Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-23T08:34:53.342Z Has data issue: false hasContentIssue false
  • English
  • Français

Excitation of interfacial waves via surface–interfacial wave interactions

Published online by Cambridge University Press:  23 January 2020

Joseph Zaleski ,
Philip Zaleski  and
Yuri V. Lvov Open the ORCID record for Yuri V. Lvov [Opens in a new window]
Joseph Zaleski
Affiliation:
Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY12180, USA
Philip Zaleski
Affiliation:
Department of Mathematics, New Jersey Institute of Technology, Newark, NJ07102, USA
Yuri V. Lvov*
Affiliation:
Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY12180, USA
*
†Email address for correspondence: lvovy@rpi.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider interactions between surface and interfacial waves in a two-layer system. Our approach is based on the Hamiltonian structure of the equations of motion, and includes the general procedure for diagonalization of the quadratic part of the Hamiltonian. Such diagonalization allows us to derive the interaction cross-section between surface and interfacial waves and to derive the coupled kinetic equations describing spectral energy transfers in this system. Our kinetic equation allows resonant and near-resonant interactions. We find that the energy transfers are dominated by the class III resonances of Alam (J. Fluid Mech., vol. 691, 2012, pp. 267–278). We apply our formalism to calculate the rate of growth for interfacial waves for different values of wind velocity. Using our kinetic equation, we also consider the energy transfer from wind-generated surface waves to interfacial waves for the case when the spectrum of the surface waves is given by the JONSWAP spectrum and interfacial waves are initially absent. We find that such energy transfer can occur along a time scale of hours; there is a range of wind speeds for the most effective energy transfer at approximately the wind speed corresponding to white capping of the sea. Furthermore, interfacial waves oblique to the direction of the wind are also generated.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering Mathematical Foundations: Hamiltonian theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 887 , 25 March 2020 , A14
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alam, M. R. 2012 A new triad resonance between co-propagating surface and interfacial waves. J. Fluid Mech. 691, 267–278.CrossRefGoogle Scholar
Ambrosi, D. 2000 Hamiltonian formulation for surface waves in a layered fluid. Wave Motion 31, 71–76.CrossRefGoogle Scholar
Ball, F. K. 1964 Energy transfer between external and internal gravity waves. J. Fluid Mech. 19, 465–478.CrossRefGoogle Scholar
Benney, D. J. & Saffmann, P. 1966 Nonlinear interaction of random waves in a dispersive medium. Proc. R. Soc. Lond. 289, 301–320.Google Scholar
Benney, J. & Newell, A. C. 1969 Random wave closure. Stud. Appl. Maths 48, 1.CrossRefGoogle Scholar
Bogoljubov, N. N. 1958 On a new method in the theory of superconductivity. Nuovo Cimento 7 (6), 794–805.CrossRefGoogle Scholar
Choi, W. 2019 Nonlinear interaction between surface and internal waves. Part I: nonlinear models and spectral formulation. J. Fluid Mech. (submitted).Google Scholar
Choi, Y., Lvov, Y. V. & Nazarenko, S. 2004 Probability densities and preservation of randomness in wave turbulence. Phys. Lett. A 332, 230–238.CrossRefGoogle Scholar
Choi, Y., Lvov, Y. V. & Nazarenko, S. 2005a Joint statistics of amplitudes and phases in wave turbulence. Physica D 201, 121–149.CrossRefGoogle Scholar
Choi, Y., Lvov, Y. V., Nazarenko, S. & Pokorni, B. 2005b Anomalous probability of large amplitudes in wave turbulence. Phys. Lett. A 339, 361–369.CrossRefGoogle Scholar
Chow, Y. 1983 A study of resonant interactions between internal and surface waves based on a two layer fluid model. Wave Motion 5, 145–155.Google Scholar
Constantin, A. & Ivanov, R. I. 2015 A Hamiltonian approach to wave–current interactions in two-layer fluids. Phys. Fluids 27, 086603.CrossRefGoogle Scholar
Dysthe, K. B. & Das, K. P. 1981 Coupling between a surface wave spectrum and an internal wave: modulational interaction. J. Fluid Mech. 104, 483–503.CrossRefGoogle Scholar
