecapping, those breaking events that produce bubbles and trails of foam, the total number of whitecaps passing a given point per unit time and, more tenuously, the whitecap coverage.

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Banner, M. L. & Melville, W. K. 1976 On the separation of air flow over water waves. J. Fluid Mech. 77, 825842.
Beal, R. C., DeLeonibus, P. S. & Katz, I. 1981 Spaceborne Synthetic Aperture Radar for Oceanography. The Johns Hopkins Press.
Cox, C. S. 1958 Measurements of slopes of high-frequency wind waves. J. Mar. Res. 16, 199225.
Donelan, M. A., Hamilton, J. & Hui, W. H. 1984 Directional spectra of wind-generated waves. Phil. Trans. R. Soc. Lond. A (in press).
Duncan, J. H. 1981 An experimental investigation of breaking waves produced by a towed hydrofoil. Proc. R. Soc. Lond. A 377, 331348.
Forristall, G. Z. 1981 Measurements of a saturated range in ocean wave spectra. J. Geophys. Res. 86, 80758084.
Fox, M. J. H. 1976 On the nonlinear transfer of energy in the peak of a gravity-wave spectrum. II. Proc. R. Soc. Lond. A 348, 467483.
Gent, P. R. & Taylor, P. A. 1976 A numerical model of the air flow above water waves. J. Fluid Mech. 77, 105128.
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity wave spectrum. Part 1. J. Fluid Mech. 12, 481500.
Hasselmann, K. 1963 On the non-linear energy transfer in a gravity wave spectrum. Parts 2 and 3. J. Fluid Mech. 15, 273281; 385–398.
Hasselmann, K. Et Al. 1973 Measurements of wind wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Herausgegeben vom Deutsch-Hydrograph. Inst., Reihe A, no. 12.
Kahma, K. K. 1981 A study of the growth of the wave spectrum with fetch. J. Phys. Oceanogr. 11, 15031515.
Kawai, S., Okuda, K. & Toba, Y. 1977 Field data support of three-seconds power law and. gu*sG−4 spectral form for growing wind waves. J. Oceanogr. Soc. Japan 33, 13750.
Keulegan, J. H. 1951 Wind tides in small closed channels. J. Res. Natl Bur. Stand. 46, 358381.
Kitaigorodskii, S. A. 1983 On the theory of the equilibrium range in the spectrum of wind-generated gravity waves. J. Phys. Oceanogr. 13, 816827.
Kitaigorodski, S. A. 1984 On the fluid dynamical theory of turbulent gas transfer across an air-sea interface in the presence of breaking waves. J. Phys. Oceanogr. 14, 960972.
Komen, G. J., Hasselmann, S. & Hasselmann, K. 1984 On the existence of a fully developed wind-sea spectrum. J. Phys. Oceanogr. 14, 12711285.
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. Lond. A 347, 311328.
Longuet-Higgins, M. S., Cartwright, D. E. & Smith, N. D. 1963 Observations of the directional spectra of sea waves using the motion of a floating buoy. In Ocean Wave Spectra, pp. 111136. Prentice-Hall.
Longuet-Higgins, M. S. & Cokelet, E. D. 1978 Growth of normal mode instabilities. Proc. R. Soc. Lond. A 364, 128.
Miles, J. W. 1957 On the generation of surface waves by shear flow. J. Fluid Mech. 3, 185204.
Mitsuyasu, H. Et Al. 1975 Observations of the directional spectra of ocean waves using a cloverleaf buoy. J. Phys. Oceanogr. 5, 750760.
Mitsuyasu, H. Et Al. 1980 Observation of the power spectrum of ocean waves using a cloverleaf buoy. J. Phys. Oceanogr. 10, 286296.
Mitsuyasu, H. & Honda, T. 1982 Wind-induced growth of water waves. J. Fluid Mech. 123, 425442.
Monahan, E. C. 1971 Oceanic whitecaps. J. Phys. Oceanogr. 1, 139144.
Monahan, E. C. & Muircheartaigh, I. 1980 Optimal power-law description of oceanic whitecap coverage dependence on wind speed. J. Phys. Oceanogr. 10, 20942099.
Phillips, O. M. 1958 The equilibrium range in the spectrum of wind-generated ocean waves. J. Fluid Mech. 4, 426434.
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, 465485.
Phillips, O. M. 1984 On the response of short ocean wave components at a fixed wave number to ocean current variations. J. Phys. Oceanogr. 14, 14251433.
Phillips, O. M. & Banner, M. L. 1974 Wave breaking in the presence of wind drift and swell. J. Fluid Mech. 66, 625640.
Pierson, W. J. (ed.) 1962 The directional spectrum of a wind-generated sea as determined from data obtained by the Stereo Wave Observation Project. Coll. Engng NYU Met. Pap. 2, no. 6.
Plant, W. J. 1982 A relationship between wind stress and wave slope. J. Geophys. Res. 87, 19611967.
Ramamonjiarisoa, A. & Coantic, M. 1976 Loi expérimental de dispersion des vagues produites par le vent sur une faible longueur d'action. C. R. Acad. Sci. Paris B 282, 111113.
Schule, J. J., Simpson, L. S. & DeLeonibus, P. S. 1971 A study of fetch limited wave spectra with an airborne laser. J. Geophys. Res. 76, 41604171.
Sell, W. & Hasselmann, K. 1972 Computation of nonlinear energy transfer for JONSWAP and empirical wind wave spectra. Rep. Inst. Geophys., Univ. Hamburg.
Snyder, R. L., Dobson, F. W., Elliott, J. A. & Long, R. B. 1981 Array measurements of atmospheric pressure fluctuations above surface gravity waves. J. Fluid Mech. 102, 159.
Tang, C. C. H. 1974 The effect of droplets in the air-sea transition zone on the mean brightness temperature. J. Phys. Oceanogr. 4, 579593.
Tang, S. & Shemdin, O. H. 1983 Measurements of high frequency waves using a wave follower. J. Geophys. Res. 88, 98329840.
Toba, Y. 1973 Local balance in the air-sea boundary processes. III. On the spectrum of wind waves. J. Oceanogr. Soc. Japan 29, 209220.
Wu, J. 1979 Oceanic whitecaps and sea state. J. Phys. Oceanogr. 9, 10641068.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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