Surface Waves and Fluxes pp 247-284 | Cite as

# Modulation of Short Wind Waves by Long Waves

## Abstract

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.

## Keywords

Wind Stress Short Wave Phase Speed Modulation Transfer Function Advection Velocity## Preview

Unable to display preview. Download preview PDF.

## References

- Abramowitz, M., and I. Stegun, 1965:
*Handbook of Mathematical Functions*, Dover.Google Scholar - 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 - Alpers, W., and I. Hennings, 1984: A theory of the imaging mechanism
*J. Geophys. Res.*,**89**(C6), 10, 529–10, 546.CrossRefGoogle Scholar - Bretherton, F.P. and C.J.R. Garrett, 1969: Wavetrains in inhomogeneous moving media.
*Proc. Roy. Soc.*A302, 529.Google Scholar - Cox, C.S., 1958: Measurements of slopes of high-frequency wind waves.
*J. Mar. Res*.**16**, 199–225.Google Scholar - 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 - Garrett, C., 1976: Generation of Langmuir circulations by surface waves — a feedback mechanism
*J. Mar. Res.***34**, 117–130.Google Scholar - Garrett, C. and J. Smith, 1976: On the interaction between long and short surface waves.
*J. Phys. Oceanography***6**, 925–930.CrossRefGoogle Scholar - 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 - 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 - 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
- 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 - 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 - Keller, W.C. and J. Wright, 1975: Microwave scattering and the straining of wind generated waves.
*Radio Science*,**10**, 139–147.CrossRefGoogle Scholar - 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 - 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 - 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 - Longuet-Higgins, M.S., 1969a: On wave breaking and the equilibrium spectrum of wind generated waves.
*Proc. Roy. Soc*, A311, 371–389.Google Scholar - Longuest-Higgins, M.S., 1969b: Action of a variable stress at the surface of water waves.
*Phys. of Fluids*,**12**, 737.CrossRefGoogle Scholar - 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 - 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 - Phillips, O.M., 1977:
*The dynamics of the upper ocean*. (2nd ed.) Cambridge University Press, 336pp.Google Scholar - 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 - 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 - Plant, W.J., 1980: On the steady-state energy balance of short gravity wave systems.
*J. Phys. Oceanog.*,**10**, 1340–1352.CrossRefGoogle Scholar - Plant, W.J., 1982: A relationship between wind stress and wave slope.
*J. Geophys. Res.*,**87**, 1961–1967.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - Smith, J.A., 1980: Waves, currents, and Langmuir circulation. Ph.B. Thesis, Dept. of Oceanography. Dalhousie University Nova Scotia, Canada.Google Scholar
- Smith, J.A., 1986: Short surface waves with Growth and Dissipation.
*J. Geophys. Res.***91**, 2616–2632.CrossRefGoogle Scholar - Smith, S.D., 1980: Wind stress and heat flux over the ocean in Gale Force Winds.
*J. Phys. Oceanog*.**10**, 709–726.CrossRefGoogle Scholar - Stewart, R.H. and J.W. Joy, 1974: HF radio measurements of surface currents.
*Deep-Sea Research*,**21**, 1039–1049.Google Scholar - Tennekes, H. and J.L. Lumley, 1972:
*A First Course in Turbulence*. MIT Press, 300pp.Google Scholar - 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 - Townsend, A.A., 1980: The response of sheared turbulence to additional distortion.
*J. Fluid Mech.*,**81**, 171–191.CrossRefGoogle Scholar - 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 - Valenzuela, G.R., 1976: The growth of gravity-capillary waves in a coupled shear flow.
*J. Fluid Mech.*,**76**b, 229–250.CrossRefGoogle Scholar - 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 - Valenzuela, G.R. and J.W. Wright, 1976: The Growth of Waves by Modulated Wind Stress.
*J. Geophys. Res.*,**81**, 5795–5796.CrossRefGoogle Scholar - 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 - 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 - Whitham, G.E., 1974 Linear and Nonlinear Waves, Wiley, N.Y.Google Scholar
- Wright, J.W., 1968: A new model for sea clutter.
*IEEE Trans. Antennas Propagation.*,**AP-16**, 217–223.CrossRefGoogle Scholar - 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 - Wu, J., 1980: Wind-stress coefficients over Sea Surface near Neutral Conditions — a Revisit.
*J. Phys. Oceanog.***10**, 727–740.CrossRefGoogle Scholar