Abstract
It is shown how the drag of the sea surface can be computed from the wind speed and the sea state. The approach, applicable both for fully developed and for developing seas, is based on conservation of momentum in the boundary layer above the sea, which allows one to relate the drag to the properties of the momentum exchange between the sea waves and the atmosphere.
The total stress is split into two parts: a turbulent part and a wave-induced part. The former is parameterized in terms of mixing-length theory. The latter is calculated by integration of the wave-induced stress over all wave numbers. Usually, the effective roughness is given in terms of the empirical Charnock relation. Here, it is shown how this relation can be derived from the dynamical balance between turbulent and wave-induced stress. To this end, the non-slip boundary conditions is assigned to the wave surface, and the local roughness parameter is determined by the scale of the molecular sublayer.
The formation of the sea drag is then described for fully developed and developing seas and for light to high winds.
For the Charnock constant, a value of about 0.018–0.030 is obtained, depending on the wind input, which is well within the range of experimental data.
It is shown that gravity-capillary waves with a wavelength less than 5 cm play a minor role in the momentum transfer from wind to waves. Most of the momentum is transferred to decimeter and meter waves, so that the drag of developing seas depends crucially on the form of the wave spectrum in the corresponding high wavenumber range.
The dependence of the drag on wave age depends sensitively on the dependence of this high wavenumbertail on wave age. If the tail is wave-age independent, the sea drag appears to be virtually independent of wave age. If the tail depends on wave age, the drag also does. There is contradictory evidence as to the actual dependence. Therefore, additional experiments are needed.
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References
Banner, M. L., Jones, I. S. F., and Trinder, J. C.: 1989, ‘Wavenumber Spectra of Short Gravity Waves’,J. Fluid Mech. 198, 321–344.
Banner, M. L.: 1990 ‘Equilibrium Spectra of Wind Waves’,J. Phys. Oceanogr. 20, 966–984.
Battjes, J. A., Zitman, T. J. and Holthuijsen, L. H.: 1987, ‘A Reanalysis of the Spectra Observed in JONSWAP’,J. Phys. Oceanogr. 17, 1288–1295.
Burgers, G. and Makin, V.: 1993, ‘Boundary-Layer Model Results for Wind-Sea Growth’,J. Phys. Oceanogr. 23, 372–385.
Caudal, G.: 1993, ‘Self-Consistency between Wind Stress, Wave Spectrum, and Wind-Induced Wave Growth for Fully Rough Air-Sea Interface’,J. Geophys. Res. 98, No. C12, 22743–22752.
Chalikov, D. and Belevich, M. Yu.: 1993, ‘One-Dimensional Theory of the Wave Boundary Layer’,Boundary-Layer Meteorol. 63, 65–96.
Chalikov, D. and Makin, V.: 1991, ‘Models of the Wave Boundary Layer’,Boundary-Layer Meteorol. 56, 83–99.
Charnock, H.: 1955 ‘Wind Stress on a Water Surface’,Q.J.R. Meteorol. Soc. 81, 639–640.
Donelan, M. A.: 1982, ‘The Dependence of the Aerodynamic Drag Coefficient on Wave Parameters’,Proc. First Int. Conf. on Meteorology and Air-Sea Interaction of the Coastal Zone, The Hague, Amer. Meteor. Soc. 381–387.
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.
Donelan, M. A. and Hui, W. H.: 1990, ‘Mechanics of Ocean Surface Waves’,Surface Waves and Fluxes, Vol. 1, G.L. Geernaert and W.J. Plant, Eds., Kluwer Academic, 336 pp.
Donelan, M. A. and Pierson, W. J.: 1987, ‘Radar Scattering and Equilibrium, Ranges in Wind-Generated Waves with Application to Scatterometry’,J. Geophys. Res. 92, No. C5, 4971–5029.
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. Oceanogr. 23, 2143–2149.
Garratt, J. R.: 1977, ‘Review of Drag Coefficients over Oceans and Continents’,Mon. Weather Rev. 105, 915–929.
Geernaert G. L., Katsaros, K. B., and Richter, K.: 1986, ‘Variation of the Drag Coefficient and its Dependence on Sea State’,J. Geophys. Res. 91, 1580–1584.
Geernaert, G. L.: 1990, ‘Bulk Parameterizations for the Wind Stress and Heat Fluxes’, in G. L. Greenaert and W. J. Plant (eds.),Surface Waves and Fluxes, Vol. 1, Kluwer Academic, 336 pp.
