The Small-Scale Instability of the Air–Water Interface Responsible for the Bag-Breakup Fragmentation

Yuliya Troitskaya aInstitute of Applied Physics, Nizhny Novgorod, Russia
bA.M. Obukhov Institute of Atmospheric Physics, Moscow, Russia

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Alexander Kandaurov aInstitute of Applied Physics, Nizhny Novgorod, Russia

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Olga Ermakova aInstitute of Applied Physics, Nizhny Novgorod, Russia

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Dmitry Kozlov aInstitute of Applied Physics, Nizhny Novgorod, Russia

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Anna Zotova aInstitute of Applied Physics, Nizhny Novgorod, Russia

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Daniil Sergeev aInstitute of Applied Physics, Nizhny Novgorod, Russia

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Abstract

The “bag breakup” fragmentation is the dominant mechanism for spume droplet production in high winds, which substantially affects the ocean–atmosphere exchange processes. The amount of droplets ejected from the surface, as well as their typical sizes, is prescribed by the surface wind velocity and fetch. The corresponding empirical correlations were obtained only for the limited parameters of the laboratory environment. The applicability range can be extended through the construction of a theoretical model that describes the initiation of the bag-breakup fragmentation, estimates the fragmenting liquid volume prescribing the droplet sizes, and determines the dependence on the wind parameters. This paper presents such a model. First, we conducted linear stability analysis of small-scale disturbances at the water surface under a high wind; this showed that the small-scale ripples (about 1 cm) propagating against the wind in the surface wind drift following the reference frame grew fast due to the Kelvin–Helmholtz instability, when the wind friction velocity u* exceeded the threshold of about 1 m s−1. Given the weak dispersion, the nonlinear stage of evolution was addressed using the Riemann simple wave equation modified to describe the increasing disturbances. The analytical solution for the equation suggested the scaling of the volume of liquid undergoing the bag-breakup fragmentation and its dependence on u* in agreement with the laboratory data. Using the scaling, we calculated the statistics of the bag-breakup fragmentation based on the lognormal size distribution of the fragmenting objects.

Significance Statement

The “bag breakup” fragmentation is the dominant mechanism for generating spray in hurricane winds. The parameters of spray droplets substantially affect the exchange processes between the ocean and the atmosphere and, thereby, the development of sea storms. The rapid process of spray generation can only be studied in laboratory environments using sophisticated experimental techniques. To apply the laboratory data to field conditions, we need a theoretical model that describes the threshold for fragmentation initiation, the fragmenting liquid volume, which scales the size and number of spray droplets, their dependence on wind parameters, etc. In the present work, we suggest a simple analytical model of the bag-breakup initiation, verify it in the laboratory experiment, and suggest the statistical description of the fragmentation events.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yuliya Troitskaya, yuliya@hydro.appl.sci-nnov.ru

Abstract

The “bag breakup” fragmentation is the dominant mechanism for spume droplet production in high winds, which substantially affects the ocean–atmosphere exchange processes. The amount of droplets ejected from the surface, as well as their typical sizes, is prescribed by the surface wind velocity and fetch. The corresponding empirical correlations were obtained only for the limited parameters of the laboratory environment. The applicability range can be extended through the construction of a theoretical model that describes the initiation of the bag-breakup fragmentation, estimates the fragmenting liquid volume prescribing the droplet sizes, and determines the dependence on the wind parameters. This paper presents such a model. First, we conducted linear stability analysis of small-scale disturbances at the water surface under a high wind; this showed that the small-scale ripples (about 1 cm) propagating against the wind in the surface wind drift following the reference frame grew fast due to the Kelvin–Helmholtz instability, when the wind friction velocity u* exceeded the threshold of about 1 m s−1. Given the weak dispersion, the nonlinear stage of evolution was addressed using the Riemann simple wave equation modified to describe the increasing disturbances. The analytical solution for the equation suggested the scaling of the volume of liquid undergoing the bag-breakup fragmentation and its dependence on u* in agreement with the laboratory data. Using the scaling, we calculated the statistics of the bag-breakup fragmentation based on the lognormal size distribution of the fragmenting objects.

