American Institute of Mathematical Sciences

Advanced
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

[1]

Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940
[2]

Alan Compelli, Rossen Ivanov. Benjamin-Ono model of an internal wave under a flat surface. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4519-4532. doi: 10.3934/dcds.2019185
[3]

Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525
[4]

Paolo Baiti, Alberto Bressan, Helge Kristian Jenssen. Instability of travelling wave profiles for the Lax-Friedrichs scheme. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 877-899. doi: 10.3934/dcds.2005.13.877
[5]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[6]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[7]

Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615
[8]

P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029
[9]

Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075
[10]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[11]

Yavar Kian. Stability of the determination of a coefficient for wave equations in an infinite waveguide. Inverse Problems & Imaging, 2014, 8 (3) : 713-732. doi: 10.3934/ipi.2014.8.713
[12]

Lili Fan, Hongxia Liu, Huijiang Zhao, Qingyang Zou. Global stability of stationary waves for damped wave equations. Kinetic & Related Models, 2013, 6 (4) : 729-760. doi: 10.3934/krm.2013.6.729
[13]

Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control & Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010
[14]

Frédéric Rousset, Nikolay Tzvetkov. On the transverse instability of one dimensional capillary-gravity waves. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 859-872. doi: 10.3934/dcdsb.2010.13.859
[15]

Hiroaki Kikuchi. Remarks on the orbital instability of standing waves for the wave-Schrödinger system in higher dimensions. Communications on Pure & Applied Analysis, 2010, 9 (2) : 351-364. doi: 10.3934/cpaa.2010.9.351
[16]

Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687
[17]

Toshi Ogawa. Degenerate Hopf instability in oscillatory reaction-diffusion equations. Conference Publications, 2007, 2007 (Special) : 784-793. doi: 10.3934/proc.2007.2007.784
[18]

Kenta Ohi, Tatsuo Iguchi. A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1205-1240. doi: 10.3934/dcds.2009.23.1205
[19]

Gennadiy Burlak, Salomon García-Paredes. Matter-wave solitons with a minimal number of particles in a time-modulated quasi-periodic potential. Conference Publications, 2015, 2015 (special) : 169-175. doi: 10.3934/proc.2015.0169
[20]

Ning-An Lai, Jinglei Zhao. Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1317-1325. doi: 10.3934/cpaa.2014.13.1317

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences