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Interaction of waves, surface currents, and turbulence: the application of surface-following coordinate systems

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Abstract

Surface waves comprise an important aspect of the interaction between the atmosphere and the ocean, so a dynamically consistent framework for modelling atmosphere-ocean interaction must take account of surface waves, either implicitly or explicitly. In order to calculate the effect of wind forcing on waves and currents, and vice versa, it is necessary to employ a consistent formulation of the energy and momentum balance within the airflow, wave field, and water column. It is very advantageous to apply surface-following coordinate systems, whereby the steep gradients in mean flow properties near the air-water interface in the cross-interface direction may be resolved over distances which are much smaller than the height of the waves themselves. We may account for the waves explicitly by employing a numerical spectral wave model, and applying a suitable theory of wave-mean flow interaction. If the mean flow is small compared with the wave phase speed, perturbation expansions of the hydrodynamic equations in a Lagrangian or generalized Lagrangian mean framework are useful: for stronger flows, such as for wind blowing over waves, the presence of critical levels where the mean flow velocity is equal to the wave phase speed necessitates the application of more general types of surface-following coordinate system. The interaction of the flow of air and water and associated differences in temperature and the concentration of various substances (such as gas species) gives rise to a complex boundary-layer structure at a wide range of vertical scales, from the sub-millimetre scales of gaseous diffusion, to several tens of metres for the turbulent Ekman layer. The balance of momentum, heat, and mass is also affected significantly by breaking waves, which act to increase the effective area of the surface for mass transfer, and increase turbulent diffusive fluxes via the conversion of wave energy to turbulent kinetic energy.

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References

  • Alpers, W., and H. Hühnerfuss, 1989. The damping of ocean waves by surface films: A new look at an old problem. J. Geophys. Res., 94(C5): 6251–6265.

    Google Scholar 

  • Anderson, J. L., S. Preiser, and E. L. Rubin, 1968. Conservation form of the equations of hydrodynamics in curvilinear coordinates. J. Comput. Phys., 2: 279–287.

    Article  Google Scholar 

  • Andrews, D. G., and M. E. McIntyre, 1978a. An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech., 89: 609–646.

    Article  Google Scholar 

  • Andrews, D. G., and M. E. McIntyre, 1978b. On wave-action and its relatives. J. Fluid Mech., 89: 647–664.

    Article  Google Scholar 

  • Ardhuin, F., B. Chapron, and T. Elfouhaily, 2003. Waves and the air-sea momentum budget, implications for ocean circulation modelling. J. Phys. Oceanogr., 34: 1741–1755.

    Article  Google Scholar 

  • Ardhuin, F., and A. D. Jenkins, 2006. On the interaction of surface waves and upper ocean turbulence. J. Phys. Oceanogr., 36: 551–557.

    Article  Google Scholar 

  • Asher, W. E., L. M. Karle, B. J. Higgins, P. J. Farley, E. C. Monahan, and I. S. Leifer, 1996. The influence of bubble plumes on air-seawater gas transfer velocities. J. Geophys. Res., 101: 12017–12026.

    Article  Google Scholar 

  • Banner, M. L., I. S. F. Jones, and J. C. Trinder, 1989. Wave number spectra of short gravity waves. J. Fluid Mech., 198: 321–344.

    Article  Google Scholar 

  • Brooke Benjamin, T., 1959. Shearing flow over a wavy boundary. J. Fluid Mech., 6: 161–205.

    Article  Google Scholar 

  • Burchard, H., 2002. Applied Turbulence Modelling in Marine Waters. Springer, Berlin, 229pp.

    Google Scholar 

  • Bonmarin, P., 1989. Geometric properties of deep-water breaking waves. J. Fluid Mech., 209: 405–433.

    Article  Google Scholar 

  • Bye, J. A. T., 1988. The coupling of wave drift and wind velocity profiles. J. Mar. Res., 46: 457–472.

    Google Scholar 

  • Chalikov, D. and V. K. Makin, 1991. Models of the wave boundary layer. Bound.-Layer Meteorol., 63: 65–96.

    Article  Google Scholar 

  • Chang, M.-S., 1969. Mass transport in deep-water long-crested random gravity waves. J. Geophys. Res., 74: 1515–1536.

    Google Scholar 

  • Charnock, H., 1955. Wind stress on a water surface. Q. J. R. Meteorol. Soc., 81: 639–640.

    Article  Google Scholar 

  • Chereskin, T. K., 1995. Direct evidence of an Ekman balance in the Califronia Current. J. Geophys. Res., 100(C9): 18261–18269.

