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Modeling “crossing sea state” wave patterns in layered and stratified fluids

H. M. Yin, Q. Pan*, and K. W. Chow

  • Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong

  • *upanqing@connect.hku.hk
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Phys. Rev. Fluids 8, 014802 – Published 17 January, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.014802

Abstract

Free surface waves with two or more spectral peaks propagating at an oblique angle to each other are commonly termed “crossing sea states”. Such crossing patterns have been suggested as possible causes for rogue waves and maritime accidents. Modulation instabilities of plane waves using coupled Schrödinger or Zakharov equations have been adopted as theoretical models in the literature. Here, extensions to layered and stratified fluids are conducted. For a two-layer fluid with long-wave–short-wave resonance, crossing patterns with two short waves will enhance instability compared with the single-wave case. Analytical treatment beyond the linear instability regime is elucidated by a cascading mechanism. Growth of the higher-order harmonics eventually leads to finite-amplitude pulsating modes or breathers. Breathers subsequently exhibit a Fermi-Pasta-Ulam-Tsingou type recurrence. The time for the first formation of breathers predicted by the cascading mechanism attains excellent agreements with the full numerical simulations. A similar study is performed for a continuously stratified fluid with constant buoyancy frequency. Triad resonance with two components as a pair of oblique waves also produces enhanced instability and a preferred inclination of maximum growth rate. These crossing patterns will likely play critical roles in many wave-propagation configurations in fluid mechanics.

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References (45)

  1. M. Onorato, A. R. Osborne, and M. Serio, Modulational Instability in Crossing Sea States: A Possible Mechanism for the Formation of Freak Waves, Phys. Rev. Lett. 96, 014503 (2006).
  2. P. K. Shukla, I. Kourakis, B. Eliasson, M. Marklund, and L. Stenflo, Instability and Evolution of Nonlinearly Interacting Water Waves, Phys. Rev. Lett. 97, 094501 (2006).
  3. S. Chakrabortty and S. Debsarma, Modulational instability of two obliquely interacting waves with different carrier frequencies, Phys. Fluids 33, 127113 (2021).
  4. S. Manna and A. K. Dhar, Modulational instability of obliquely interacting capillary-gravity waves over infinite depth, Arch. Mech. 73, 583 (2021).
  5. J. N. Steer, A. G. L. Borthwick, M. Onorato, A. Chabchoub, and T. S. van den Bremer, Hydrodynamic X Waves, Phys. Rev. Lett. 123, 184501 (2019).
  6. O. Gramstad, E. Bitner-Gregersen, K. Trulsen, and J. C. N. Borge, Modulational instability and rogue waves in crossing sea states, J. Phys. Oceanogr. 48, 1317 (2018).
  7. F. E. Laine-Pearson, Instability growth rates of crossing sea states, Phys. Rev. E 81, 036316 (2010).
  8. T. A. A. Adcock, P. H. Taylor, S. Yan, Q. W. Ma, and P. A. E. M. Janssen, Did the Draupner wave occur in a crossing sea? Proc. R. Soc. A 467, 3004 (2011).
  9. V. P. Ruban, Two different kinds of rogue waves in weakly crossing sea states, Phys. Rev. E 79, 065304(R) (2009).
  10. S. Kundu, S. Debsarma, and K. P. Das, Wind effect on the evolution of two obliquely interacting random wave trains in deep water, Wave Motion 89, 14 (2019).
  11. J. F. Luxmoore, S. Ilic, and N. Mori, On kurtosis and extreme waves in crossing directional seas: A laboratory experiment, J. Fluid Mech. 876, 792 (2019).
  12. M. L. McAllister, S. Draycott, T. A. A. Adcock, P. H. Taylor, and T. S. van den Bremer, Laboratory recreation of the Draupner wave and the role of breaking in crossing seas, J. Fluid Mech. 860, 767 (2019).
  13. L. Cavaleri, L. Bertotti, L. Torrisi, E. Bitner-Gregersen, M. Serio, and M. Onorato, Rogue waves in crossing seas: The Louis Majesty accident, J. Geophys. Res. 117, C00J10 (2012).
  14. K. Trulsen, J. C. Nieto Borge, O. Gramstad, L. Aouf, and J.-M. Lefèvre, Crossing sea state and rogue wave probability during the Prestige accident, J. Geophys. Res. Oceans 120, 7113 (2015).
  15. D. J. Benney, A general theory for interactions between short and long waves, Stud. Appl. Math. 56, 81 (1977).
  16. R. Grimshaw, Modulation of an internal gravity wave packet in a stratified shear flow, Wave Motion 3, 81 (1981).
  17. M. Oikawa, M. Okamura, and M. Funakoshi, Two-dimensional resonant interaction between long and short waves, J. Phys. Soc. Jpn. 58, 4416 (1989).
  18. C. G. Koop and L. G. Redekopp, The interaction of long and short internal gravity waves: Theory and experiment, J. Fluid Mech. 111, 367 (1981).
  19. K. W. Chow, H. N. Chan, D. J. Kedziora, and R. H. J. Grimshaw, Rogue wave modes for the long wave–short wave resonance model, J. Phys. Soc. Jpn. 82, 074001 (2013).
  20. S. Chen, P. Grelu, and J. M. Soto-Crespo, Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance, Phys. Rev. E 89, 011201(R) (2014).
  21. S. Chen, J. M. Soto-Crespo, and P. Grelu, Coexisting rogue waves within the (2+1)−component long-wave–short-wave resonance, Phys. Rev. E 90, 033203 (2014).
  22. K. Sakkaravarthi, T. Kanna, M. Vijayajayanthi, and M. Lakshmanan, Multicomponent long-wave–short-wave resonance interaction system: Bright solitons, energy-sharing collisions, and resonant solitons, Phys. Rev. E 90, 052912 (2014).
  23. J. Chen, Y. Chen, B.-F. Feng, and K. Maruno, Multi-dark soliton solutions of the two-dimensional multi-component Yajima–Oikawa systems, J. Phys. Soc. Jpn. 84, 034002 (2015).
  24. J. Chen, Y. Chen, B.-F. Feng, and K. Maruno, Rational solutions to two- and one-dimensional multicomponent Yajima-Oikawa systems, Phys. Lett. A 379, 1510 (2015).
  25. H. N. Chan, E. Ding, D. J. Kedziora, R. Grimshaw, and K. W. Chow, Rogue waves for a long wave-short wave resonance model with multiple short waves, Nonlinear Dyn. 85, 2827 (2016).
  26. J. Rao, T. Kanna, and J. He, A study on resonant collision in the two-dimensional multi-component long-wave–short-wave resonance system, Proc. R. Soc. A 478, 2258 (2022).
  27. A. D. D. Craik, Wave Interactions and Fluid Flows (Cambridge University Press, Cambridge, 1985).
  28. A. K. Liu and D. J. Benney, The evolution of nonlinear wave trains in stratified shear flows, Stud. Appl. Math. 64, 247 (1981).
  29. T. Dauxois, S. Joubaud, P. Odier, and A. Venaille, Instabilities of internal gravity wave beams, Annu. Rev. Fluid Mech. 50, 131 (2018).
  30. B. R. Sutherland and R. Jefferson, Triad resonant instability of horizontally periodic internal modes, Phys. Rev. Fluids 5, 034801 (2020).
  31. L. Biswas and P. Shukla, Resonant triad interactions in a stably stratified uniform shear flow, Phys. Rev. Fluids 7, 023904 (2022).
  32. S. A. Chin, O. A. Ashour, and M. R. Belić, Anatomy of the Akhmediev breather: Cascading instability, first formation time, and Fermi-Pasta-Ulam recurrence, Phys. Rev. E 92, 063202 (2015).
  33. H. M. Yin, Q. Pan, and K. W. Chow, Four-wave mixing and coherently coupled Schrödinger equations: Cascading processes and Fermi–Pasta–Ulam–Tsingou recurrence, Chaos 31, 083117 (2021).
  34. G. Van Simaeys, P. Emplit, and M. Haelterman, Experimental study of the reversible behavior of modulational instability in optical fibers, J. Opt. Soc. Am. B 19, 477 (2002).
  35. C. C. Mei, The Applied Dynamics of Ocean Surface Waves (World Scientific, Singapore, 1983).
  36. Z. Mu, Haochen Li, and Y. Wang, A novel energy-preserving scheme for the coupled nonlinear Schrödinger equations, Int. J. Comput. Math. 95, 61 (2018).
  37. H. N. Chan and K. W. Chow, Rogue waves for an alternative system of coupled Hirota equations: Structural robustness and modulation instabilities, Stud. Appl. Math. 139, 78 (2017).
  38. D. J. Kaup, A. Reiman, and A. Bers, Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium, Rev. Mod. Phys. 51, 275 (1979).
  39. F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, Rogue Waves Emerging from the Resonant Interaction of Three Waves, Phys. Rev. Lett. 111, 114101 (2013).
  40. S. Chen, F. Baronio, J. M. Soto-Crespo, P. Grelu, and D. Mihalache, Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems, J. Phys. A: Math. Theor. 50, 463001 (2017).
  41. D. O. Mora, E. Monsalve, M. Brunet, T. Dauxois, and P.-P. Cortet, Three-dimensionality of the triadic resonance instability of a plane inertial wave, Phys. Rev. Fluids 6, 074801 (2021).
  42. A. Purkait and S. Debsarma, Modulational instability of two obliquely interacting waves in presence of a thin pycnocline, Eur. J. Mech. B Fluids 84, 517 (2020).
  43. Z. W. Xu, G.-F. Yu, and Z. N. Zhu, Bright-dark soliton solutions of the multi-component AB system, Wave Motion 83, 134 (2018).
  44. X. G. Geng, J. Shen, and B. Xue, Dynamical behaviour of rogue wave solutions in a multi-component AB system, Wave Motion 89, 1 (2019).
  45. K. Nakayama, T. Kakinuma, and H. Tsuji, Oblique reflection of large internal solitary waves in a two-layer fluid, Eur. J. Mech. B Fluids 74, 81 (2019).

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Phys. Rev. Fluids 8, 014802– Published 17 January, 2023
  • Received 3 May 2022
  • Accepted 13 December 2022
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