Boundary-Layer Meteorology

, Volume 138, Issue 1, pp 1–41 | Cite as

Coupled Numerical Modelling of Wind and Waves and the Theory of the Wave Boundary Layer

  • D. ChalikovEmail author
  • S. Rainchik


The description of a coupled wind and wave model in conformal coordinates is given. The wave model is based on potential equations for the flow with a free surface, extended with the algorithm of breaking dissipation. The wave boundary-layer (WBL) model is based on the Reynolds equations with the Kε closure scheme with the solutions for air and water matched through the interface. The structure of the WBL and vertical profiles of the wave-produced momentum flux (WPMF) in a long-term simulation of the coupled dynamics are investigated and parameterized. The shape of the β function connecting elevation and surface pressure is studied up to high nondimensional wave frequencies. The errors of a linear presentation of the surface pressure are estimated. The β function and the universal shape of the WPMF profile obtained in coupled simulations allow a formulation of the one-dimensional theory of the WBL, and the carrying out of a detailed study of the WBL structure including the dependence of the drag coefficient on the wind speed. It is shown that a wide scatter of the experimental data on the drag coefficient can be explained, taking into account the age of waves. It is suggested that a reduction of the drag coefficient at high wind speeds can be qualitatively explained by the high-frequency wave suppression.


Boundary layer Numerical modelling Sea waves Wind–wave interaction 


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  1. Andreas EL (2004) Spray stress revised. J Phys Oceanogr 34: 1429–1440CrossRefGoogle Scholar
  2. Babanin AV (2009) Breaking of ocean surface waves. Acta Phys Slovaca 594: 305–535Google Scholar
  3. Babanin AV, Soloviev YP (1998) Investigation of transformation of the wind wave frequency spectrum with fetch and the stage of development. J Phys Oceanogr 28: 563–576CrossRefGoogle Scholar
  4. Babanin AV, Young IR, Banner ML (2001) Breaking probabilities for dominant surface waves on water of finite constant depth. J Geophys Res 106(C6): 11659–11676CrossRefGoogle Scholar
  5. Banner ML, Pierson WL (2007) Wave breaking onset and strength for two-dimensional and deep-water wave groups. J Fluid Mech 585: 93–115CrossRefGoogle Scholar
  6. Belcher S, Harris JA, Street RL (1994) Linear dynamics of wind waves in coupled turbulent air-flow. Part 1. Theory. J Fluid Mech 271: 119–151CrossRefGoogle Scholar
  7. Benjamin TB, Feir JE (1967) The disintegration of wave trains in deep water. J Fluid Mech 27: 417–430CrossRefGoogle Scholar
  8. Chalikov DV (1976) A mathematical model of wind-induced waves. Dokl Acad Sci USSR 229: 121–126Google Scholar
  9. Chalikov DV (1978) Numerical simulation of wind–wave interaction. J Fluid Mech 87: 561–582CrossRefGoogle Scholar
  10. Chalikov DV (1986) Numerical simulation of the boundary layer above waves. Boundary-Layer Meteorol 34: 63–98CrossRefGoogle Scholar
  11. Chalikov D (1993) Comments on “Wave induced stress and the drag of air flow over sea waves” and “Quasi-linear theory of wind–wave generation applied to wave/forecasting”. J Phys Oceanogr 1(423): 11597–11600Google Scholar
  12. Chalikov D (1995) The parameterization of the wave boundary layer. J Phys Oceanogr 25: 1335–1349CrossRefGoogle Scholar
  13. Chalikov D (1998) Interactive modelling of surface waves and boundary layer. Ocean wave measurements and analysis. ASCE. In: Proceedings of the third international symposium WAVES, vol 97, pp 1525–1540Google Scholar
  14. Chalikov D (2005) Statistical properties of nonlinear one-dimensional wave fields. Nonlinear Process Geophys 12: 1–19CrossRefGoogle Scholar
  15. Chalikov D (2007) Simulation of Benjamin–Feir instability and its consequences. Phys Fluids 19: 016602–016615CrossRefGoogle Scholar
  16. Chalikov D (2009) Freak waves: their occurrence and probability. Phys Fluids 21: 076602. doi: 10.1063/1.3175713 CrossRefGoogle Scholar
  17. Chalikov D, Belevich M (1993) One-dimensional theory of the wave boundary layer. Boundary-Layer Meteorol 63: 65–96CrossRefGoogle Scholar
  18. Chalikov D, Sheinin D (1996) Numerical modelling of surface waves based on principal equations of potential wave dynamics. Technical Note. NOAA/NCEP/OMB, 54 ppGoogle Scholar
  19. Chalikov D, Sheinin D (1998) Direct modelling of one-dimensional nonlinear potential waves. Nonlinearoceanwaves. In: Perrie W (eds) Advances in fluid mechanics, vol 17. WIT Press, Southampton, pp 207–222Google Scholar
  20. Chalikov D, Sheinin D (2005) Modelling of extreme waves based on equations of potential flow with a free surface. J Comput Phys 210: 247–273CrossRefGoogle Scholar
  21. Crapper GD (1957) An exact solution for progressive capillary waves of arbitrary amplitude. J Fluid Mech 96: 417–445Google Scholar
  22. Dold JW (1992) An efficient surface-integral algorithm applied to unsteady gravity waves. J Comput Phys 103: 90–115CrossRefGoogle Scholar
  23. Donelan MA (1982) The dependence of the aerodynamic drag coefficient on wave parameters. In: Proceedings of the first international conference on the meteorology and air–sea interaction in coastal zone, Hague, The Netherlands. American Meteorological Society, Boston, pp 381–387Google Scholar
  24. Donelan MA (1990) Air–sea interaction. In: LeMehaute B, Hanes DM (eds) The sea: ocean engineering science, vol 9. Wiley-Interscience, New York, pp 239–292Google Scholar
  25. Donelan MA, Haus BK, Reul N, Plant WJ, Stiassnie M, Graber HC, Brown OB, Saltzman ES (2004) On the limiting aerodynamic roughness of the ocean in very strong winds. Geophys Res Lett 31: L18306. doi: 10.1029/2004GL019460 CrossRefGoogle Scholar
  26. Donelan MA, Babanin AV, Young R, Banner ML (2006) Wave-follower field measurements of the wind-input spectral function. Part II: Parameterization of the wind input. J Phys Oceanogr 36: 1672–1689CrossRefGoogle Scholar
  27. Emanuel KA (1995) Sensitivity of tropical cyclones to surface exchange coefficients and a revised steady-state model incorporating eye dynamics. J Atmos Sci 52: 3969–3976CrossRefGoogle Scholar
  28. Fornberg B (1980) A numerical method for conformal mapping. SIAM J Sci Comput 1: 386–400CrossRefGoogle Scholar
  29. Garratt JR (1977) Review of drag coefficient over oceans and continent. Mon Weather Rev 105: 915CrossRefGoogle Scholar
  30. Geernaert GL, Katsaros KB, Richter K (1986) Variation of the drag coefficient and its dependence on sea state. J Geophys Res 91(C6): 7667–7679CrossRefGoogle Scholar
  31. Gent PR, Taylor PA (1976) A numerical model of the air flow above water waves. J Fluid Mech 77: 105–128CrossRefGoogle Scholar
  32. Hasselmann K (1962) On the nonlinear energy transfer in a gravity wave spectrum. P.1. General theory. J Fluid Mech 12: 481–500CrossRefGoogle Scholar
  33. Janssen PAEM (1991) Quasi-linear theory of wind wave generation applied to wave forecasting. J Phys Oceanogr 21: 1631–1642CrossRefGoogle Scholar
  34. Kahma KK, Calkoen CJ (1992) Reconciling discrepancies in the observed growth of wind generated waves. J Phys Oceanogr 22: 52–78CrossRefGoogle Scholar
  35. Kano T, Nishida T (1979) Sur le ondes de surface de l’eau avec une justification mathematique des equations des ondes en eau peu profonde. J Math Kyoto Univ (JMKYAZ) 19-2: 335–370Google Scholar
  36. Keller LV, Friedmann AA (1924) Differentialgleichung für die turbulente Bewegung einer kompressiblen. Flüssigkeit. In: Proceedings of the 1st international congress for applied mechanics, vol 31, issue 4, pp 789–799Google Scholar
  37. Kudryavtsev VN (2006) On effect of sea drops on atmospheric boundary layer. J Geophys Res 111: C07020. doi: 10.1029/2005JC002970 CrossRefGoogle Scholar