    Article  Google Scholar 

  • Craig, P. D., and M. L. Banner, 1994. Modeling wave-enhanced turbulence in the ocean surface layer. J. Phys. Oceanogr., 24: 2546–2559.

    Article  Google Scholar 

  • Craik, A. D. D., 1985. Wave Interactions and Fluid Flows. Cambridge University Press, Cambridge, U.K., 322pp.

    Google Scholar 

  • Craik, A. D. D., and S. Leibovich, 1976. A rational model for Langmuir circulations. J. Fluid Mech., 73: 401–426.

    Article  Google Scholar 

  • Csanady, G. T., 1990. The role of breaking wavelets in air-sea gas transfer. J. Geophys. Res., 95: 749–759.

    Google Scholar 

  • Dommermuth, D. G., D. K. P. Yue, W. M. Lin, R. J. Rapp, E. S. Chan, et al., 1988. Deep-water plunging breakers: A comparison between potential theory and experiments. J. Fluid Mech., 189: 423–442.

    Article  Google Scholar 

  • Dorrestein, R., 1951. General linearized theory of the effect of surface films on water ripples, I–II. Proc. K. Nederl. Akad. Wet., Ser. B, 54: 260–272 & 350–356.

    Google Scholar 

  • Farmer, D., and M. Li, 1995. Patterns of bubble clouds organized by Langmuir circulations. J. Phys. Oceanogr., 25: 1426–1440.

    Article  Google Scholar 

  • Farmer, D. M., C. L. McNeil, and B. D. Johnson, 1993. Evidence for the importance of bubbles in increasing air-sea gas flux. Nature, 361: 620–623.

    Article  Google Scholar 

  • Foster, T. D., 1971. Intermittent convection. Geophys. Fluid Dyn., 2: 201–217.

    Article  Google Scholar 

  • Gerstner, F. J., 1804. Theorie der Wellen. Abhandl. Kgl. Böhm. Ges. Wiss., Prague, Vol. 1, 1–65.

    Google Scholar 

  • Groeneweg, J., and G. Klopman, 1998. Changes of the mean velocity profiles in the combined wave-current motion described in a GLM formulation. J. Fluid Mech., 370: 271–296.

    Article  Google Scholar 

  • Hasselmann, K., 1970. Wave-driven inertial oscillations. Geophys. Fluid Dyn., 1: 463–502.

    Google Scholar 

  • Hasselmann, K., 1974. On the spectral dissipation of ocean waves due to white capping. Bound.-Layer Meteorol., 6: 107–127.

    Article  Google Scholar 

  • Janssen, P. A. E. M., 1989. Wave-induced stress and the drag of air flow over sea waves. J. Phys. Oceanogr., 19: 745–754.

    Article  Google Scholar 

  • Jeffreys, H., 1924. On the formation of water waves by wind. Proc. R. Soc. Lond., A107: 189–206.

    Google Scholar 

  • Jenkins, A. D., 1986. A theory for steady and variable wind and wave induced currents. J. Phys. Oceanogr., 16: 1370–1377.

    Article  Google Scholar 

  • Jenkins, A. D., 1987. Wind and wave induced currents in a rotating sea with depth-varying eddy viscosity. J. Phys. Oceanogr., 17: 938–951.

    Article  Google Scholar 

  • Jenkins, A. D., 1989a. Conservation form of the momentum equation in a general curvilinear coordinate system. Ocean Modelling (newsletter), 84: 6–8 (Unpublished manuscript, available from the Robert Hooke Institute, Dept. of Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, U.K. Manuscript available for download at URL http://www.gfi.uib.no/:_jenkins/papers/JenkinsAD_OM-1989-6.ps.gz).

    Google Scholar 

  • Jenkins, A. D., 1989b. The use of a wave prediction model for driving a near-surface current model. Dt. Hydrogr. Z., 42: 133–149.

    Article  Google Scholar 

  • Jenkins, A. D., 1992. A quasi-linear eddy-viscosity model for the flux of energy and momentum to wind waves, using conservation-law equations in a curvilinear coordinate system. J. Phys. Oceanogr., 22: 843–858.

    Article  Google Scholar 

  • Jenkins, A. D., 1993. A simplified quasilinear model for wave generation and air-sea momentum flux. J. Phys. Oceanogr., 23: 2001–2018.

    Article  Google Scholar 

  • Jenkins, A. D., 1994. A stationary potential-flow approximation for a breaking-wave crest. J. Fluid Mech., 280: 335–347.

