March  2015, 35(3): 909-916. doi: 10.3934/dcds.2015.35.909

Instability of equatorial water waves in the $f-$plane

1. 

School of Mathematical Sciences, University College Cork, Cork

2. 

Tainan Hydraulics Laboratory, National Cheng Kung Univ, Tainan 701, Taiwan

Received  March 2014 Revised  April 2014 Published  October 2014

This paper addresses the hydrodynamical stability of nonlinear geophysical equatorial waves in the $f-$plane approximation. By implementing the short-wavelength perturbation approach, we show that certain westward-propagating equatorial waves are linearly unstable when the wave steepness exceeds a specific threshold.
Citation: David Henry, Hung-Chu Hsu. Instability of equatorial water waves in the $f-$plane. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 909-916. doi: 10.3934/dcds.2015.35.909
References:
[1]

B. J. Bayly, Three-dimensional instabilities in quasi-two dimensional inviscid flows,, in Nonlinear Wave Interactions in Fluids, (1987), 71.   Google Scholar

[2]

A. Bennett, Lagrangian Fluid Dynamics,, Cambridge University Press, (2006).  doi: 10.1017/CBO9780511734939.  Google Scholar

[3]

A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[4]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A, 34 (2001), 9723.  doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[5]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[6]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[7]

A. Constantin, An exact solution for equatorially trapped waves,, J. Geophys. Res., 117 (2012).  doi: 10.1029/2012JC007879.  Google Scholar

[8]

A. Constantin, On the modelling of Equatorial waves,, Geophys. Res. Lett., 39 (2012).  doi: 10.1029/2012GL051169.  Google Scholar

[9]

A. Constantin, Some three-dimensional nonlinear Equatorial flows,, J. Phys. Oceanogr., 43 (2013), 165.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[10]

A. Constantin, On equatorial wind waves,, Differential and Integral equations, 26 (2013), 237.   Google Scholar

[11]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves,, J. Phys. Oceanogr., 44 (2014), 781.  doi: 10.1175/JPO-D-13-0174.1.  Google Scholar

[12]

A. Constantin and P. Germain, Instability of some equatorially trapped waves,, J. Geophys. Res. Oceans, 118 (2013), 2802.  doi: 10.1002/jgrc.20219.  Google Scholar

[13]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 63 (2010), 533.  doi: 10.1002/cpa.20299.  Google Scholar

[14]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects,, Academic, (2011).   Google Scholar

[15]

P. G. Drazin, Introduction to Hydrodynamic Stability,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511809064.  Google Scholar

[16]

P. G. Drazin and W. H. Reid, Hydrodynamic Stability,, Cambridge University Press, (2004).  doi: 10.1017/CBO9780511616938.  Google Scholar

[17]

A. V. Fedorov and J. N. Brown, Equatorial waves,, in Encyclopedia of Ocean Sciences, (2009), 271.  doi: 10.1016/B978-012374473-9.00610-X.  Google Scholar

[18]

S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid,, Phys. Rev. Lett., 66 (1991), 2204.  doi: 10.1103/PhysRevLett.66.2204.  Google Scholar

[19]

S. Friedlander and V. Yudovich, Instabilities in fluid motion,, Not. Am. Math. Soc., 46 (1999), 1358.   Google Scholar

[20]

S. Friedlander, Lectures on stability and instability of an ideal fluid,, in Hyperbolic Equations and Frequency Interactions, 5 (1999), 227.   Google Scholar

[21]

I. Gallagher and L. Saint-Raymond, On the influence of the Earth's rotation on geophysical flows,, in Handbook of Mathematical Fluid Mechanics, 4 (2007), 201.   Google Scholar

[22]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current,, J. Math. Fluid Mech., (2014), 1.  doi: 10.1007/s00021-014-0175-4.  Google Scholar

[23]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., 2 (1809), 412.   Google Scholar

[24]

D. Henry, On the deep-water Stokes flow,, Int. Math. Res. Not. IMRN, (2008).   Google Scholar

[25]

D. Henry, On Gerstner's water wave,, J. Nonl. Math. Phys., 15 (2008), 87.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[26]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current,, Eur. J. Mech. B Fluids, 38 (2013), 18.  doi: 10.1016/j.euromechflu.2012.10.001.  Google Scholar

[27]

D. Henry and A. Matioc, On the existence of equatorial wind waves,, Nonlinear Anal., 101 (2014), 113.  doi: 10.1016/j.na.2014.01.018.  Google Scholar

[28]

H.-C. Hsu, An exact solution of equatorial waves,, Monatsh. Math., (2014), 1.  doi: 10.1007/s00605-014-0618-2.  Google Scholar

[29]

H.-C. Hsu, An exact solution for nonlinear internal equatorial waves in the $f-$plane approximation,, J. Math. Fluid Mech., 16 (2014), 463.  doi: 10.1007/s00021-014-0168-3.  Google Scholar

[30]

H.-C. Hsu, Some nonlinear internal equatorial waves with a strong underlying current,, Appl. Math. Letters, 34 (2014), 1.  doi: 10.1016/j.aml.2014.03.005.  Google Scholar

[31]

T. Izumo, The equatorial current, meridional overturning circulation, and their roles in mass and heat exchanges during the El Niño events in the tropical Pacific Ocean,, Ocean Dyn., 55 (2005), 110.   Google Scholar

[32]

S. Leblanc, Local stability of Gerstner's waves,, J. Fluid Mech., 506 (2004), 245.  doi: 10.1017/S0022112004008444.  Google Scholar

[33]

