Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-20T05:16:56.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Multi-wave interaction theory for wind-generated surface gravity waves

Published online by Cambridge University Press:  26 April 2006

Roman E. Glazman
Roman E. Glazman
Affiliation:
Jet Propulsion Laboratory. California Institute of Technology, Pasadena. CA 91109. USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Consistently employing the assumption of localness of wave–wave interactions in the wavenumber space, the Kolmogorov treatment of the energy cascade is applied to the case of wind-generated surface gravity waves. The effective number v of resonantly interacting wave harmonics is not limited to four but is found as a solution of a coupled system of equations expressing: (i) the dependence of the spectrum shape on the degree of the wave nonlinearity, and (ii) the continuity of the wave action flux through the spectrum given a continuous positive input from wind. The latter is specified in a Miles-type fashion, and a simple scaling relationship based on the concept of the turnover time is derived in place of the kinetic equation. The mathematical problem is reduced to an ordinary differential equation of first order. The exponent in the ‘power law’ for the spectral density of the wave potential energy and the effective number of resonantly interacting wave harmonics are found as functions of the wave frequency and of external factors of wind—wave interaction. The solution is close to the Zakharov—Filonenko spectrum at low frequencies and low wind input while approaching the Phillips spectrum at high frequencies and sufficiently high wind.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 243 , October 1992 , pp. 623 - 635
Copyright
© 1992 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.)

References

Donelan, M. A., Hamilton, J. & Hui, W. H. 1985 Directional spectra of wind-generated waves.. Phil. Trans. R. Soc. Lond. A 315, 509–562.Google Scholar
Forristall, Z. 1981 Measurements of a saturated range in ocean wave spectra. J. Geophys. Res. 86, 8075–8084.Google Scholar
Frisch, U., Sulem, P.-L. & Nelkin, M. 1978 A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87, 719–736.Google Scholar
Glazman, R. E. 1986 Statistical characterization of sea surface geometry for a wave slope field discontinuous in the mean square. J. Geophys. Res. 91, 6629–6641.Google Scholar
Glazman, R. E. 1990 Near-nadir radar backscatter from a well-developed sea. Radio Sci. 25, 1211–1219.Google Scholar
Glazman, R. E. 1991 Statistical problems of wind-generated gravity waves arising in microwave remote sensing of surface winds. Trans. Geosci. Rem. Sensing 29, 135–142.Google Scholar
Glazman, R. E., Pihos, G. G. & Ip, J. 1988 Scatterometer wind-speed bias induced by the large-scale component of the wave field. J. Geophys. Res. 93, 1317–1328.Google Scholar
Glazman, R. E. & Srokosz, M. A. 1991 Equilibrium wave spectrum and sea state bias in satellite altimetry. J. Phys. Oceanogr. 21, 1609–1621.Google Scholar
Glazman, R. E. & Weichman, P. 1989 Statistical geometry of a small surface patch in a developed sea. J. Geophys. Res. 94, 4998–5010.Google Scholar
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity-wave spectrum. J. Fluid Mech. 12, 481–500.Google Scholar
Kitaigorodskii, S. A. 1983 On the theory of the equilibrium range in the spectrum of wind-generated gravity waves. J. Phys. Oceanogr. 13, 816–827.Google Scholar
Klemens, P. G. 1965 Effect of thermal and phonon processes on ultrasonic attenuation. In Physical Acoustics (ed. W. P. Mason), pp. 201–236, vol. IIIB. Academic.
Kolmogorov, A. N. 1941 Dissipation of energy in locally isotropic turbulence. C.R. Acad. Sci. USSR 32, 16.Google Scholar
Kraichnan, R. H. 1972 In Statistical Mechanics: New Concepts, New Problems, New Applications (ed. S. A. Rice, K. F. Freed & J. C. Light), p. 201. University of Chicago.
Larraza, A. 1987 Universal power spectra for wave turbulence: applications to wind waves, flicker noise, solar wind spectrum, and classical second sound. Ph.D. thesis, University of California at Los Angeles.
Larraza, A., Garrett, S. L. & Putterman, S. 1990 Dispersion relations for gravity waves in a deep fluid: second sound in a stormy sea.. Phys. Rev. A 41, 3144–3155.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. 1958 The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mech. 4, 426–434.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean, 2nd edn. Cambridge University Press.
Phillips, O. M. 1985 Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech. 156, 505–531.Google Scholar
Plant, W. J. 1982 A relationship between wind stress and wave slope. J. Geophys. Res. 87, 1961–1967.Google Scholar
Snyder, R. L., Obson, F. W., Elliot, J. A. & Long, R. B. 1981 Array measurements of atmospheric pressure fluctuations above surface gravity waves. J. Fluid Mech. 102, 1–59.Google Scholar
West, B. J., Brueckner, K. A., Janda, R. S., Milder, D. M. & Milton, R. L. 1987 A new numerical method for surface hydrodynamics. J. Geophys. Res. 92, 11803–11824.Google Scholar
Zakharov, V. E. 1984 Kolmogorov spectra in weak turbulence problems. In Handbook of Plasma Physics (ed. M. N. Rosenbluth & R. Z. Sagdeev), Vol. 2: Basic Plasma Physics II, Ch. 5.1, pp. 3–36. Elsevier.
Zakharov, V. E. & Filonenko, N. N. 1966 The energy spectrum for stochastic oscillation of a fluid's surface. Dokl. Acad. Nauk SSSR 170, 1292–1295 (in Russian).Google Scholar
Zakharov, V. E. & L'vov, V. S. 1975 Statistical description of nonlinear wave fields. Radiophys. Quantum Electron. 18, 1084–1097 (English transl.).Google Scholar
Zakharov, V. E. & Zaslavskii, M. M. 1982 The kinetic equation and Kolmogorov spectra in the weak turbulence theory of wind waves. Izv. Atmos. Ocean. Phys. 18, 747–753 (English transl.).Google Scholar