Harris, J. A., Belcher, S. E., and Street, R. L.: 1994, ‘Linear Dynamics of Wind-Waves in Coupled Turbulent Air-Water Flow. Part 2. Numerical Model’,J. Fluid Mech., (accepted for publication).
Hasselmann, D. and Bösenberg, J.: 1991, ‘Field Measurements of Wave-Induced Pressure over Wind-Sea and Swell’,J. Fluid Mech. 230, 391–428.
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.
Jähne, B. and Riemer, K. S.: 1990, ‘Two-Dimensional Wave Number Spectra of Small-Scale Water Surface Waves’,J. Geophys. Res. 95, 11531–11546.
Janssen, P. A. E. M.: 1989, ‘Wave-Induced Stress and the Drag of Air Flow over Sea Waves’,J. Phys. Oceanogr. 19, 745–754.
Janssen, P. A. E. M.: 1991, ‘Quasi-Linear Theory of Wind-Wave Generation Applied to Wave Forecasting’,J. Phys. Oceanogr. 21, 1631–1642.
Janssen, P. A. E. M.: 1992, ‘Experimental Evidence of the Effect of Surface Waves on the Airflow’,J. Phys. Oceanogr. 22, 1600–1604.
JONSWAP Group (Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright, D. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E., Kruseman, P., Meerburg, A., Müller, P., Olbers, D. J., Richter, K., Sell, W. and Walden, H.: 1971, ‘Measurements of Wind-Wave Growth and Swell Decay During the Joint North Sea Wave Project (JONSWAP)’,Dtsch. Hydrogr. Z. A8(12), 1–95.
Kitaigorodskii, S. A.: 1970,The physics of air-sea interaction, Israel Program for Scientific Translations, Jerusalem, 1973.
Kitaigorodskii, S. A.: 1992, ‘The Dissipation Subrange in Wind Wave Spectra’,Mat. Fys. Medd. 42(5), 3–24.
Larson, T. R. and Wright, J. W.: 1975, ‘Wind-Generated Gravity-Capillary Waves: Laboratory Measurements of Temporal Growth Rates using Microwave Backscatter’,J. Fluid Mech. 70, 417–436.
Maat, N., Kraan, C., and Oost, W. A.: 1991, ‘The Roughness of Wind Waves’,Boundary-Layer Meteorol. 54, 89–103.
Makin, V.: 1989, ‘The Dynamics and Structure of the Boundary Layer Above Sea’, Senior doctorate thesis, Inst. of Oceanology, Acad. of Sci. USSR, Moscow, 417 pp.
Makin, V.: 1990, ‘Deviation of the Mean Wind Speed Profile Above Waves from the Logarithmic Distribution’,Izv. Atm. Oc. Phys. 26, 322–324.
Mastenbroek, C., Burgers, G., and Janssen, P. A. E. M.: 1993, ‘The Dynamical Coupling of a Wave Model and a Storm surge Model through the Atmospheric Boundary Layer’,J. Phys. Oceanogr. 23, 1856–1866.
Monin, A. S. and Yaglom, A. M.: 1971,Statistical Fluid Mechanics, Vol. 1, Cambridge: MIT Press, 769 pp..
Phillips, O. M.: 1977,Dynamics of the Upper Ocean, 2nd ed. Cambridge University Press, 336 pp.
Plant, W. J.: 1982, ‘A Relationship between Wind Stress and Wave Slope’,J. Geophys. Res. 87, 1961–1967.
Snyder, R. L., Dobson, F. W., Elliott, J. A., and Long, R. B.: 1981, ‘Array Measurements of Atmospheric Pressure Fluctuations Above Surface Gravity Waves’,J. Fluid Mech. 102, 1–59.
Stewart, R. W.: 1974, ‘The Air-Sea Momentum Exchange’,Boundary-Layer Meteorol. 6, 151–167.
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.
Toba, Y., Iida, N., Kawamura, H., Ebuchi, N., and Jones, I. S. F.: 1990, ‘Wave Dependence of Sea-Surface Wind Stress’,J. Phys. Oceanogr. 20, 705–721.
Wu, J.: 1980, ‘Wind-Stress Coefficient Over Sea Surface Near Neutral Conditions — A Revisit’,J. Phys. Oceanogr. 10, 727–740.
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The investigation was in part supported by the Netherlands Geosciences Foundation (GOA) with financial aid from the Netherlands Organization for Scientific Research (NWO).
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Makin, V.K., Kudryavtsev, V.N. & Mastenbroek, C. Drag of the sea surface. Boundary-Layer Meteorol 73, 159–182 (1995). https://doi.org/10.1007/BF00708935
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DOI: https://doi.org/10.1007/BF00708935