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On the Dependence of the Equilibrium Range of Sea Wave Spectra with Wave Age

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Abstract

Donelan and Pierson have proposed a semiphysical model of the equilibrium sea wave spectrum, based upon a parameterization of wave growth and dissipation terms. Their model is applicable for fully developed seas only. In the framework of Donelan and Pierson's approach, this paper explores the dependence of the equilibrium spectrum upon wave age. To this end, we examine how the dissipation through wave breaking is expected to vary with wave age, according to the approach proposed by Longuet-Higgins in 1969. The constraint imposed by Longuet-Higgins' theory requires an increase of the equilibrium spectrum F(k,0) in the wind direction with increasing inverse wave age U/Cp. This is in accordance with Banner's empirically deduced statement that F(k,0) is proportional to (U/Cp)0.5 in the equilibrium range. Our inferred F(k,0) tends to increase more or less linearly with U/Cp (we find F(k,0) proportional to 1 + 0.25(U/Cp - 0.83), rather than through a power law. If a power law is fitted we obtain F(k,0) approximately proportional to (U/Cp)0.35 for the range 0.83 < U/Cp > 5. Finally, the roughness length of the air-sea interface is inferred from our modelled spectrum through integration of the form drag over wave number under rough conditions. This shows a wave age dependence that is compatible with measurements of wind stress performed in the field at various wave ages.

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References

  • Al-Zanaidi, M. A. and Hui, W. H.: 1984, 'Turbulent Air Flow Over Water Waves - A Numerical Study', J. Fluid Mech. 148, 225-246.

    Google Scholar 

  • Banner, M. L.: 1990, 'Equilibrium Spectra of Wind Waves', J. Phys. Oceanog. 20, 966-984.

    Google Scholar 

  • Banner, M. L., Jones, I. S. F., and Trinder, J. C.: 1989, 'Wavenumber Spectra of Short GravityWaves', J. Fluid Mech. 198, 321-344.

    Google Scholar 

  • Battjes, J. A., Zitman, T. J., and Holthuijsen, L. H.: 1987, 'A Reanalysis of the Spectra Observed in JONSWAP', J. Phys. Oceanog. 17, 1288-1295.

    Google Scholar 

  • Caudal, G.: 1993, 'Self-Consistency BetweenWind Stress, Wave Spectrum, and Wind-Induced Wave Growth for Fully Rough Air-Sea Interface', J. Geophys. Res. 98(C12), 22743-22752.

    Google Scholar 

  • Donelan, M. A.: 1990, 'Air-Sea Interaction', in B. Le Méhauté and D. M. Hanes (eds.), The Sea, J. Wiley and Sons, New York, pp. 239-292.

    Google Scholar 

  • Donelan, M. A., Hamilton, J., and Hui, W. H.: 1985, 'Directional Spectra of Wind Generated Waves', Philos. Trans. R. Soc. London, Ser. A 315, 509-562.

    Google Scholar 

  • Donelan, M. A. and Pierson, Jr., W. J.: 1987, 'Radar Scattering and Equilibrium Ranges in Wind-Generated Waves with Application to Scatterometry', J. Geophys. Res. 92(C5), 4971-5029.

    Google Scholar 

  • Donelan, M. A., Dobson, F. W., Smith, S. D., and Anderson, R. J.: 1993, 'On the Dependence of Sea Surface Roughness on Wave Development', J. Phys. Oceanog. 23(9), 2143-2149.

    Google Scholar 

  • Donelan, M. and Yuan, Y.: 1994, 'Wave Dissipation by Surface Processes', in G. J. Komen, L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann, and P. A. E. M. Janssen (eds.), Dynamics and modelling of ocean waves, Cambridge University Press, Cambridge, pp. 143-155.

    Google Scholar 

  • Forristall, G. Z.: 1981, 'Measurements of a Saturated Range in Ocean Wave Spectra', J. Geophys. Res. 86, 8075-8084.

    Google Scholar 

  • Glazman, R. E.: 1987, 'Wind-Fetch Dependence of Seasat Scatterometer Measurements', Int. J. Remote Sens. 8, 1641-1647.

    Google Scholar 

  • Hasselmann, K.: 1974, 'On the Spectral Dissipation of Ocean Waves Due to Whitecapping', Boundary-Layer Meteorol. 6, 107-127.

    Google Scholar 

  • Hasselmann, K., Janssen, P. A. E. M., and Komen, G. J.: 1994, 'Wave-Wave Interaction', in G. J. Komen, L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann, and P. A. E. M. Janssen (eds.), Dynamics andModelling of Ocean Waves, Cambridge University Press, Cambridge, pp.113-143.

    Google Scholar 

  • Hsiao, S. V. and Shemdin, O. H.: 1983, 'Measurements of Wind Velocity and Pressure With a Wave Follower During MARSEN', J. Geophys. Res. 88(C14), 9841-9849.

    Google Scholar 

  • Hsu, S. A.: 1974, 'A Dynamical Roughness Equation and Its Application to Wind Stress Determination at the Air-Sea Interface', J. Phys. Oceanog. 4, 116-120.

    Google Scholar 

  • Janssen, P. A. E. M.: 1989, 'Wave-Induced Stress and the Drag of Air Flow Over Sea Waves', J. Phys. Oceanog. 19, 745-754.

    Google Scholar 

  • Kahma, K. K.: 1981, 'A Study of the Growth of the Wave Spectrum With Fetch', J. Phys. Oceanog. 11, 1503-1515.

    Google Scholar 

  • Kawai, S., Okuda, K., and Toba, Y.: 1977, 'Field Data Support of Three Seconds Power Law and gu * ω-4 Spectral Form for Growing Wind Waves', J. Oceanogr. Soc. Japan 33, 137-150.

    Google Scholar 

  • Komen, G. J., Hasselmann, S., and Hasselmann, K.: 1984, 'On the Existence of Fully Developed Wind-Sea Spectrum', J. Phys. Oceanog. 14, 1271-1285.

    Google Scholar 

  • Lamb, H.: 1932, Hydrodynamics, 6th. ed., Cambridge University Press, Cambridge, reprinted by Dover Publications, New York, 1945, 738 pp.

    Google Scholar 

  • Larson, T. R. and Wright, J. W.: 1975, 'Wind-Generated Gravity-Capillary Waves: Laboratory Measurements of Temporal Growth Rates Using Microwave Backscatter', J. FluidMech. 70, 417-436.

    Google Scholar 

  • Leykin, I. A. and Rozenberg, A. D.: 1984, 'Sea-Tower Measurements of Wind-Wave Spectra in the Caspian Sea', J. Phys. Oceanog. 14(1), 168-176.

    Google Scholar 

  • Longuet-Higgins, M. S.: 1969, 'On Wave Breaking and the Equilibrium Spectrum of Wind-Generated Waves', Proc. Roy. Soc. London A310, 151-159.

    Google Scholar 

  • Makin, V. K., Kudryavtsev, V. N., and Mastenbroek, C.: 1995, 'Drag of the Sea Surface', Boundary-Layer Meteorol. 73, 159-182.

    Google Scholar 

  • Ochi, M. K. and Tsai, C. H.: 1983, 'Prediction of Occurrence of Breaking Waves in Deep Water', J. Phys. Oceanog. 13, 2008-2019.

    Google Scholar 

  • Phillips, O. M.: 1958, 'The Equilibrium Range in the Spectra of Wind-Generated Waves', J. Fluid Mech. 4, 426-434.

    Google Scholar 

  • Phillips, O. M.: 1980, The Dynamics of the Upper Ocean, 2nd. ed., Cambridge University Press, Cambridge, 336 pp.

    Google Scholar 

  • Plant, W. J.: 1982, 'A Relationship Between Wind Stress and Wave Slope', J. Geophys. Res. 87(C3), 1961-1967.

    Google Scholar 

  • Smith, S. D., Anderson, R. J., Oost, W. A., Kraan, C., Maat, N., DeCosmo, J., Katsaros, K. B., Davidson, K. L., Bumke, K., Hasse, L., and Chadwick, H. M.: 1992, 'Sea Surface Wind Stress and Drag Coefficients: The HEXOS Results', Boundary-Layer Meteorol. 60, 109-142.

    Google Scholar 

  • Snyder, R. L., Dobson, F. W., Elliot, J. A., and Long, R. B.: 1981, 'Array Measurements of Atmospheric Pressure Fluctuations Above Surface GravityWaves', J. Fluid Mech. 102, 1-59.

    Google Scholar 

  • Toba, Y.: 1973, 'Local Balance in the Air-Sea Boundary Processes. Part III: On the Spectrum of Wind Waves', J. Oceanog. Soc. Japan 29, 209-220.

    Google Scholar 

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  1. Centre détude des Environnements Terrestre et Planétaires (CETP), 10-12 Avenue de l'Europe, 78140, Vélizy, France

    Gerard Caudal

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Caudal, G. On the Dependence of the Equilibrium Range of Sea Wave Spectra with Wave Age. Boundary-Layer Meteorology 86, 29–45 (1998). https://doi.org/10.1023/A:1000553829844

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