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October  2012, 17(7): 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability

Zhi-Min Chen 1, and  Philip A. Wilson 2,

1. 

School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ
2. 

Ship Science, University of Southampton, Southampton SO17 1BJ, United Kingdom

Received  November 2010 Revised  March 2012 Published  July 2012

The Benjamin-Feir instability describes the instability of a uniform oscillatory wave train in an irrotational flow subject to small perturbation of wave number, amplitude and frequency. Their instability analysis is based on the perturbation around the second order Stokes wave which satisfies the dynamic and kinematic free-surface boundary conditions up to the second order. In the same irrotational flow and perturbation framework of the Benjamin-Feir analysis, the perturbation in the present paper is around a nonlinear oscillatory wave train which solves exactly the dynamic free-surface boundary condition and satisfies the kinematic free-surface boundary condition up to the third order. It is shown that the nonlinear oscillatory wave train is stable with respect to the perturbation when the irrotational flow involves small Rayleigh energy dissipation.
Keywords: Benjamin-Feir instability, gravity wave, stability., oscillatory wave, potential flow.
Mathematics Subject Classification: 35Q35, 76B15, 76D05, 76E9.
Citation: Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329
References:
[1]

N. F. Bondarenko, M. Z. Gak and F. V. Dolzhansky, Laboratory and theoretical models of a plane periodic flow, Izv. Atmos. Oceanic Phys., 15 (1979), 711-716.

[2]

N. E. Canney and J. D. Carter, Stability of plane waves on deep water with dissipation, Math. Comp. Simul., 74 (2007), 159-167. doi: 10.1016/j.matcom.2006.10.010.

[3]

J. G. Charney and J. G. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216. doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2.

[4]

B. Chen and P. G. Saffman, Numerical evidence for the existence of new types of gravity waves of permanent form on deep water, Stud. Appl. Math., 62 (1980), 1-21.

[5]

Z.-M. Chen, A vortex based panel method for potential flow simulation around a hydrofoil, J. Fluids Stuct., 28 (2012), 378-391. doi: 10.1016/j.jfluidstructs.2011.10.003.

[6]

T. B. Benjamin, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. R. Soc. A, 299 (1967), 59-75. doi: 10.1098/rspa.1967.0123.

[7]

T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech., 27 (1967), 417-430. doi: 10.1017/S002211206700045X.

[8]

T. J. Bridges and A. Mielke, A proof of the Benjamin-Feir instability, Arch. Ratianal Mech. Anal., 133 (1995), 145-198. doi: 10.1007/BF00376815.

[9]

H. Lamb, "Hydrodynamics,'' Cambridge University Press, Cambridge, 1932.

[10]

M. S. Longuet-Higgins, On the stability of steep gravity waves, Proc. R. Soc. Lond. Ser. A, 396 (1984), 269-280. doi: 10.1098/rspa.1984.0122.

[11]

J. C. Luke, A variational principle for a fluid with a free surface, J. Fluid Mech., 27 (1967), 395-397. doi: 10.1017/S0022112067000412.

[12]

J. Pedlosky, "Geophysical Fluid Dynamics,'' Springer-Verlag, New York, 1979.

[13]

H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff and K. Socha, Stabilizing the Benjamin-Feir instability, J. Fluid Mech., 539 (2005), 229-271.

[14]

G. G. Stokes, On the theory of osillatory waves, Trans. Cambridge Philos. Soc., 8 (1847), 441-473.

[15]

M. Tanaka, The stability of steep gravity waves, J. Phys. Soc. Japan, 52 (1983), 3047-3055. doi: 10.1143/JPSJ.52.3047.

[16]

M. Tanaka, The stability of steep gravity waves. II, J. Fluid Mech., 156 (1985), 281-289. doi: 10.1017/S0022112085002099.

[17]

G. B. Whitham, "Linear and Nonlinear Waves,'' Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[18]

G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Comm. Pure Appl. Math., 41 (1988), 19-46. doi: 10.1002/cpa.3160410104.

[19]

G. Wu, Y. Liu, and D. K. P. Yue, A note on stabilizing the Benjamin-Feir instability, J. Fluid Mech., 556 (2006), 45-54. doi: 10.1017/S0022112005008293.

[20]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of adeep fluid, J. Appl. Mech. Tech. Phys., 2 (1968), 190-194.

show all references

References:
[1]

N. F. Bondarenko, M. Z. Gak and F. V. Dolzhansky, Laboratory and theoretical models of a plane periodic flow, Izv. Atmos. Oceanic Phys., 15 (1979), 711-716.

[2]

N. E. Canney and J. D. Carter, Stability of plane waves on deep water with dissipation, Math. Comp. Simul., 74 (2007), 159-167. doi: 10.1016/j.matcom.2006.10.010.

[3]

J. G. Charney and J. G. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216. doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2.

[4]

B. Chen and P. G. Saffman, Numerical evidence for the existence of new types of gravity waves of permanent form on deep water, Stud. Appl. Math., 62 (1980), 1-21.

[5]

Z.-M. Chen, A vortex based panel method for potential flow simulation around a hydrofoil, J. Fluids Stuct., 28 (2012), 378-391. doi: 10.1016/j.jfluidstructs.2011.10.003.

[6]

T. B. Benjamin, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. R. Soc. A, 299 (1967), 59-75. doi: 10.1098/rspa.1967.0123.

[7]

T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech., 27 (1967), 417-430. doi: 10.1017/S002211206700045X.

[8]

T. J. Bridges and A. Mielke, A proof of the Benjamin-Feir instability, Arch. Ratianal Mech. Anal., 133 (1995), 145-198. doi: 10.1007/BF00376815.

[9]

H. Lamb, "Hydrodynamics,'' Cambridge University Press, Cambridge, 1932.

[10]

M. S. Longuet-Higgins, On the stability of steep gravity waves, Proc. R. Soc. Lond. Ser. A, 396 (1984), 269-280. doi: 10.1098/rspa.1984.0122.

[11]

J. C. Luke, A variational principle for a fluid with a free surface, J. Fluid Mech., 27 (1967), 395-397. doi: 10.1017/S0022112067000412.

[12]

J. Pedlosky, "Geophysical Fluid Dynamics,'' Springer-Verlag, New York, 1979.

[13]

H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff and K. Socha, Stabilizing the Benjamin-Feir instability, J. Fluid Mech., 539 (2005), 229-271.

[14]

G. G. Stokes, On the theory of osillatory waves, Trans. Cambridge Philos. Soc., 8 (1847), 441-473.

[15]

M. Tanaka, The stability of steep gravity waves, J. Phys. Soc. Japan, 52 (1983), 3047-3055. doi: 10.1143/JPSJ.52.3047.

[16]

M. Tanaka, The stability of steep gravity waves. II, J. Fluid Mech., 156 (1985), 281-289. doi: 10.1017/S0022112085002099.

[17]

G. B. Whitham, "Linear and Nonlinear Waves,'' Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[18]

G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Comm. Pure Appl. Math., 41 (1988), 19-46. doi: 10.1002/cpa.3160410104.

[19]

G. Wu, Y. Liu, and D. K. P. Yue, A note on stabilizing the Benjamin-Feir instability, J. Fluid Mech., 556 (2006), 45-54. doi: 10.1017/S0022112005008293.

[20]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of adeep fluid, J. Appl. Mech. Tech. Phys., 2 (1968), 190-194.

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