American Institute of Mathematical Sciences

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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 & 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.   Google Scholar

[2]

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

[3]

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

[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.   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

H. Lamb, "Hydrodynamics,'', Cambridge University Press, (1932).   Google Scholar

[10]

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

[11]

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

[12]

J. Pedlosky, "Geophysical Fluid Dynamics,'', Springer-Verlag, (1979).   Google Scholar

[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.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

G. B. Whitham, "Linear and Nonlinear Waves,'', Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar

[18]

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

[19]

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

[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.   Google Scholar

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.   Google Scholar

[2]

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

[3]

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

[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.   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

H. Lamb, "Hydrodynamics,'', Cambridge University Press, (1932).   Google Scholar

[10]

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

[11]

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

[12]

J. Pedlosky, "Geophysical Fluid Dynamics,'', Springer-Verlag, (1979).   Google Scholar

[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.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

G. B. Whitham, "Linear and Nonlinear Waves,'', Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar

[18]

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

[19]

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

[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.   Google Scholar

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