    Article  Google Scholar 

  • Jenkins, A. D., 1996. A quasi-stationary irrotational solution for a breaking wave crest. In Donelan, M. A., W. H. Hui, and W. J. Plant (eds.), The Air-Sea Interface. Proc. Sympos. Air-Sea Interface, Radio & Acoust. Sensing, Turbulence & Wave Dynamics, Marseilles, France, 24–30 June 1993. University of Miami, Florida, U.S.A., 247–252.

    Google Scholar 

  • Jenkins, A. D., 2001. Geometrical and kinematic properties of breaking waves in the framework of a stationary flow approximation. In: Olagnon, M., and Athanassoulis, G. eds., Rogue Waves 2000: Proc. Workshop, Brest, France, 29–30 November 2000. Ifremer, Brest, 221–226.

    Google Scholar 

  • Jenkins, A. D., and F. Ardhuin, 2004. Interaction of ocean waves and currents: How different approaches may be reconciled. Proc. 14th Int. Offshore & Polar Engng Conf., Toulon, France, 23–28 May 2004, Int. Soc. of Offshore & Polar Engrs, Vol. 3, 105–111.

  • Jenkins, A. D., and K. B. Dysthe, 1997. The effective film viscosity coefficients of a thin floating fluid layer. J. Fluid Mech., 344: 335–337.

    Article  Google Scholar 

  • Jenkins, A. D., and S. J. Jacobs, 1997. Wave damping by a thin layer of viscous fluid. Phys. Fluids, 9: 1256–1264.

    Article  Google Scholar 

  • Jenkins, A. D., R. B. Olsen, and S. Christianidis, 1986. Intercomparison trials: Near-surface current measurements over the Norwegian continental shelf. Proc. IEEE Third Working Conf. on Current Measurement, Airlie, Virginia, January 1986. IEEE, New York, 20–25.

    Chapter  Google Scholar 

  • Komen, G. J., L. Cavaleri, M. A. Donelan, K. Hasselmann, S. Hasselmann, and P. A. E. M. Janssen, 1994. Dynamics and Modelling of Ocean Waves. Cambridge University Press, Cambridge, U.K., 540pp.

    Google Scholar 

  • Kudryavtsev, V. N., V. K. Makin, and B. Chapron, 1999. Coupled sea-surface-atmosphere model. 2. Spectrum of short wind waves. J. Geophys. Res., 104: 7625–7639.

    Article  Google Scholar 

  • Kudryavtsev, V. N., V. K. Makin, and J. F. Meirink, 2001. Simplified model of the air flow above waves. Bound.-Layer Meteorol., 100: 63–90.

    Article  Google Scholar 

  • Lamb, H., 1932. Hydrodynamics. 6th edition, Cambridge University Press, Cambridge, U.K., 738pp.

    Google Scholar 

  • Leibovich, S., 1980. On wave-current interaction theories of Langmuir circulations. J. Fluid Mech., 99: 715–724.

    Article  Google Scholar 

  • Li, Z., and A. G. Davies, 1996. Towards predicting sediment transport in combined wave-current flow. J. Wtrwy., Port, Coast., and Ocean Engng, 122: 157–164.

    Article  Google Scholar 

  • Longuet-Higgins, M. S., 1953. Mass transport in water waves. Philos. Trans. R. Soc. Lond., A245: 535–581.

    Google Scholar 

  • Longuet-Higgins, M. S., 1958. The mechanics of the boundary-layer near the bottom in a progressive wave. —Appendix to Russell, R. C. H., and J. D. C. Osorio, ‘An experimental investigation of drift profiles in a closed channel’. Proc. 6th Conf. on Coastal Engng. Council on Wave Research, Univ. of California, Berkeley, 171–193.

    Google Scholar 

  • McIntyre, M. E., 1988. A note on the divergence effect and the Lagrangian-mean surface elevation in periodic water waves. J. Fluid Mech., 189: 235–242.

    Article  Google Scholar 

  • McGillis, W. R., and R. Wanninkhof, 2006: Aqueous CO2 gradients for air-sea flux estimates. Marine Chem., 98: 100–108.

    Article  Google Scholar 

  • Madsen, O. S., 1977. A realistic model of the wind-induced Ekman boundary layer. J. Phys. Oceanogr., 7: 248–255.

    Article  Google Scholar 

  • Makin, V. K., and V. N. Kudryavtsev, 1999. Coupled sea-surface-atmosphere model. 1. Wind over wave coupling. J. Geophys. Res., 104: 7613–7623.

    Article  Google Scholar 

  • Makin, V. K., V. N. Kudryavtsev, and C. Mastenbroek, 1995. Drag of the sea surface. Bound.-Layer Meteorol., 73: 159–182.

