Modulation of Short Wind Waves by Long Waves

  • Jerome A. Smith
Part of the Environmental Fluid Mechanics book series (EFME, volume 7)


The behavior of short surface waves (wavelength less than a meter or so) riding on longer wind waves or swell has sparked interest for several decades now. In the problem as posed here, the energetic waves near the peak of a wind-wave or swell spectrum are treated as a large-scale, slowly varying “medium” in which short gravity-capillary waves evolve. This “WKB approximation” should be well founded, since the time and space scales of the long and short waves are widely separated, and no reflections of the short waves occur. Short wave “packets” or “components” are examined independently as they evolve under the influence of slowly varying winds and currents, and later reassembled into a larger picture. Resonant non-linear exchanges are not explicitly included, although (for example) a “packet” might be regarded as a set of tightly coupled wavenumbers rather than as a pure sine-wave component.


Wind Stress Short Wave Phase Speed Modulation Transfer Function Advection Velocity 
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. Abramowitz, M., and I. Stegun, 1965: Handbook of Mathematical Functions, Dover.Google Scholar
  2. Alpers, W., and K. Hasselmann, 1978: The two-frequency microwave technique for measuring ocean-wave spectra from an airplane or satellite. Bound. Layer Met., 13, 215–230.CrossRefGoogle Scholar
  3. Alpers, W., and I. Hennings, 1984: A theory of the imaging mechanism J. Geophys. Res., 89(C6), 10, 529–10, 546.CrossRefGoogle Scholar
  4. Bretherton, F.P. and C.J.R. Garrett, 1969: Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. A302, 529.Google Scholar
  5. Cox, C.S., 1958: Measurements of slopes of high-frequency wind waves. J. Mar. Res. 16, 199–225.Google Scholar
  6. Donelan, M.A., 1987: Radar Scattering and Equilibrium Ranges in Wind-Generated Waves With Applications to Scatterometry. J. Geophys. Res. 92, C5, 4971–5030.CrossRefGoogle Scholar
  7. Garrett, C., 1976: Generation of Langmuir circulations by surface waves — a feedback mechanism J. Mar. Res. 34, 117–130.Google Scholar
  8. Garrett, C. and J. Smith, 1976: On the interaction between long and short surface waves. J. Phys. Oceanography 6, 925–930.CrossRefGoogle Scholar
  9. Gent, P.R. and P.A. Taylor, 1976: A numerical model of the air flow above water waves. J. Fluid Mech. 77, 105–128.CrossRefGoogle Scholar
  10. Hasse, L., 1986: On Charnock’s relation for roughness at sea Oceanic Whitecaps and their role in air-sea exchange processes. E.C. Monahan and G. Mac-Niocaill, eds.Google Scholar
  11. Henyey, F.S., D.B. Creamer, K.B. Dysthe, R.L., and J.A. Wright, 1988: The energy and action of small waves riding on large waves, (submitted to J. Fluid Mech.)Google Scholar
  12. Hsiao, S.V., and O.H. Shemdin, 1983: Measurements of wind velocity and pressure with a wave follower during Marsen. J. Geophys. Res., 88, 9841–9850.CrossRefGoogle Scholar
  13. Huang, N.E., L.F. Bliven, S.R. Long, and C. Tung, 1986: An Analytical Model for Oceanic Whitecap Coverage. J. Phys. Oceanog., 16 No. 10, 1597–1604.CrossRefGoogle Scholar
  14. Keller, W.C. and J. Wright, 1975: Microwave scattering and the straining of wind generated waves. Radio Science, 10, 139–147.CrossRefGoogle Scholar
  15. Keller, W.C., WJ. Plant, and D.E. Weissman, 1985: The dependence of X band microwave sea return on atmospheric stability and sea state. J. Geophys. Res. 90, 1019–1029.CrossRefGoogle Scholar
  16. Landahl, M.T., J.A. Smith, and S.E. Widnall, 1986: The interaction between long and short wind-generated waves in Wave Dynamics and Radio Probing of the Ocean Surface, (eds. O.M. Phillips and Klaus Hasselmann) 59–47.Google Scholar
  17. Larson, T.R., and J. Wright, 1975: Wind generated gravity-capillary waves: laboratory measurements of temporal growth rates using microwave backscatter J. Fluid Mech., 70, 417.CrossRefGoogle Scholar
  18. Longuet-Higgins, M.S., 1969a: On wave breaking and the equilibrium spectrum of wind generated waves. Proc. Roy. Soc, A311, 371–389.Google Scholar
  19. Longuest-Higgins, M.S., 1969b: Action of a variable stress at the surface of water waves. Phys. of Fluids, 12, 737.CrossRefGoogle Scholar
  20. Monaldo, F.M., and R.S. Kasevich, 1981: Measurement of short-wave modulation using fine time-series optical spectra. J. Phys. Oceanography, 11, 1034–1036.CrossRefGoogle Scholar