  38. Large WG, Pond S (1982) Sensible and latent heat flux measurements over the ocean. J Phys Oceanogr 12: 464–482CrossRefGoogle Scholar
  39. Launder BE, Spalding DB (1974) The numerical computation of turbulent flows. Comput Methods Appl Mech Eng 3: 269–289CrossRefGoogle Scholar
  40. Longuet-Higgins MS, Tanaka M (1997) On the crest instabilities of steep surface waves. J Fluid Mech 336: 51–68CrossRefGoogle Scholar
  41. Mastenbroek CV, Makin VK, Garrat MH, Giovanangeli JP (1996) Experimental evidence of the rapid distortion of the turbulence in the air flow over water waves. J Fluid Mech 318: 273–302CrossRefGoogle Scholar
  42. Miles JW (1957) On the generation of surface waves by shear flows. J Fluid Mech 3: 185CrossRefGoogle Scholar
  43. Monin AS, Yaglom AM (1971) Statistical fluid mechanics: mechanics of turbulence, vol 1. The MIT Press, Cambridge, p 769Google Scholar
  44. Orszag SA (1970) Transform method for calculation of vector coupled sums. Application to the spectral form of vorticity equation. J Atmos Sci 27: 890–895CrossRefGoogle Scholar
  45. Ovsyannikov LV (1973) Dinamika sploshnoi sredy, no 15. M.A. Lavrent’ev Institute of Hydrodynamics Sib Branch USSR Acad Sci, pp 104–125 (in Russian)Google Scholar
  46. Phillips OM (1977) Dynamics of upper ocean, 2nd edn. Cambridge University Press, Cambridge, p 336Google Scholar
  47. Powell MD, Vickery PJ, Reinhold TA (2003) Reduced drag coefficient for high wind speeds in tropical cyclones. Nature 422: 279–283CrossRefGoogle Scholar
  48. Sheinin D, Chalikov D (2005) Hydrodynamical modelling of potential surface waves. In: Problems of hydrometeorology and environment on the eve of XXI century. In: Proceedings of international theoretical conference, St. Petersburg, June 24–25, 1999. Hydrometeoizdat, St. Petersburg, pp 305–337Google Scholar
  49. Simonov VV (1982) On the calculation the drag of wavy surface. Izv Atmos Ocean Phys 18: 269–275Google Scholar
  50. Smedman AS, Larsen XG, Höström U (2003) Is the logarithmic wind law valid over the sea? In: Sajjadi SG, Hunt JCR (eds) Wind over waves II: forecasting and fundamentals of applications. Horwood Publishing Ltd., ChichesterGoogle Scholar
  51. Smith SD, Banke EG (1975) Variation of the sea surface drag coefficient with wind speed. Q J Roy Meteorol Soc 101: 665CrossRefGoogle Scholar
  52. Tanveer S (1991) Singularities in water waves and Rayleigh–Taylor instability. Proc Roy Soc Lond A435: 137–158Google Scholar
  53. Tanveer S (1993) Singularities in the classical Rayleigh–Taylor flow: formation and subsequent motion. Proc Roy Soc Lond A441: 501–525Google Scholar
  54. Tolman H, Chalikov D (1994) Development of a third-generation ocean wave model at NOAA-NMC. In: Isaacson M, Quick MC (eds) Physical and numerical modeling. University of British Columbia, Vancover, pp 724–733Google Scholar
  55. Tolman H, Chalikov D (1996) On the source terms in a third-generation wind wave model. J Phys Oceanogr 26(11): 2497–2518CrossRefGoogle Scholar
  56. Wu J (1980) Wind-stress coefficient over sea surface near neutral conditions—a revisit. J Phys Oceanogr 10: 727CrossRefGoogle Scholar
  57. Yelland MJ, Taylor PK (1996) Wind stress measurements from the open ocean. J Phys Oceanogr 26: 541–558CrossRefGoogle Scholar
  58. Zakharov VE, Dyachenko AI, Vasilyev OA (2002) New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface. Eur J Mech B 21: 283–291CrossRefGoogle Scholar
  59. Zakharov VE, Dyachenko AI, Prokofiev AO (2006) Freak waves as nonlinear stage of Stokes wave modulation instability. Eur J Mech B 25: 677–692CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Faculty of Engineering and Industrial SciencesSwinburne University of TechnologyHawthornAustralia
  2. 2.P.P. Shirshov Institute of Oceanography, RAS, St. Petersburg BranchSt. PetersburgRussia
  3. 3.Russian State Hydrometeorological UniversitySt. PetersburgRussia

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