A. Lifschitz and E. Hameiri, Local stability conditions in fluid dynamics,, Phys. Fluids, 3 (1991), 2644.  doi: 10.1063/1.858153.  Google Scholar

[34]

A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach,, J. Phys. A, 45 (2012).  doi: 10.1088/1751-8113/45/36/365501.  Google Scholar

[35]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach,, J. Nonlinear Math. Phys., 18 (2011), 127.  doi: 10.1142/S1402925111001210.  Google Scholar

show all references

References:
[1]

B. J. Bayly, Three-dimensional instabilities in quasi-two dimensional inviscid flows,, in Nonlinear Wave Interactions in Fluids, (1987), 71.   Google Scholar

[2]

A. Bennett, Lagrangian Fluid Dynamics,, Cambridge University Press, (2006).  doi: 10.1017/CBO9780511734939.  Google Scholar

[3]

A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[4]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A, 34 (2001), 9723.  doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[5]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[6]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[7]

A. Constantin, An exact solution for equatorially trapped waves,, J. Geophys. Res., 117 (2012).  doi: 10.1029/2012JC007879.  Google Scholar

[8]

A. Constantin, On the modelling of Equatorial waves,, Geophys. Res. Lett., 39 (2012).  doi: 10.1029/2012GL051169.  Google Scholar

[9]

A. Constantin, Some three-dimensional nonlinear Equatorial flows,, J. Phys. Oceanogr., 43 (2013), 165.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[10]

A. Constantin, On equatorial wind waves,, Differential and Integral equations, 26 (2013), 237.   Google Scholar

[11]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves,, J. Phys. Oceanogr., 44 (2014), 781.  doi: 10.1175/JPO-D-13-0174.1.  Google Scholar

[12]

A. Constantin and P. Germain, Instability of some equatorially trapped waves,, J. Geophys. Res. Oceans, 118 (2013), 2802.  doi: 10.1002/jgrc.20219.  Google Scholar

[13]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 63 (2010), 533.  doi: 10.1002/cpa.20299.  Google Scholar

[14]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects,, Academic, (2011).   Google Scholar

[15]

P. G. Drazin, Introduction to Hydrodynamic Stability,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511809064.  Google Scholar

[16]

P. G. Drazin and W. H. Reid, Hydrodynamic Stability,, Cambridge University Press, (2004).  doi: 10.1017/CBO9780511616938.  Google Scholar

[17]

A. V. Fedorov and J. N. Brown, Equatorial waves,, in Encyclopedia of Ocean Sciences, (2009), 271.  doi: 10.1016/B978-012374473-9.00610-X.  Google Scholar

[18]

S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid,, Phys. Rev. Lett., 66 (1991), 2204.  doi: 10.1103/PhysRevLett.66.2204.  Google Scholar

[19]

S. Friedlander and V. Yudovich, Instabilities in fluid motion,, Not. Am. Math. Soc., 46 (1999), 1358.   Google Scholar

[20]

S. Friedlander, Lectures on stability and instability of an ideal fluid,, in Hyperbolic Equations and Frequency Interactions, 5 (1999), 227.   Google Scholar

[21]

I. Gallagher and L. Saint-Raymond, On the influence of the Earth's rotation on geophysical flows,, in Handbook of Mathematical Fluid Mechanics, 4 (2007), 201.   Google Scholar

[22]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current,, J. Math. Fluid Mech., (2014), 1.  doi: 10.1007/s00021-014-0175-4.  Google Scholar

[23]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., 2 (1809), 412.   Google Scholar

[24]

D. Henry, On the deep-water Stokes flow,, Int. Math. Res. Not. IMRN, (2008).   Google Scholar

[25]

D. Henry, On Gerstner's water wave,, J. Nonl. Math. Phys., 15 (2008), 87.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[26]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current,, Eur. J. Mech. B Fluids, 38 (2013), 18.  doi: 10.1016/j.euromechflu.2012.10.001.  Google Scholar

[27]

D. Henry and A. Matioc, On the existence of equatorial wind waves,, Nonlinear Anal., 101 (2014), 113.  doi: 10.1016/j.na.2014.01.018.  Google Scholar

[28]

H.-C. Hsu, An exact solution of equatorial waves,, Monatsh. Math., (2014), 1.  doi: 10.1007/s00605-014-0618-2.  Google Scholar

[29]

H.-C. Hsu, An exact solution for nonlinear internal equatorial waves in the $f-$plane approximation,, J. Math. Fluid Mech., 16 (2014), 463.  doi: 10.1007/s00021-014-0168-3.  Google Scholar

[30]

H.-C. Hsu, Some nonlinear internal equatorial waves with a strong underlying current,, Appl. Math. Letters, 34 (2014), 1.  doi: 10.1016/j.aml.2014.03.005.  Google Scholar

[31]

T. Izumo, The equatorial current, meridional overturning circulation, and their roles in mass and heat exchanges during the El Niño events in the tropical Pacific Ocean,, Ocean Dyn., 55 (2005), 110.   Google Scholar

[32]

S. Leblanc, Local stability of Gerstner's waves,, J. Fluid Mech., 506 (2004), 245.  doi: 10.1017/S0022112004008444.  Google Scholar

[33]

A. Lifschitz and E. Hameiri, Local stability conditions in fluid dynamics,, Phys. Fluids, 3 (1991), 2644.  doi: 10.1063/1.858153.  Google Scholar

[34]

A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach,, J. Phys. A, 45 (2012).  doi: 10.1088/1751-8113/45/36/365501.  Google Scholar

[35]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach,, J. Nonlinear Math. Phys., 18 (2011), 127.  doi: 10.1142/S1402925111001210.  Google Scholar

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