  21. Okuda, K., S. Kawai, and Y. Toba, 1977: Measurement of Skin Friction Distribution along the Surface of Wind Waves. J. Oceanog. Soc. of Japan 33, 190–198.CrossRefGoogle Scholar
  22. Phillips, O.M., 1977: The dynamics of the upper ocean. (2nd ed.) Cambridge University Press, 336pp.Google Scholar
  23. Phillips, O.M., 1984: On the response of short ocean wave components at a fixed wavenumber to ocean current variations. J. Phys. Oceanog., 14, 1425–1433.CrossRefGoogle Scholar
  24. Phillips, O.M. and M.L. Banner, 1974: Wave breaking in the presence of wind drift and swell. J. Fluid Mech., 66, 625–640.CrossRefGoogle Scholar
  25. Plant, W.J., 1980: On the steady-state energy balance of short gravity wave systems. J. Phys. Oceanog., 10, 1340–1352.CrossRefGoogle Scholar
  26. Plant, W.J., 1982: A relationship between wind stress and wave slope. J. Geophys. Res., 87, 1961–1967.CrossRefGoogle Scholar
  27. Plant, W.J. and W.C. Keller, 1983: The two-scale RADAR wave probe and SAR imagery of the ocean. J. Geophys. Res., 88, 9776–9784.CrossRefGoogle Scholar
  28. Plant, W.J., W.C. Keller, and A. Cross, 1983: Parametric dependence of ocean wave-radar Modulation Transfer Functions. J. Geophys. Res., 88, 9747–9756.CrossRefGoogle Scholar
  29. Plant, W.J. and J. Wright, 1977: Growth and equilibrium of short gravity waves in a wind wave tank. J. Fluid Mech., 82, 767–793.CrossRefGoogle Scholar
  30. Plant, W.J. and J. Wright, 1980: Phase speeds of upwind and downwind travelling short gravity waves. J. Geophys. Res., 85, 3304–3310.CrossRefGoogle Scholar
  31. Smith, J.A., 1980: Waves, currents, and Langmuir circulation. Ph.B. Thesis, Dept. of Oceanography. Dalhousie University Nova Scotia, Canada.Google Scholar
  32. Smith, J.A., 1986: Short surface waves with Growth and Dissipation. J. Geophys. Res. 91, 2616–2632.CrossRefGoogle Scholar
  33. Smith, S.D., 1980: Wind stress and heat flux over the ocean in Gale Force Winds. J. Phys. Oceanog. 10, 709–726.CrossRefGoogle Scholar
  34. Stewart, R.H. and J.W. Joy, 1974: HF radio measurements of surface currents. Deep-Sea Research, 21, 1039–1049.Google Scholar
  35. Tennekes, H. and J.L. Lumley, 1972: A First Course in Turbulence. MIT Press, 300pp.Google Scholar
  36. Thompson, T.W., D.E. Weissman, and F.I. Gonzalez, 1983: L-band radar backscatter dependence upon surface wind stress: a summary of new SEAS AT-1 and aircraft observations. J. Geophys. Res., 88, 1727–1735.CrossRefGoogle Scholar
  37. Townsend, A.A., 1980: The response of sheared turbulence to additional distortion. J. Fluid Mech., 81, 171–191.CrossRefGoogle Scholar
  38. Trowbridge, J. and O.S. Madsen, 1984: Turbulent wave boundary layers, 1, model formulation and first-order solution. J. Geophys. Res., 898, 7989–7998.CrossRefGoogle Scholar
  39. Valenzuela, G.R., 1976: The growth of gravity-capillary waves in a coupled shear flow. J. Fluid Mech., 76b, 229–250.CrossRefGoogle Scholar
  40. Valenzuela, G.R., and M.B. Laing, 1972: Non-linear energy transfer in gravity-capillary wave spectra with applications. J. Fluid Mech., 54, 507–520.CrossRefGoogle Scholar
  41. Valenzuela, G.R. and J.W. Wright, 1976: The Growth of Waves by Modulated Wind Stress. J. Geophys. Res., 81, 5795–5796.CrossRefGoogle Scholar
  42. Valenzuela, G.R. and J.W. Wright, 1979: Modulation of short gravity-capillary waves by longer scale periodic flows—a higher order theory. Radio Science, 14, 1099–1110.CrossRefGoogle Scholar
  43. van Gastel, K., P.A.E. Janssen, and G.J. Komen, 1985: On phase velocity and growth rate of wind induced gravity-capillary waves. J. Fluid Mech., 161, 199.CrossRefGoogle Scholar
  44. Whitham, G.E., 1974 Linear and Nonlinear Waves, Wiley, N.Y.Google Scholar
  45. Wright, J.W., 1968: A new model for sea clutter. IEEE Trans. Antennas Propagation., AP-16, 217–223.CrossRefGoogle Scholar
  46. Wright, J.W., W.J. Plant, W.C. Keller, and W.L. Jones, 1980: Ocean wave-radar Modulation Transfer Functions from the West Coast Experiment, J. Geophys. Res., 85, 4957–4966.CrossRefGoogle Scholar
  47. Wu, J., 1975: Wind-induced drift currents. J. Fluid Mech., 68, 49–70.CrossRefGoogle Scholar
  48. Wu, J., 1980: Wind-stress coefficients over Sea Surface near Neutral Conditions — a Revisit. J. Phys. Oceanog. 10, 727–740.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Jerome A. Smith
    • 1
  1. 1.The Marine Physical Laboratory of The Scripps Institution of OceanographyUniversity of CaliforniaLa JollaUSA

Personalised recommendations