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Third-order resonant wave interactions under the influence of background current fields

  • Takuji Waseda (a1), T. Kinoshita (a2), L. Cavaleri (a3) and A. Toffoli (a4)


A series of experiments were conducted in a wave basin (50 m long, 10 m wide and 5 m deep) generating two waves propagating at an angle by a directional wavemaker. When the two waves were selected from a resonant triplet, an initially non-existing wave grew as the waves propagated down the tank. The linear growth rate of the resonating wave agreed well with third-order resonance theory based on Zakharov’s reduced gravity equation. Additional experiments with opposing and coflowing mean current with large temporal and spatial variations were conducted. As the flow rate increased, the linear growth was suppressed. As reproduced numerically with Zakharov’s equation, the resonant interaction saturated at time scales inversely proportional to the magnitude of the forced random resonance detuning. It is conjectured that the resonance is detuned by the variation and not by the mean of the current field due to wavelength-dependent Doppler shift and to the refraction of wave rays. Further analysis of the spectral evolution revealed that while discrete peaks appear at high frequencies as a result of dynamical cascading, a continuously saturated spectrum develops in the background as the current speed increases. Additional experiments were conducted studying the evolution of the random directional wave on a dynamical time scale under the influence of current. Due to random resonance detuning by the current, the spectral tail tended to be suppressed.


