Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T18:21:55.277Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  29 October 2015

Takuji Waseda ,
T. Kinoshita ,
L. Cavaleri  and
A. Toffoli
Takuji Waseda*
Affiliation:
Graduate School of Frontier Sciences, University of Tokyo, Kashiwa, Chiba 277-8563, Japan
T. Kinoshita
Affiliation:
Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan
L. Cavaleri
Affiliation:
Institute of Marine Sciences, Castello 2737/F, 30122 Venice, Italy
A. Toffoli
Affiliation:
Centre for Ocean Engineering, Science and Technology, Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia
*
†Email address for correspondence: waseda@k.u-tokyo.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 784 , 10 December 2015 , pp. 51 - 73
Check for updates
Copyright
© 2015 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

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, 181–207.Google Scholar
Badulin, S. I., Pushkarev, A. N., Resio, D. & Zakharov, V. E. 2005 Self-similarity of wind-driven seas. Nonlinear Process. Geophys. 12 (6), 891–945.Google Scholar
Benney, D. J. 1962 Non-linear gravity wave interactions. J. Fluid Mech. 14 (4), 577–584.Google Scholar
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), 603–674.Google Scholar
Dalrymple, R. A. 1989 Directional wavemaker theory with sidewall reflection. J. Hydraul. Res. 27 (1), 23–34.CrossRefGoogle Scholar
Fuhrman, D. R., Madsen, P. A. & Bingham, H. B. 2006 Numerical simulation of lowest-order short-crested wave instabilities. J. Fluid Mech. 563, 415–441.Google Scholar
Hammack, J. L., Henderson, D. M. & Segur, H. 2005 Progressive waves with persistent two-dimensional surface patterns in deep water. J. Fluid Mech. 532, 1–52.Google Scholar
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity-wave spectrum. J. Fluid Mech. 12 (15), 481–500.Google Scholar
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), 1369–1377.2.0.CO;2>CrossRefGoogle Scholar
Hirobe, T.2013 Numerical study of nonlinear ineteraction of ocean waves and wind influence. PhD thesis, The University of Tokyo.Google Scholar
Hjelmervik, K. B. & Trulsen, K. 2009 Freak wave statistics on collinear currents. J. Fluid Mech. 637, 267–284.Google Scholar
Janssen, P. 2004 The Interaction of Ocean Waves and Wind. Cambridge University Press.CrossRefGoogle Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33 (4), 863–884.Google Scholar
Jones, A. F. 1984 The generation of cross-waves in a long deep channel by parametric resonance. J. Fluid Mech. 138, 53–74.Google Scholar
Kartashova, E. 2009 Discrete wave turbulence. Eur. Phys. Lett. 87 (4), 44001.Google Scholar
Kartashova, E. & Shugan, I. V. 2011 Dynamical cascade generation as a basic mechanism of Benjamin–Feir instability. Eur. Phys. Lett. 95 (3), 30003.CrossRefGoogle Scholar
Kit, E., Shemer, L. & Miloh, T. 1987 Experimental and theoretical investigation of nonlinear sloshing waves in a rectangular channel. J. Fluid Mech. 181, 265–291.CrossRefGoogle Scholar
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
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), 1274–1303.Google Scholar
Liu, Z. & Liao, S.-J. 2014 Steady-state resonance of multiple wave interactions in deep water. J. Fluid Mech. 742, 664–700.CrossRefGoogle Scholar
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, 1–23.Google Scholar
Longuet-Higgins, M. S. 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12 (3), 321–332.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Smith, N. D. 1966 An experiment on third-order resonant wave interactions. J. Fluid Mech. 25 (3), 417–435.Google Scholar
Madsen, P. A. & Fuhrman, D. R. 2006 Third-order theory for bichromatic bi-directional water waves. J. Fluid Mech. 557, 369–397.CrossRefGoogle Scholar
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), 437–456.Google Scholar
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves: Nonlinear Aspects, vol. 23. World Scientific.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3 (2), 185–204.Google Scholar
Onorato, M., Proment, D. & Toffoli, A. 2011 Triggering rogue waves in opposing currents. Phys. Rev. Lett. 107 (18), 184502.Google Scholar
Phillips, O. M. 1958 The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mech. 4 (4), 426–434.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9 (2), 193–217.CrossRefGoogle Scholar
Phillips, O. M. 1985 Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech. 156, 505–531.Google Scholar
Pushkarev, A., Resio, D. & Zakharov, V. 2003 Weak turbulent approach to the wind-generated gravity sea waves. Physica D 184 (1), 29–63.Google Scholar
Qingpu, Z. 1996 Nonlinear instability of wavetrain under influences of shear current with varying vorticity and air pressure. Acta Mechanica Sin. 12 (1), 24–38.Google Scholar
Stewart, R. H. & Joy, J. W. 1974 HF radio measurements of surface currents. In Deep Sea Research and Oceanographic Abstracts, vol. 21, pp. 1039–1049. Elsevier.Google Scholar
Stiassnie, M. & Shemer, L. 2005 On the interaction of four water-waves. Wave Motion 41 (4), 307–328.Google Scholar
Takezawa, S., Kobayashi, K. & Kasahara, A. 1988 Directional irregular waves generated in a long tank. J. Soc. Naval Architects of Japan 163 (6), 222–232.CrossRefGoogle Scholar
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.Google Scholar
Tanaka, M. 2001 Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations. J. Fluid Mech. 444, 199–221.CrossRefGoogle Scholar
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, 277–297.Google Scholar
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.Google Scholar
Tomita, H.1989 Theoretical and experimental investigations of interaction among deep-water gravity waves. PhD thesis, The University of Tokyo.Google Scholar
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), 235–237.CrossRefGoogle Scholar
Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197–232.Google Scholar
Waseda, T., Kinoshita, T. & Tamura, H. 2009 Evolution of a random directional wave and freak wave occurrence. J. Phys. Oceanogr. 39 (3), 621–639.Google Scholar
White, B. S. & Fornberg, B. 1998 On the chance of freak waves at sea. J. Fluid Mech. 355, 113–138.Google Scholar
Wu, C. H. & Yao, A. 2004 Laboratory measurements of limiting freak waves on currents. J. Geophys. Res. 109, C12002.Google Scholar
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, 379–418.Google Scholar
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, 211–232.Google Scholar
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), 190–194.Google Scholar