Abstract
The evolution of a short-wave (SW) spectrum along a long wave (LW) isstudied. The evolution of the SW spectrum variation is treated in therelaxation time approximation. The variation of the SW spectrum is caused bythe LW orbital velocities and by the variation of the wind stress along thesurface of a LW. The latter is due to the distortion of the flow by a LW, andto the variation of the roughness induced by the modulated short waves. Thisintroduces a feedback mechanism: more SWs give rise to a larger roughness,which by increasing the local stress stimulates the growth of more SWs. It isshown that this aerodynamic feedback effect dominates the modulation of theSW spectrum for moderate and strong winds. The feedback mechanism is mosteffective for SWs in the gravity-capillary range, increasing its dominancewith increasing windspeed and decreasing frequency of a LW. The maximum ofthe SW amplitude modulation is situated at the crest of a LW. The results arein agreement with laboratory and field measurements of the short-wavemodulation.
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References
Alpers, W. R. and Hasselmann, K.: 1978, ‘The Two-Frequency Microwave Technique for Measuring Ocean-Wave Spectra from an Airplane or Satellite’, Boundary-Layer Meteorol. 13, 215–230.
Belcher, S. E. and Hunt, J. C. R.: 1993, ‘Turbulent Shear Flow over Slowly Moving Waves’, J. Fluid Mech. 251, 109–148.
Chalikov, D. V. and Makin, V. K.: 1991, ‘Models of the Wave Boundary Layer’, Boundary-Layer Meteorol. 56, 83–99.
Donelan, M. A. and Pierson, W. J.: 1987, ‘Radar Scattering and Equilibrium Ranges in Wind-Generated Waves With Applications to Scatterometry’, J. Geophys.Res. 92, 4971–5029.
Donelan, M. A., Hamilton, J.. and Hui,W. H.: Directional Spectra of Wind Generated Waves’, Phil. Trans. Roy. Soc. London, Ser. A, 315, 509-562. d ai]Grodsky, S.A., Kudryavtsev, V. N., and Makin, V. K.: 1991, ‘The Estimation of the Air Flow Influence on the Radar MTF of the Sea Surface’, Morskoy Hydroph. J. N1, 15–22.
Hara, T. and Plant, W. J.: 1994, ‘Hydrodynamic Modulation of Short Wind-wave Spectra by Long Waves and its Measurement using Microwave Backscatter’, J. Geophys.Res. 99, 9767–9784.
Janssen, P. A. E. M.: 1989, ‘Wave-Induced Stress and the Drag of Air Flow over Sea Waves’, J. Phys. Oceanogr. 19, 745–754.
Keller, W. C. and Wright, J. W.: 1975, ‘Microwave Scattering and Straining of Wind-generated Waves’, Radio Sci. 10, 139–147.
Keller, W. C. and Plant, W. J.: 1985, ‘The Dependence of X-Band Microwave Sea Return on Atmospheric Stability and Sea State’, J. Geophys.Res. 90, 1019–1029.
Komen, G. J., Cavaleri, L., Donelan, M., Hasselmann, K., Hasselmann, S., and Janssen, P. A. E. M.: 1994, Dynamics and Modelling of Ocean Waves, Cambridge University Press, 540 pp.
Kudryavtsev, V. N.: 1994, ‘The Coupling of Wind and Internal Waves: Modulation and Friction Mechanisms’, J. Fluid Mech. 278, 33–62.
Makin, V. K., Kudryavtsev, V. N., and Mastenbroek, C.: 1995, ‘Drag of the Sea Surface’, Boundary-Layer Meteorol. 73, 159–182.
Mastenbroek, C., Makin, V. K., Garat, M. H., and Giovanangeli, J. P.: 1996, ‘Experimental Evidence of the Rapid Distortion of Turbulence in the Air Flow over Water Waves’, J. Fluid Mech. 318, 273–302.
Miller, S. J. and Shemdin, O. H.: 1991, ‘Measurements of the Hydrodynamic Modulation of Centimetre Waves’, J. Geophys.Res. 96, 2749–2759.
Plant, W. J.: 1982, ‘A Relationship between Wind Stress and Wave Slope’, J. Geophys.Res. 87, 1961–1967.
Plant, W. J., Keller, W. C. and Cross, A.: 1983, ‘Parametric Dependence of Ocean Wave-Radar Modulation Transfer Functions’, J. Geophys. Res. 88, 9747–9756.
Romeiser, R., Schmidt, A., and Alpers W.: 1994, ‘A Three-Scale Composite Surface Model for the Ocean Wave-Radar Modulation Transfer Function’, J. Geophys.Res 99, 9785–9801.
Schröter, J., Feindt, F., Alpers, W. R., and Keller, W. C.: 1986, ‘Measurements of the Ocean Wave-Radar Modulation Transfer Function at 4.3 GHz’, J. Geophys.Res. 91, 923–932.
Smith, J. A.: 1990, ‘Modulation of Short Wind Waves by Long Waves’, in G. L. Geernaert and W. J. Plant (ed.), Surface Waves and Fluxes, Vol. 1, Kluwer Academic Publishers, 336 pp.
Valenzuela, G. R. and Wright J.W.: 1979, ‘Modulation of Short Gravity-Capillary Waves by Longer-Scale Periodic Flows.-A Higher-Order Theory’, Radio Sci. 14, 1099–1110.
Wright, J. W., Plant, W. J., Keller, W. C., and Jones, W. L.: 1980, ‘Ocean Wave-Radar Modulation Transfer Functions From the West Coast Experiment’, J. Geophys. Res. 85, 4957–4966.
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KUDRYAVTSEV, V.N., MASTENBROEK, C. & MAKIN, V.K. MODULATION OF WIND RIPPLES BY LONG SURFACE WAVES VIA THE AIR FLOW: A FEEDBACK MECHANISM. Boundary-Layer Meteorology 83, 99–116 (1997). https://doi.org/10.1023/A:1000278225089
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DOI: https://doi.org/10.1023/A:1000278225089