Significance Statement

The “bag breakup” fragmentation is the dominant mechanism for generating spray in hurricane winds. The parameters of spray droplets substantially affect the exchange processes between the ocean and the atmosphere and, thereby, the development of sea storms. The rapid process of spray generation can only be studied in laboratory environments using sophisticated experimental techniques. To apply the laboratory data to field conditions, we need a theoretical model that describes the threshold for fragmentation initiation, the fragmenting liquid volume, which scales the size and number of spray droplets, their dependence on wind parameters, etc. In the present work, we suggest a simple analytical model of the bag-breakup initiation, verify it in the laboratory experiment, and suggest the statistical description of the fragmentation events.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yuliya Troitskaya, yuliya@hydro.appl.sci-nnov.ru
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  • Alekseenko, S. V., and V. E. Nakoryakov, 1995: Instability of a liquid film moving under the effect of gravity and gas flow. Int. J. Heat Mass Transf., 38, 21272134, https://doi.org/10.1016/0017-9310(94)00326-Q.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Al-Zanaidi, M. A., and W. H. Hui, 1984: Turbulent airflow over water waves – A numerical study. J. Fluid Mech., 148, 225246, https://doi.org/10.1017/S0022112084002329.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Andreas, E. L, 1998: A new sea spray generation function for wind speeds up to 32 m s−1. J. Phys. Oceanogr., 28, 21752184, https://doi.org/10.1175/1520-0485(1998)028<2175:ANSSGF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Andreas, E. L, 2011: Fallacies of the enthalpy transfer coefficient over the ocean in high winds. J. Atmos. Sci., 68, 14351445, https://doi.org/10.1175/2011JAS3714.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Andreas, E. L, and K. A. Emanuel, 2001: Effects of sea spray on tropical cyclone intensity. J. Atmos. Sci., 58, 37413751, https://doi.org/10.1175/1520-0469(2001)058<3741:EOSSOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Azzopardi, B. J., 1997: Drops in annular two-phase flow. Int. J. Multiphase Flow, 23, 153, https://doi.org/10.1016/S0301-9322(97)90087-2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Banner, M. L., and W. L. Peirson, 1998: Tangential stress beneath wind driven air–water interfaces. J. Fluid Mech., 364, 115145, https://doi.org/10.1017/S0022112098001128.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bao, J.-W., C. W. Fairall, S. A. Michelson, and L. Bianco, 2011: Parameterizations of sea-spray impact on the air–sea momentum and heat fluxes. Mon. Wea. Rev., 139, 37813797, https://doi.org/10.1175/MWR-D-11-00007.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Belcher, S. E., and J. C. R. Hunt, 1993: Turbulent shear flow over slowly moving waves. J. Fluid Mech., 251, 109148, https://doi.org/10.1017/S0022112093003350.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Belcher, S. E., T. M. J. Newleyt, and J. C. R. Hunt, 1993: The drag on an undulating surface induced by the flow of a turbulent boundary layer. J. Fluid Mech., 249, 557596, https://doi.org/10.1017/S0022112093001296.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Belcher, S. E., J. A. Harris, and R. L. Street, 1994: Linear dynamics of wind waves in coupled turbulent air—water flow. Part 1. Theory. J. Fluid Mech., 271, 119151, https://doi.org/10.1017/S0022112094001710.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Benney, D. J., 1966: Long waves on liquid films. J. Math. Phys., 45, 150155, https://doi.org/10.1002/sapm1966451150.

  • Betchov, R., and W. O. Criminale, 1967: Stability of Parallel Flows. Academic Press, 345 pp.

  • Bianco, L., J. W. Bao, C. W. Fairall, and S. A. Michelson, 2011: Impact of sea-spray on the atmospheric surface layer. Bound.-Layer Meteor., 140, 361381, https://doi.org/10.1007/s10546-011-9617-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chalikov, D. V., 1986: Numerical simulation of the boundary layer above waves. Bound.-Layer Meteor., 34, 6398, https://doi.org/10.1007/BF00120909.

