Journal of Oceanography

, Volume 52, Issue 4, pp 509–537 | Cite as

A new scheme of nonlinear energy transfer among wind waves: RIAM method-algorithm and performance-

  • Kosei Komatsu
  • Akira Masuda


A numerical scheme for calculating the nonlinear energy transfer among wind waves (RIAM method) was developed on the basis of the rigorous method of Masuda. Then the performance of the RIAM method was examined by applying it to various forms of wind-wave spectra and different situations of wind-wave evolution, in comparison mainly with the WAM method. The computational time of the Masuda method was reduced by a factor of 300 by the RIAM method, which is still 2000 times slower than the WAM method simply because the RIAM method processes thousands of resonance configurations whereas the WAM method does only one. The RIAM method proves to give accurate results even for spectra of narrow band widths or bimodal spectra, whereas the WAM method often calculates an unrealistic magnitude and pattern of nonlinear energy transfer functions. In the duration-limited evolution of wind-wave spectra, the RIAM method yields a unimodal directional distribution on the low-frequency side of the spectral peak, whereas the WAM method produces a spurious bimodal one there. At higher frequencies, however, both methods give a bimodal directional distribution with two oblique maxima. The RIAM method enhances the growth of the total energy and peak period of wind waves in comparison with the WAM method. Nevertheless, Toba's constant of his 3/2-power law approaches almost the same standard value of 0.06 in both methods. For spectra of a narrow band width or for those perturbed by a small hump or depression, the RIAM method tends to recover the monotonic smoother form of spectrum whereas the WAM method often yields unrealistic humps or depressions.


Depression Transfer Function Accurate Result Numerical Scheme Spectral Peak 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Banner, M. L. and I. R. Young (1994): Modeling spectral dissipation in the evolution of wind waves. Part I: Assessment of existing model performance.J. Phys. Oceanogr.,24, 1550–1571.CrossRefGoogle Scholar
  2. Barnett, T. P. and A. J. Sutherland (1968): A note on an overshoot effect in wind-generated waves.J. Geophys. Res.,73, 6879–6885.Google Scholar
  3. Donelan, M. A., J. Hamilton and W. H. Hui (1985): Directional spectra of wind-generated waves.Phil. Trans. Roy. Soc. London,A315, 509–562.Google Scholar
  4. Dungey, J. C. and W. H. Hui (1979): Nonlinear energy transfer in a narrow gravity-wave spectrum.Proc. Roy. Soc. London,A368, 239–265.Google Scholar
  5. Fox, M. J. (1976): On the nonlinear transfer of energy in the peak of a gravity-wave spectrum II.Proc. Roy. Soc. London,A348, 467–483.Google Scholar
  6. Hasselmann, D. E., M. Dunckel and J. A. Ewing (1980): Directional wave spectra observed during JONSWAP 1973.J. Phys. Oceanogr.,10, 1264–1280.CrossRefGoogle Scholar
  7. Hasselmann, K. (1962): On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory.J. Fluid Mech.,12, 481–500.Google Scholar
  8. Hasselmann, K. (1963): On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility.J. Fluid Mech.,15, 273–281.Google Scholar
  9. Hasselmann, K., T. P. Barnett, H. Bouws, H. Carlson, D. E. Cartright, K. Enke, J. A. Ewing, H. Gineapp, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. Muller, D. J. Olbers, K. Richter, W. Sell and H. Walden (1973): Measurements of wind wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP).Dtsch. Hydrogr. Z.,Suppl.,A 8, No. 12, 95 pp.Google Scholar
  10. Hasselmann, S. and K. Hasselmann (1981): A symmetrical method of computing the nonlinear transfer in a gravity wave spectrum.Hamb. Geophys. Einzelschriften, Reihe A: Wiss. Abhand.,52, 138 pp.Google Scholar
  11. Hasselmann, S. and K. Hasselmann (1985): Computations and parameterizations of the nonlinear energy transfer in a gravity-wave spectrum. Part I: A new method for efficient computations of the exact nonlinear transfer integral.J. Phys. Oceanogr.,15, 1369–1377.CrossRefGoogle Scholar
  12. Hasselmann, S., K. Hasselmann, J. H. Allender and T. P. Barnett (1985): Computations and parameterizations of the nonlinear energy transfer in a gravity-wave spectrum. Part II: Parameterizations of the nonlinear energy transfer for application in wave models.J. Phys. Oceanogr.,15, 1378–1391.CrossRefGoogle Scholar
  13. Komatsu, K., T. Kusaba and A. Masuda (1993): An efficient method for computing nonlinear energy transfer among wind waves.Bull. Res. Inst. Appl. Mech. Kyushu Univ.,75, 121–146 (in Japanese).Google Scholar
  14. Komen, G. J., S. Hasselmann and K. Hasselmann (1984): On the existence of a fully developed wind-sea spectrum.J. Phys. Oceanogr.,14, 1271–1285.CrossRefGoogle Scholar
  15. Kusaba, T. and A. Masuda (1988): Wind-wave spectra based on the hypothesis of local equilibrium.J. Oceanogr. Soc. Japan,45, 45–64.CrossRefGoogle Scholar
  16. Longuet-Higgins, M. S. (1962): Resonant interactions between two trains of gravity waves.J. Fluid Mech.,12, 321–332.Google Scholar
  17. 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. London A347, 311–328.Google Scholar
  18. Masson, D. (1993): On the nonlinear coupling between swell and wind waves.J. Phys. Oceanogr.,23, 1249–1258.CrossRefGoogle Scholar
  19. Masuda, A. (1980): Nonlinear energy transfer between wind waves.J. Phys. Oceanogr.,10, 2082–2092.CrossRefGoogle Scholar
  20. Masuda, A. (1986): Nonlinear energy transfer between random gravity waves. p. 41–57. InWave Dynamics and Radio Probing of the Ocean Surface, ed. by O. M. Phillips and K. Hasselmann, Plenum Press, New York.Google Scholar
  21. Mitsuyasu, H. (1969): On the growth of the spectrum of wind-generated waves (II).Rep. Res. Inst. Appl. Mech. Kyushu Univ.,17, 235–243.Google Scholar
  22. Mitsuyasu, H., F. Tasai, T. Suhara, S. Mizuno, M. Ohkusu, T. Honda and K. Rikiishi (1975): Observations of the directional spectra of ocean waves using a cloverleaf bouy.J. Phys. Oceanogr.,5, 750–760.CrossRefGoogle Scholar
  23. Phillips, O. M. (1977):The Dynamics of the Upper Ocean. Cambridge Univ. Press, Cambridge, 336 pp.Google Scholar
  24. Resio, D. and W. Perrie (1991): A numerical study of nonlinear energy fluxes due to wave-wave interactions.J. Fluid Mech.,223, 603–629.Google Scholar
  25. Sell, W. and K. Hasselmann (1972): Computations of nonlinear energy transfer for JONSWAP and empirical wind-wave spectra.Rep. Inst. Geophys. Univ. Hamburg, 1–6.Google Scholar
  26. Snyder, R. L., F. W. Dobson, J. A. Elliot and R. B. Long (1981): Array measurements of atmospheric pressure fluctuations above surface gravity waves.J. Fluid Mech.,102, 1–59.Google Scholar
  27. The SWAMP Group (24 Authors) (1985):Ocean Wave Modeling. Plenum Press, New York 256 pp.Google Scholar
  28. The WAMDI Group (13 authors) (1988): The WAM model—A third generation ocean wave prediction model.J. Phys. Oceanogr.,18, 1378–1391.Google Scholar
  29. Toba, Y. (1972): Local balance in the air-sea boundary processes. I. On the growht process of wind waves.J. Oceanogr. Soc. Japan,28, 109–120.CrossRefGoogle Scholar
  30. Tolman, H. L. (1992): Effects of numerics on the physics in a third-generation wind-wave model.J. Phys. Oceanogr.,22, 1095–1111.CrossRefGoogle Scholar
  31. Webb, D. J. (1978): Non-linear transfers between sea waves.Deep-Sea Res.,25, 279–298.CrossRefGoogle Scholar
  32. Wilson, B. W. (1965): Numerical, prediction of ocean waves in the North Atlantic for December 1959.Deut. Hydrogr. Zeit., Jahrgang18 Heft 3, 114–130.Google Scholar
  33. Wu, J. (1982): Wind-stress coefficients over sea surface from breeze to hurricane.J. Geophys. Res.,87, 9704–9706.Google Scholar
  34. Young, I. R. and G. Ph. Van Vledder (1993): A review of the central role of nonlinear interactions in wind-wave evolution.Phil. Trans. Roy. Soc. London,A342, 505–524.Google Scholar
  35. Young, I. R., S. Hasselmann and K. Hasselmann (1987): Computations of a wave spectrum to a sudden change in wind direction.J. Phys. Oceanogr.,17, 1317–1338.CrossRefGoogle Scholar

Copyright information

© Journal of the Oceanographic Society of Japan 1996

Authors and Affiliations

  • Kosei Komatsu
    • 1
  • Akira Masuda
    • 2
  1. 1.Department of Earth System Science & Technology, Graduate School of Engineering SciencesKyushu UniversityKasugaJapan
  2. 2.Research Institute for Applied MechanicsKyushu UniversityKasugaJapan

Personalised recommendations