    Article  Google Scholar 

  • Marangoni, C., 1872. Sul principio della viscosità superficiale dei liquidi stabili. Nuovo Cimento, Ser. 2, 5/6: 239–273.

    Article  Google Scholar 

  • Mellor, G., 2003. The three-dimensional current and surface wave equations. J. Phys. Oceanogr., 33: 1978–1989.

    Article  Google Scholar 

  • Miles, J. W., 1957. On the generation of surface waves by shear flows. J. Fluid Mech., 3: 185–204.

    Article  Google Scholar 

  • Pierson, W. J., 1962. Perturbation analysis of the Navier-Stokes equations in Lagrangian form with selected linear solutions. J. Geophys. Res., 67: 3151–3160.

    Google Scholar 

  • Pollard, R. T., 1970. Surface waves with rotation: An exact solution. J. Geophys. Res., 75: 5895–5898.

    Google Scholar 

  • Pollard, R. T., 1973. Interpretation of near-surface current meter observations. Deep-Sea Res., 20: 261–268.

    Google Scholar 

  • Rapp, R. J., and W. K. Melville, 1990. Laboratory measurements of deep-water breaking waves. Philos. Trans. R. Soc. London, A331: 735–800.

    Google Scholar 

  • Sjöblom, A., and A. Smedman, 2003. Vertical structure in the marine atmospheric boundary layer and its implication to the inertial dissipation method. Bound.-Layer Meteorol., 190: 1–25.

    Article  Google Scholar 

  • Sjöblom, A., and A. Smedman, 2004. Comparison between eddy-correlation and inertial dissipation methods in the marine atmospheric surface layer. Bound.-Layer Meteorol., 110: 141–164.

    Article  Google Scholar 

  • Stewart, R. H., and J. W. Joy, 1974. HF radio measurements of surface currents. Deep-Sea Res., 21: 1039–1049.

    Google Scholar 

  • Stokes, G. G., 1847. On the theory of oscillatory waves. Trans. Cambridge Philos. Soc., 8: 441–455.

    Google Scholar 

  • Soloviev, A. V., and P. Schlüssel, 1994. Parameterization of the cool skin of the ocean and of the air-ocean gas transfer on the basis of modelling surface renewal. J. Phys. Oceanogr., 24: 1339–1346.

    Article  Google Scholar 

  • Thorpe, S. A., 1984. On the determination of in the near-surface ocean from acoustic measurements of bubbles. J. Phys. Oceanogr., 14: 855–863.

    Article  Google Scholar 

  • Ünlüata, Ü., and C. C. Mei, 1970. Mass transport in water waves. J. Geophys. Res., 75: 7611–7618.

    Google Scholar 

  • Ursell, F., 1950. On the theoretical form of ocean swell on a rotating earth. Mon. Not. Roy. Astron. Soc. (Geophys. Suppl.), 6: 1–8.

    Google Scholar 

  • Weber, J. E., 1983. Steady wind-and wave-induced currents in the open ocean. J. Phys. Oceanogr., 13: 524–530.

    Article  Google Scholar 

  • Weber, J. E., 1985. Friction-induced roll motion in short-crested surface gravity waves. J. Phys. Oceanogr., 15: 936–942.

    Article  Google Scholar 

  • Weber, J. E., 1987. Wave attenuation and wave drift in the marginal ice zone. J. Phys. Oceanogr., 17: 2351–2361.

    Article  Google Scholar 

  • Weber, J. E., 1990. Eulerian versus Lagrangian approach to wave-drift in a rotating ocean. Kungl. Vetenskaps-og VitterhetsSamhället, Göteborg, Acta: Geophysica, 3: 155–170.

    Google Scholar 

  • Weber, J. E., and Ø. Sætra, 1995. Effects of film elasticity on the drift velocity of capillary-gravity waves. Phys. Fluids, 7: 307–314.

    Article  Google Scholar 

  • Woolf, D. K., and S. A. Thorpe, 1991. Bubbles and the air-sea exchange of gases in near-saturation conditions. J. Mar. Res., 49: 435–466.

    Article  Google Scholar 

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  1. Bjerknes Centre for Climate Research, Geophysical Institute, Allegaten 70, N-5007, Bergen, Norway

    Jenkins Alastair D.

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Jenkins, A.D. Interaction of waves, surface currents, and turbulence: the application of surface-following coordinate systems. J Ocean Univ. China 6, 319–331 (2007). https://doi.org/10.1007/s11802-007-0319-8

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