Gargettt, A. E. & Hughes, B. A. 1972 On the interaction of surface and internal waves. J. Fluid Mech. 52 (1), 179–191.CrossRefGoogle Scholar
Haney, S. & Young, W. R. 2017 Radiation of internal waves from groups of surface gravity waves. J. Fluid Mech. 829, 280–303.CrossRefGoogle Scholar
Hassleman, K., Barnett, T. P., Bouws, E. & Carlson, H. 1973 Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Deut. Hydrogr. Z. 8, 1–95.Google Scholar
Janssen, P. A. E. M. 2004 The Interaction of Ocean Waves and Wind. Cambridge University Press.CrossRefGoogle Scholar
Kadomtsev, B. B. 1965 Plasma Turbulence. Academic Press.Google Scholar
Lvov, V., Lvov, Y. V., Newell, A. & Zakharov, V. 1997 Statistical description of acoustic turbulence. Phys. Rev. E 56, 390–405.Google Scholar
Lvov, Y. V. & Nazarenko, S. 2004 Noisy spectra, long correlations, and intermittency in wave turbulence. Phys. Rev. E 69, 066608.Google ScholarPubMed
Lvov, Y. V., Polzin, K. L. & Tabak, E. G. 2004 Energy spectra of the ocean’s internal wave field: theory and observations. Phys. Rev. Lett. 92, 128501.CrossRefGoogle ScholarPubMed
Lvov, Y. V., Polzin, K. L. & Yokoyama, N. 2012 Resonant and near-resonant internal wave interactions. J. Phys. Oceanogr. 42, 669–691.CrossRefGoogle Scholar
Lvov, Y. V. & Tabak, E. 2001 Hamiltonian formalism and the Garrett-Munk spectrum of internal waves in the ocean. Phys. Rev. Lett. 87, 168501.CrossRefGoogle ScholarPubMed
Lvov, Y. V. & Tabak, E. 2004 A Hamiltonian formulation for long internal waves. Physica D 195, 106–122.CrossRefGoogle Scholar
Lvov, Y. V., Tabak, E., Polzin, K. L. & Yokoyama, N. 2010 The oceanic internal wavefield: theory of scale invariant spectra. J. Phys. Oceanogr. 40, 2605–2623.CrossRefGoogle Scholar
Lvov, Y. V. & Yokoyama, N. 2009 Wave–wave interactions in stratified flows: direct numerical simulations. Physica D 238, 803–815.CrossRefGoogle Scholar
Muller, P., Henyey, F. & Pomphrey, N. 1986 Nonlinear interactions among internal gravity waves. Rev. Geophys. 24 (3), 493–536.CrossRefGoogle Scholar
Nazarenko, S. 2011 Wave Turbulence. Springer.CrossRefGoogle Scholar
Newell, A. C. 1968 The closure problem in a system of random gravity waves. Rev. Geophys. 6, 1–31.CrossRefGoogle Scholar
Olbers, D. & Eden, C. 2016 Revisiting the generation of internal waves by resonant interaction with surface waves. J. Phys. Oceanogr. 46, 2335–2350.CrossRefGoogle Scholar
Olbers, D. & Herterich, K. 1979 The spectral energy transfer from surface waves to internal waves. J. Fluid Mech. 92 (2), 349–379.CrossRefGoogle Scholar
Pierson, W. J. & Moskowitz, L. 1964 A proposed spectral form for fully developed seas based on the similarity theory of of S. A. Kitaigorodskii. J. Fluid Mech. 69 (24), 5181–5190.Google Scholar
Polzin, K. L. & Lvov, Y. V. 2011 Toward regional characterizations of the oceanic internal wavefield. Rev. Geophys. 49, RG4003.CrossRefGoogle Scholar
Polzin, K. L. & Lvov, Y. 2017 An oceanic ultra-violet catastrophe, wave-particle duality and a strongly nonlinear concept for geophysical turbulence. Fluids 2, 36.CrossRefGoogle Scholar
Segur, H. 1980 Resonant interactions of surface and internal gravity waves. Phys. Fluids 23, 2556.CrossRefGoogle Scholar
Tanaka, M. & Wakayama, K. 2015 The spectral energy transfer from surface waves to internal waves. J. Fluid Mech. 763, 202–217.CrossRefGoogle Scholar
Thorpe, S. A. 1966 On wave interactions in a stratified fluid. J. Fluid Mech. 24 (4), 737–751.CrossRefGoogle Scholar
Valatin, J. G. 1958 Comments on the theory of superconductivity. Nuovo Cimento 7 (6), 843–857.CrossRefGoogle Scholar
Watson, K. 1989 The coupling of surface and internal gravity waves: revisited. J. Phys. Oceanogr. 20, 1233–1247.2.0.CO;2>CrossRefGoogle Scholar
Watson, K. 1994 Energy transfer between surface and internal waves in the North Pacific Ocean. J. Geophys. Res. 99, 12549–12560.CrossRefGoogle Scholar
Watson, K., West, B. J. & Cohen, B. I. 1976 Coupling of surface and internal gravity waves: a mode coupling model. J. Fluid Mech. 77, 185–193.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of period waves of finite amplitude on surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence. Springer.CrossRefGoogle Scholar