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Experimental investigation on the evolution of the modulation instability with dissipation

Published online by Cambridge University Press:  03 September 2012

Y. Ma ,
G. Dong ,
M. Perlin ,
X. Ma  and
G. Wang
Y. Ma*
Affiliation:
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China
G. Dong
Affiliation:
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China
M. Perlin
Affiliation:
Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA
X. Ma
Affiliation:
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China
G. Wang
Affiliation:
School of Harbor, Coastal and Offshore Eningeering, Hohai University, Nanjing, 210098, China
*
†Email address for correspondence: yuxma@126.com
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Abstract

An experimental investigation focusing on the effect of dissipation on the evolution of the Benjamin–Feir instability is reported. A series of wave trains with added sidebands, and varying initial steepness, perturbed amplitudes and frequencies, are physically generated in a long wave flume. The experimental results directly confirm the stabilization theory of Segur et al. (J. Fluid Mech., vol. 539, 2005, pp. 229–271), i.e. dissipation can stabilize the Benjamin–Feir instability. Furthermore, the experiments reveal that the effect of dissipation on modulational instability depends strongly on the perturbation frequency. It is found that the effect of dissipation on the growth rates of the sidebands for the waves with higher perturbation frequencies is more evident than on those of waves with lower perturbation frequencies. In addition, numerical simulations based on Dysthe’s equation with a linear damping term included, which is estimated from the experimental data, can predict the experimental results well if the momentum integral of the wave trains is conserved during evolution.

JFM classification

Geophysical and Geological Flows: Ocean processes Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 711 , 25 November 2012 , pp. 101 - 121
Copyright
Copyright © Cambridge University Press 2012

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References

1. Alber, I. E. 1978 The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. R. Soc. Lond. A 363, 525–546.Google Scholar
2. Banner, M. L. & Peirson, W. L. 2007 Wave breaking onset and strength for two-dimensional deep-water wave groups. J. Fluid Mech. 585, 93–115.CrossRefGoogle Scholar
3. Benjamin, T. B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299, 59–75.Google Scholar
4. Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part I. Theory. J. Fluid Mech. 27, 417–430.CrossRefGoogle Scholar
5. Canney, N. E. & Carter, J. 2007 Stability of plane waves on deep water with dissipation. Math. Comput. Simul. 74, 159–167.CrossRefGoogle Scholar
6. Carter, J. D. & Contreras, C. C. 2008 Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrodinger equation. Physica D 237 (24), 3292–3296.CrossRefGoogle Scholar
7. Chiang, W. S. & Hwung, H. H. 2007 Steepness effect on modulation instability of the nonlinear wave train. Phys. Fluids 19, 1–13.CrossRefGoogle Scholar
8. Dysthe, K. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 105–114.Google Scholar
9. Gramstad, O. & Trulsen, K. 2011 Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth. J. Fluid Mech. 670, 404–426.CrossRefGoogle Scholar
10. Hammack, J. L. & Henderson, D. M. 1993 Resonant interactions among surface-water waves. Annu. Rev. Fluid Mech. 25, 55–97.CrossRefGoogle Scholar
11. Henderson, D. M., Segur, H. & Carter, J. D. 2010 Experimental evidence of stable wave patterns on deep water. J. Fluid Mech. 658, 247–278.CrossRefGoogle Scholar
12. Hwung, H. H., Chiang, W. S. & Hsiao, S. C. 2007 Observations on the evolution of wave modulation. Proc. R. Soc. Lond. A 463 (2077), 85–112.Google Scholar
13. Islas, A. & Schober, C. M. 2011 Rogue waves, dissipation, and downshifting. Physica D 240 (12), 1041–1054.CrossRefGoogle Scholar
14. Janssen, P. A. E. M. 2003 Nonlinear four wave interaction and freak waves. J. Phys. Oceanogr. 33, 863–884.2.0.CO;2>CrossRefGoogle Scholar
15. Kharif, C., Kraenkel, R. A., Manna, M. A. & Thomas, R. 2010 The modulational instability in deep water under the action of wind and dissipation. J. Fluid Mech. 664, 138–149.CrossRefGoogle Scholar
16. Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
17. Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves – theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83 (November), 49–74.CrossRefGoogle Scholar
18. Lo, E. & Mei, C. C. 1985 A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger-equation. J. Fluid Mech. 150, 395–416.CrossRefGoogle Scholar
19. Mei, C. C. & Hancock, M. J. 2003 Weakly nonlinear surface waves over a random seabed. J. Fluid Mech. 475, 247–268.CrossRefGoogle Scholar
20. Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165–185.CrossRefGoogle Scholar
21. Segur, H., Henderson, D., Carter, J., Hammack, J., Li, C., Pheiff, D. & Socha, K. 2005 Stabilizing the Benjamin–Feir instability. J. Fluid Mech. 539, 229–271.CrossRefGoogle Scholar
22. Tian, Z., Perlin, M. & Choi, W. 2010 Energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model. J. Fluid Mech. 655, 217–257.CrossRefGoogle Scholar
23. Touboul, J. & Kharif, C. 2010 Nonlinear evolution of the modulational instability under weak forcing and damping. Nat. Haz. Earth Syst. Sci. 10 (12), 2589–2597.CrossRefGoogle Scholar
24. Trulsen, K. & Dysthe, K. B. 1996 A modified nonlinear Schrodinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24 (3), 281–289.CrossRefGoogle Scholar
25. Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197–232.CrossRefGoogle Scholar
26. Waseda, T. & Tulin, M. P. 1999 Experimental study of the stability of deep-water wave trains including wind effects. J. Fluid Mech. 401, 55–84.CrossRefGoogle Scholar
27. Wu, G. Y., Liu, Y. M. & Yue, D. K. P. 2006 A note on stabilizing the Benjamin–Feir instability. J. Fluid Mech. 556, 45–54.CrossRefGoogle Scholar
28. Yuen, H. C. & Lake, B. M. 1980 Instabilities of waves on deep-water. Annu. Rev. Fluid Mech. 12, 303–334.CrossRefGoogle Scholar
29. Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 19.Google Scholar