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Annenkov, S. Y. & Shrira, V. I. 2006 Role of non-resonant interactions in the evolution of nonlinear random water wave fields. J. Fluid Mech. 561, 181207.
Badulin, S. I., Pushkarev, A. N., Resio, D. & Zakharov, V. E. 2005 Self-similarity of wind-driven seas. Nonlinear Process. Geophys. 12 (6), 891945.
Benney, D. J. 1962 Non-linear gravity wave interactions. J. Fluid Mech. 14 (4), 577584.
Cavaleri, L., Alves, J.-H. G. M., Ardhuin, F., Babanin, A., Banner, M., Belibassakis, K., Benoit, M., Donelan, M., Groeneweg, J., Herbers, T. H. C., Hwang, P., Janssen, P. A. E. M., Janssen, T., Lavrenov, I. V., Magne, R., Monbaliu, J., Onorato, M., Polnikov, V., Resio, D., Rogers, W. E., Sheremet, A., McKee Smith, J., Tolman, H. L., van Vledder, G., Wolf, J. & Young, I. 2007 Wave modelling – the state of the art. Prog. Oceanogr. 75 (4), 603674.
Dalrymple, R. A. 1989 Directional wavemaker theory with sidewall reflection. J. Hydraul. Res. 27 (1), 2334.
Fuhrman, D. R., Madsen, P. A. & Bingham, H. B. 2006 Numerical simulation of lowest-order short-crested wave instabilities. J. Fluid Mech. 563, 415441.
Hammack, J. L., Henderson, D. M. & Segur, H. 2005 Progressive waves with persistent two-dimensional surface patterns in deep water. J. Fluid Mech. 532, 152.
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity-wave spectrum. J. Fluid Mech. 12 (15), 481500.
Hasselmann, S. & Hasselmann, K. 1985 Computations and parameterizations of the nonlinear energy transfer in a gravity-wave spectrum. Part I: a new method for efficient computations of the exact nonlinear transfer integral. J. Phys. Oceanogr. 15 (11), 13691377.
Hirobe, T.2013 Numerical study of nonlinear ineteraction of ocean waves and wind influence. PhD thesis, The University of Tokyo.
Hjelmervik, K. B. & Trulsen, K. 2009 Freak wave statistics on collinear currents. J. Fluid Mech. 637, 267284.
Janssen, P. 2004 The Interaction of Ocean Waves and Wind. Cambridge University Press.
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33 (4), 863884.
Jones, A. F. 1984 The generation of cross-waves in a long deep channel by parametric resonance. J. Fluid Mech. 138, 5374.
Kartashova, E. 2009 Discrete wave turbulence. Eur. Phys. Lett. 87 (4), 44001.
Kartashova, E. & Shugan, I. V. 2011 Dynamical cascade generation as a basic mechanism of Benjamin–Feir instability. Eur. Phys. Lett. 95 (3), 30003.
Kit, E., Shemer, L. & Miloh, T. 1987 Experimental and theoretical investigation of nonlinear sloshing waves in a rectangular channel. J. Fluid Mech. 181, 265291.
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 120.
Liao, S.-J. 2011 On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves. Commun. Nonlinear Sci. Numer. Simul. 16 (3), 12741303.
Liu, Z. & Liao, S.-J. 2014 Steady-state resonance of multiple wave interactions in deep water. J. Fluid Mech. 742, 664700.
Liu, Z., Xu, D., Li, J., Peng, T., Alsaedi, A. & Liao, S. J. 2015 On the existence of steady-state resonant waves in experiments. J. Fluid Mech. 763, 123.
Longuet-Higgins, M. S. 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12 (3), 321332.
Longuet-Higgins, M. S. & Smith, N. D. 1966 An experiment on third-order resonant wave interactions. J. Fluid Mech. 25 (3), 417435.
Madsen, P. A. & Fuhrman, D. R. 2006 Third-order theory for bichromatic bi-directional water waves. J. Fluid Mech. 557, 369397.
McGoldrick, L. F., Phillips, O. M., Huang, N. E. & Hodgson, T. H. 1966 Measurements of third-order resonant wave interactions. J. Fluid Mech. 25 (3), 437456.
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves: Nonlinear Aspects, vol. 23. World Scientific.
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3 (2), 185204.
Onorato, M., Proment, D. & Toffoli, A. 2011 Triggering rogue waves in opposing currents. Phys. Rev. Lett. 107 (18), 184502.
Phillips, O. M. 1958 The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mech. 4 (4), 426434.
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9 (2), 193217.
Phillips, O. M. 1985 Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech. 156, 505531.
Pushkarev, A., Resio, D. & Zakharov, V. 2003 Weak turbulent approach to the wind-generated gravity sea waves. Physica D 184 (1), 2963.
Qingpu, Z. 1996 Nonlinear instability of wavetrain under influences of shear current with varying vorticity and air pressure. Acta Mechanica Sin. 12 (1), 2438.
Stewart, R. H. & Joy, J. W. 1974 HF radio measurements of surface currents. In Deep Sea Research and Oceanographic Abstracts, vol. 21, pp. 10391049. Elsevier.
Stiassnie, M. & Shemer, L. 2005 On the interaction of four water-waves. Wave Motion 41 (4), 307328.
Takezawa, S., Kobayashi, K. & Kasahara, A. 1988 Directional irregular waves generated in a long tank. J. Soc. Naval Architects of Japan 163 (6), 222232.
Tamura, H., Waseda, T. & Miyazawa, Y. 2010 Impact of nonlinear energy transfer on the wave field in Pacific hindcast experiments. J. Geophys. Res. 115, C12036.
Tanaka, M. 2001 Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations. J. Fluid Mech. 444, 199221.
Toffoli, A., Waseda, T., Houtani, H., Cavaleri, L., Greaves, D. & Onorato, M. 2015 Rogue waves in opposing currents: an experimental study on deterministic and stochastic wave trains. J. Fluid Mech. 769, 277297.
Toffoli, A., Waseda, T., Houtani, H., Kinoshita, T., Collins, K., Proment, D. & Onorato, M. 2013 Excitation of rogue waves in a variable medium: an experimental study on the interaction of water waves and currents. Phys. Rev. E 87 (5), 051201.
Tomita, H.1989 Theoretical and experimental investigations of interaction among deep-water gravity waves. PhD thesis, The University of Tokyo.
Trulsen, K., Stansberg, C. T. & Velarde, M. G. 1999 Laboratory evidence of three-dimensional frequency downshift of waves in a long tank. Phys. Fluids 11 (1), 235237.
Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197232.
Waseda, T., Kinoshita, T. & Tamura, H. 2009 Evolution of a random directional wave and freak wave occurrence. J. Phys. Oceanogr. 39 (3), 621639.
White, B. S. & Fornberg, B. 1998 On the chance of freak waves at sea. J. Fluid Mech. 355, 113138.
Wu, C. H. & Yao, A. 2004 Laboratory measurements of limiting freak waves on currents. J. Geophys. Res. 109, C12002.
Xu, D., Lin, Z., Liao, S. & Stiassnie, M. 2012 On the steady-state fully resonant progressive waves in water of finite depth. J. Fluid Mech. 710, 379418.
Yao, Y., Tulin, M. P. & Kolaini, A. R. 1994 Theoretical and experimental studies of three-dimensional wavemaking in narrow tanks, including nonlinear phenomena near resonance. J. Fluid Mech. 276, 211232.
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190194.
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Third-order resonant wave interactions under the influence of background current fields

  • Takuji Waseda (a1), T. Kinoshita (a2), L. Cavaleri (a3) and A. Toffoli (a4)


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