- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Modulational instability of nonuniformly damped, broad-banded waves: Applications to waves in sea ice

Raphael Stuhlmeier* and Conor Heffernan

Alberto Alberello and Emilian Părău

  • *Contact author: raphael.stuhlmeier@plymouth.ac.uk
PDF

Phys. Rev. Fluids 9, 094802 – Published 12 September, 2024

DOI: https://doi.org/10.1103/PhysRevFluids.9.094802

Abstract

This paper sets out to explore the modulational (or Benjamin-Feir) instability of a monochromatic wave propagating in the presence of damping such as that induced by sea ice on the ocean surface. The fundamental wave motion is modelled using the spatial Zakharov equation, to which either uniform or nonuniform (frequency-dependent) damping is added. By means of mode truncation the spatial analog of the classical Benjamin-Feir instability can be studied analytically using dynamical systems techniques. The formulation readily yields the free surface and its envelope, giving insight into the physical implications of damping on the modulational instability. The evolution of an initially unstable mode is also studied numerically by integrating the damped, spatial Zakharov equation, in order to complement the analytical theory. This sheds light on the effects of damping on spectral broadening arising from this instability.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (46)

  1. R. S. Johnson, The classical problem of water waves: A reservoir of integrable and nearly-integrable equations, J. Nonlin. Math. Phys. 10, 72 (2003).
  2. J. F. Toland, Stokes waves, Top. Methods Nonlin. Anal. 7, 1 (1996).
  3. T. Levi-Civita, Détermination rigoureuse des ondes permanentes d'ampleur finie, Math. Ann. 93, 264 (1925).
  4. T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water Part 1. Theory, J. Fluid Mech. 27, 417 (1967).
  5. G. Vanderhaegen, C. Naveau, P. Szriftgiser, A. Kudlinski, M. Conforti, A. Mussot, M. Onorato, S. Trillo, A. Chabchoub, and N. Akhmediev, “Extraordinary” modulation instability in optics and hydrodynamics, Proc. Natl. Acad. Sci. USA 118, e2019348118 (2021).
  6. F. Snodgrass, K. Hasselmann, G. Miller, W. H. Munk, and W. Powers, Propagation of ocean swell across the Pacific, Philos. Trans. R. Soc. London A 259, 431 (1966).
  7. H. Segur, D. Henderson, J. D. Carter, J. L. Hammack, C.-M. Li, D. Pheiff, and K. Socha, Stabilizing the Benjamin–Feir instability, J. Fluid Mech. 539, 229 (2005).
  8. D. Henderson and H. Segur, The Benjamin-Feir instability and propagation of swell across the Pacific, Math. Comput. Simul. 82, 1172 (2012).
  9. M. H. Meylan, L. G. Bennetts, J. E. M. Mosig, W. E. Rogers, M. J. Doble, and M. A. Peter, Dispersion relations, power laws, and energy loss for waves in the marginal ice zone, J. Geophys. Res.: Oceans 123, 3322 (2018).
  10. B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, Nonlinear deep-water waves: Theory and experiment. Part 2. Evolution of a continuous wave train, J. Fluid Mech. 83, 49 (1977).
  11. F. Dias, A. I. Dyachenko, and V. E. Zakharov, Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions, Phys. Lett. A 372, 1297 (2008).
  12. T. Hara and C. C. Mei, Frequency downshift in narrowbanded surface waves under the influence of wind, J. Fluid Mech. 230, 429 (1991).
  13. J. D. Carter and A. Govan, Frequency downshift in a viscous fluid, Eur. J. Mech. B Fluids 59, 177 (2016).
  14. A. Armaroli, D. Eeltink, M. Brunetti, and J. Kasparian, Nonlinear stage of Benjamin-Feir instability in forced/damped deep-water waves, Phys. Fluids 30, 017102 (2018).
  15. M. Stiassnie, Note on the modified nonlinear Schrödinger equation for deep water waves, Wave Motion 6, 431 (1984).
  16. L. Shemer, H. Jiao, E. Kit, and Y. Agnon, Evolution of a nonlinear wave field along a tank: Experiments and numerical simulations based on the spatial Zakharov equation, J. Fluid Mech. 427, 107 (2001).
  17. A. Alberello and E. I. Părău, A dissipative nonlinear Schrödinger model for wave propagation in the marginal ice zone, Phys. Fluids 34, 061702 (2022).
  18. A. Alberello, E. Părău, and A. Chabchoub, The dynamics of unstable waves in sea ice, Sci. Rep. 13, 13654 (2023).