  • Cherdantsev, A. V., D. B. Hann, and B. J. Azzopardi, 2014: Study of gas-sheared liquid film in horizontal rectangular duct using high-speed LIF technique: Three-dimensional wavy structure and its relation to liquid entrainment. Int. J. Multiphase Flow, 67, 5264, https://doi.org/10.1016/j.ijmultiphaseflow.2014.08.003.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cheung, T. K., and R. L. Street, 1988: The turbulent layer in the water at an air-water interface. J. Fluid Mech., 194, 133, https://doi.org/10.1017/S0022112088002927.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chou, W.-H., and G. M. Faeth, 1998: Temporal properties of secondary drop breakup in the bag breakup regime. Int. J. Multiphase Flow, 24, 889912, https://doi.org/10.1016/S0301-9322(98)00015-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fairall, C. W., J. D. Kepert, and G. J. Holland, 1994: The effect of sea spray on surface energy transports over the ocean. Global Atmos. Ocean Syst., 2, 121142.

    • Search Google Scholar
    • Export Citation
  • Gent, P. R., 1977: A numerical model of the air flow above water waves. Part 2. J. Fluid Mech., 82, 349369, https://doi.org/10.1017/S0022112077000706.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gent, P. R., and P. A. Taylor, 1976: A numerical model of the air flow above water waves. J. Fluid Mech., 77, 105128, https://doi.org/10.1017/S0022112076001158.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harris, J. A., S. E. Belcher, and R. L. Street, 1996: Linear dynamics of wind waves in coupled turbulent air-water flow. Part 2. Numerical model. J. Fluid Mech., 308, 219254, https://doi.org/10.1017/S0022112096001462.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hunt, J. C. R., S. Leibovich, and K. J. Richards, 1988: Turbulent shear flows over low hills. Quart. J. Roy. Meteor. Soc., 114, 14351471, https://doi.org/10.1002/qj.49711448405.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 843858, https://doi.org/10.1175/1520-0485(1992)022<0843:AQLEVM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kandaurov, A., 2021: TSWiWaT spray generation shadowgraph data, version 1. Open Science Framework, accessed 27 May 2021, https://osf.io/pszjw/.

    • Search Google Scholar
    • Export Citation
  • Kawai, S., 1979: Generation of initial wavelets by instability of a coupled shear flow and their evolution to wind waves. J. Fluid Mech., 93, 661, https://doi.org/10.1017/S002211207900197X.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Keller, W. C., T. R. Larson, and J. W. Wright, 1974: Mean speeds of wind waves at short fetch. Radio Sci., 9, 10911100, https://doi.org/10.1029/RS009i012p01091.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Koga, M., 1981: Direct production of droplets from breaking wind-waves—Its observation by a multi-colored overlapping exposure photographing technique. Tellus, 33, 552563, https://doi.org/10.3402/tellusa.v33i6.10776.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kolmogorov, A. N., 1941: On the lognormal distribution of the fragment sizes undergrinding. Dokl. Akad. Nauk SSSR, 31, 99101.

  • Lin, C. C., 1955: The Theory of Hydrodynamic Stability. Cambridge University Press, 126 pp.

  • Makin, V. K., 1979: The wind field above waves. Oceanology, 19, 206212.

  • Miles, J. W., 1957: On the generation of surface waves by shear flows. J. Fluid Mech., 3, 185, https://doi.org/10.1017/S0022112057000567.

  • Miles, J. W., 1959a: On the generation of surface waves by shear flows. Part 2. J. Fluid Mech., 6, 568582, https://doi.org/10.1017/S0022112059000830.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Miles, J. W., 1959b: On the generation of surface waves by shear flows. Part 3. Kelvin-Helmholtz instability. J. Fluid Mech., 6, 583598, https://doi.org/10.1017/S0022112059000842.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Miles, J. W., 1962: On the generation of surface waves by shear flows. Part 4. J. Fluid Mech., 13, 433448, https://doi.org/10.1017/S0022112062000828.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Miles, J., 1996: Surface-wave generation: A viscoelastic model. J. Fluid Mech., 322, 131145, https://doi.org/10.1017/S002211209600273X.

  • Monin, A .S., and A. M. Yaglom, 1975: Mechanics of Turbulence. Statistical Fluid Mechanics, Vol. 1. MIT Press, 769 pp.

  • Okuda, K., S. Kawai, and Y. Toba, 1977: Measurement of skin friction distribution along the surface of wind waves. J. Oceanogr. Soc. Japan, 33, 190198, https://doi.org/10.1007/BF02109691.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Plant, W. J., and J. W. Wright, 1980: Phase speeds of upwind and downwind traveling short gravity waves. J. Geophys. Res., 85, 3304, https://doi.org/10.1029/JC085iC06p03304.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pruppacher, H. R., and J. D. Klett, 1978: Microphysics of Clouds and Precipitation. D. Reidel, 714 pp.

  • Reutov, V. P., and Y. I. Troitskaya, 1995: On nonlinear effects due to water wave interaction with turbulent wind. Izv. Akad. Nauk. Fiz. Atmos. Ocean, 31, 825834.

    • Search Google Scholar
    • Export Citation
  • Riley, D. S., M. A. Donelan, and W. H. Hui, 1982: An extended Miles’ theory for wave generation by wind. Bound.-Layer Meteor., 22, 209225, https://doi.org/10.1007/BF00118254.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rodi, W., 1980: Turbulence Models and Their Application in hydraulics: A State of the Art Review. International Association for Hydraulic Research, 124 pp.

    • Search Google Scholar
    • Export Citation
  • Saetra, Ø., 1998: Effects of surface film on the linear stability of an air-sea interface. J. Fluid Mech., 357, 5981, https://doi.org/10.1017/S002211209700801X.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schlichting, H., 1955: Boundary Layer Theory. McGraw-Hill, 535 pp.

  • Shkadov, V. Y., 1967: Wave flow regimes of a thin layer of viscous fluid subject to gravity. Fluid Dyn., 2, 2934, https://doi.org/10.1007/BF01024797.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Shrira, V., 1993: Surface waves on shear currents: Solution of the boundary-value problem. J. Fluid Mech., 252, 565584, https://doi.org/10.1017/S002211209300388X.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Siddiqui, M. H. K., and M. R. Loewen, 2007: Characteristics of the wind drift layer and microscale breaking waves. J. Fluid Mech., 573, 417456, https://doi.org/10.1017/S0022112006003892.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Smolyakov, A. V., 1973: Spectrum of the quadruple radiation of the plane turbulent boundary layer. Acoust. Phys., 19, 420425.

  • Soloviev, A. V., R. Lukas, M. A. Donelan, B. K. Haus, and I. Ginis, 2014: The air-sea interface and surface stress under tropical cyclones. Sci. Rep., 4, 5306, https://doi.org/10.1038/srep05306.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stiassnie, M., Y. Agnon, and P. A. E. M. Janssen, 2007: Temporal and spatial growth of wind waves. J. Phys. Oceanogr., 37, 106114, https://doi.org/10.1175/JPO2982.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sullivan, P. P., J. C. McWilliams, and W. K. Melville, 2004: The oceanic boundary layer driven by wave breaking with stochastic variability. Part 1. Direct numerical simulations. J. Fluid Mech., 507, 143174, https://doi.org/10.1017/S0022112004008882.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Takagaki, N., S. Komori, N. Suzuki, K. Iwano, T. Kuramoto, S. Shimada, R. Kurose, and K. Takahashi, 2012: Strong correlation between the drag coefficient and the shape of the wind sea spectrum over a broad range of wind speeds. Geophys. Res. Lett., 39, L23604, https://doi.org/10.1029/2012GL053988.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Takagaki, N., S. Komori, N. Suzuki, K. Iwano, and R. Kurose, 2016: Mechanism of drag coefficient saturation at strong wind speeds. Geophys. Res. Lett., 43, 98299835, https://doi.org/10.1002/2016GL070666.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Teixeira, M. A. C., 2018: A model for the wind-driven current in the wavy oceanic surface layer: Apparent friction velocity reduction and roughness length enhancement. J. Phys. Oceanogr., 48, 27212736, https://doi.org/10.1175/JPO-D-18-0086.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Townsend, A. A., 1972: Flow in a deep turbulent boundary layer over a surface distorted by water waves. J. Fluid Mech., 55, 719735, https://doi.org/10.1017/S0022112072002101.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Townsend, A. A., 1980: The response of sheared turbulence to additional distortion. J. Fluid Mech., 98, 171191, https://doi.org/10.1017/S0022112080000092.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Troitskaya, Y. I., 1991: The viscous-diffusion nonlinear critical layer in a stratified shear flow. J. Fluid Mech., 233, 2548, https://doi.org/10.1017/S002211209100037X.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Troitskaya, Y. I., 1997: Wind excitation of surface waves in the coupled turbulent shear flow: A simple model of visco-elastic turbulence. Preprints, IAP RAS 425, Institute of Applied Physics, Russian Academy of Sciences, 44 pp.

    • Search Google Scholar
    • Export Citation
  • Troitskaya, Y. I., D. Sergeev, A. A. Kandaurov, G. A. Baidakov, M. A. Vdovin, and V. I. Kazakov, 2012: Laboratory and theoretical modeling of air-sea momentum transfer under severe wind conditions. J. Geophys. Res., 117, C00J21, https://doi.org/10.1029/2011JC007778.

    • Search Google Scholar
    • Export Citation
  • Troitskaya, Y. I., D. Sergeev, O. Druzhinin, A. A. Kandaurov, O. S. Ermakova, E. V. Ezhova, I. Esau, and S. Zilitinkevich, 2014: Atmospheric boundary layer over steep surface waves. Ocean Dyn., 64, 11531161, https://doi.org/10.1007/s10236-014-0743-4.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Troitskaya, Y. I., A. Kandaurov, O. Ermakova, D. Kozlov, D. Sergeev, and S. Zilitinkevich, 2017: Bag-breakup fragmentation as the dominant mechanism of sea-spray production in high winds. Sci. Rep., 7, 1614, https://doi.org/10.1038/s41598-017-01673-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Troitskaya, Y. I., A. Kandaurov, O. Ermakova, D. Kozlov, D. Sergeev, and S. Zilitinkevich, 2018a: The “bag breakup” spume droplet generation mechanism at high winds. Part I: Spray generation function. J. Phys. Oceanogr., 48, 21672188, https://doi.org/10.1175/JPO-D-17-0104.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Troitskaya, Y. I., O. Druzhinin, D. Kozlov, and S. Zilitinkevich, 2018b: The “Bag Breakup” spume droplet generation mechanism at high winds. Part II: Contribution to momentum and enthalpy transfer. J. Phys. Oceanogr., 48, 21892207, https://doi.org/10.1175/JPO-D-17-0105.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Troitskaya, Y. I., D. Sergeev, A. Kandaurov, M. Vdovin, and S. Zilitinkevich, 2019: The effect of foam on waves and the aerodynamic roughness of the water surface at high winds. J. Phys. Oceanogr., 49, 959981, https://doi.org/10.1175/JPO-D-18-0168.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tsai, W. T., and M. Y. Lin, 2004: Stability analysis on the initial surface-wave generation within an air-sea coupled shear flow. J. Mar. Sci. Technol., 12, 9, http://dx.doi.org/10.51400/2709-6998.2239.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Valenzuela, G. R., 1976: The growth of gravity-capillary waves in a coupled shear flow. J. Fluid Mech., 76, 229250, https://doi.org/10.1017/S0022112076000608.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Van Duin, C. A., and P. A. E. M. Janssen, 1992: An analytic model of the generation of surface gravity waves by turbulent air flow. J. Fluid Mech., 236, 197215, https://doi.org/10.1017/S0022112092001393.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • van Gastel, K., P. A. M. Janssen, and G. Komen, 1985: On phase velocity and growth rate of wind-induced gravity—Capillary waves. J. Fluid Mech., 161, 199, https://doi.org/10.1017/S0022112085002889.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Veron, F., G. Saxena, and S. K. Misra, 2007: Measurements of the viscous tangential stress in the airflow above wind waves. Geophys. Res. Lett., 34, L19603, https://doi.org/10.1029/2007GL031242.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Veron, F., C. Hopkins, E. L. Harrison, and J. A. Mueller, 2012: Sea spray spume droplet production in high wind speeds. Geophys. Res. Lett., 39, L16602, https://doi.org/10.1029/2012GL052603.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitham, G. B., 1974: Linear and Nonlinear Waves. Wiley, 635 pp.

  • Wu, J., 1975: Wind-induced drift currents. J. Fluid Mech., 68, 49, https://doi.org/10.1017/S0022112075000687.

  • Wu, J., 1984: Viscous sublayer below a wind-disturbed water surface. J. Phys. Oceanogr., 14, 138144, https://doi.org/10.1175/1520-0485(1984)014<0138:VSBAWD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zeisel, A., M. Stiassnie, and Y. Agnon, 2008: Viscous effects on wave generation by strong winds. J. Fluid Mech., 597, 343369, https://doi.org/10.1017/S0022112007009858.

    • Crossref
    • Search Google Scholar
    • Export Citation
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