  19. V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 9, 190 (1968).
  20. V. P. Krasitskii, On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves, J. Fluid Mech. 272, 1 (1994).
  21. V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer-Verlag, Berlin, 1992).
  22. R. Stuhlmeier, An introduction to the Zakharov equation for modelling deep water waves, in Nonlinear Dispersive Waves, edited by D. Henry (Birkhäuser, Cham, 2024), pp. 99–131.
  23. E. Kit and L. Shemer, Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves, J. Fluid Mech. 450, 201 (2002).
  24. L. Shemer, K. Goulitski, and E. Kit, Evolution of wide-spectrum unidirectional wave groups in a tank: An experimental and numerical study, Eur. J. Mech. B 26, 193 (2007).
  25. L. Shemer and A. Chernyshova, Spatial evolution of an initially narrow-banded wave train, J. Ocean Eng. Mar. Energy 3, 333 (2017).
  26. V. I. Shrira, S. I. Badulin, and C. Kharif, A model of water ‘horse-shoe’ patterns, J. Fluid Mech. 318, 375 (1996).
  27. V. A. Squire, Ocean wave interactions with sea ice: A reappraisal, Annu. Rev. Fluid Mech. 52, 37 (2020).
  28. O. Kimmoun, H. C. Hsu, H. Brager, M. Li, Y. Y. Chen, C. Kharif, M. Onorato, E. J. Kelleher, B. Kibler, N. Akhmediev, and A. Chabchoub, Modulational instability and phase-shifted Fermi-Pasta-Ulam recurrence, Sci. Rep. 6, 28516 (2016).
  29. R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge University Press, Cambridge, 1997).
  30. D. Andrade and R. Stuhlmeier, Instability of waves in deep water—A discrete Hamiltonian approach, Eur. J. Mech. B 101, 320 (2023).
  31. G. Cappellini and S. Trillo, Third-order three-wave mixing in single-mode fibers: Exact solutions and spatial instability effects, J. Opt. Soc. Am. B 8, 824 (1991).
  32. D. Andrade and R. Stuhlmeier, The nonlinear Benjamin-Feir instability—Hamiltonian dynamics, discrete breathers, and steady solutions, J. Fluid Mech. 958, A17 (2023).
  33. C. Heffernan, A. Chabchoub, and R. Stuhlmeier, Nonlinear spatial evolution of degenerate quartets of water waves, Wave Motion 130, 103381 (2024).
  34. C. C. Mei, M. A. Stiassnie, and D. K.-P. Yue, Theory and Applications of Ocean Surface Waves, 3rd ed. (World Scientific, Singapore, 2018).
  35. H. C. Yuen and B. M. Lake, in Nonlinear Dynamics of Deep-Water Gravity Waves, Advances in Applied Mechanics Vol. 22 (Academic Press, London, 1982), pp. 68–229.
  36. R. Stuhlmeier and M. Stiassnie, Nonlinear dispersion for ocean surface waves, J. Fluid Mech. 859, 49 (2019).
  37. A. Armaroli, M. Brunetti, and J. Kasparian, Recurrence in the high-order nonlinear Schrödinger equation: A low dimensional analysis, Phys. Rev. E 96, 012222 (2017).
  38. P. Janssen, The Interaction of Ocean Waves and Wind (Cambridge University Press, Cambridge, 2004).
  39. M. Galvagno, D. Eeltink, and R. Stuhlmeier, Spatial deterministic wave forecasting for nonlinear sea-states, Phys. Fluids 33, 102116 (2021).
  40. L. Shemer, Quantitative analysis of nonlinear water-waves: A perspective of an experimentalist, in New Approaches to Nonlinear Waves (Springer International Publishing, Cham, 2016), pp. 211–293.
  41. D. Kachulin, A. Dyachenko, and A. Gelash, Interactions of coherent structures on the surface of deep water, Fluids 4, 83 (2019).
  42. A. I. Dyachenko, D. I. Kachulin, and V. E. Zakharov, Super compact equation for water waves, J. Fluid Mech. 828, 661 (2017).
  43. S. Leblanc, On elementary four-wave interactions in dispersive media, J. Fluid Mech. 983, A27 (2024).
  44. T. Waseda, A. Alberello, T. Nose, T. Toyota, T. Kodaira, and Y. Fujiwara, Observation of anomalous spectral downshifting of waves in the Okhotsk Sea marginal ice zone, Philos. Trans. R. Soc. A 380, 20210256 (2022).
  45. R. I. Ivanov, C. I. Martin, and M. D. Todorov, Hamiltonian approach to modelling interfacial internal waves over variable bottom, Physica D 433, 133190 (2022).
  46. C. I. Martin, Resonant interactions of capillary-gravity water waves, J. Math. Fluid Mech. 19, 807 (2017).

Outline

Information

Phys. Rev. Fluids 9, 094802– Published 12 September, 2024
  • Received 28 February 2024
  • Accepted 21 August 2024
Crossmark - Check for updates button

©2024 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation