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 and Continuous Dynamical Systems, 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, edited by R. W. Miksad et al., ASME, New York, (1987), 71-77.

[2]

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

[3]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313.

[4]

A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311.

[5]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[6]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, Vol. 81, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[7]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879.

[8]

A. Constantin, On the modelling of Equatorial waves, Geophys. Res. Lett., 39 (2012), L05602. doi: 10.1029/2012GL051169.

[9]

A. Constantin, Some three-dimensional nonlinear Equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175. doi: 10.1175/JPO-D-12-062.1.

[10]

A. Constantin, On equatorial wind waves, Differential and Integral equations, 26 (2013), 237-252.

[11]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. doi: 10.1175/JPO-D-13-0174.1.

[12]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219.

[13]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.

[14]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011.

[15]

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

[16]

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

[17]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences, edited by J. Steele, Academic, San Diego, Calif., (2009), 271-287. doi: 10.1016/B978-012374473-9.00610-X.

[18]

S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett., 66 (1991), 2204-2206. doi: 10.1103/PhysRevLett.66.2204.

[19]

S. Friedlander and V. Yudovich, Instabilities in fluid motion, Not. Am. Math. Soc., 46 (1999), 1358-1367.

[20]

S. Friedlander, Lectures on stability and instability of an ideal fluid, in Hyperbolic Equations and Frequency Interactions, IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 5 (1999), 227-304.

[21]

I. Gallagher and L. Saint-Raymond, On the influence of the Earth's rotation on geophysical flows, in Handbook of Mathematical Fluid Mechanics, edited by S. Friedlander and D. Serre, North-Holland, Amsterdam, 4 (2007), 201-329.

[22]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., (2014), 1-2. doi: 10.1007/s00021-014-0175-4.

[23]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445.

[24]

D. Henry, On the deep-water Stokes flow, Int. Math. Res. Not. IMRN, (2008), 7 pp.

[25]

D. Henry, On Gerstner's water wave, J. Nonl. Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.S2.7.

[26]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21. doi: 10.1016/j.euromechflu.2012.10.001.

[27]

D. Henry and A. Matioc, On the existence of equatorial wind waves, Nonlinear Anal., 101 (2014), 113-123. doi: 10.1016/j.na.2014.01.018.

[28]

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

[29]

H.-C. Hsu, An exact solution for nonlinear internal equatorial waves in the $f-$plane approximation, J. Math. Fluid Mech., 16 (2014), 463-471. doi: 10.1007/s00021-014-0168-3.

[30]

H.-C. Hsu, Some nonlinear internal equatorial waves with a strong underlying current, Appl. Math. Letters, 34 (2014), 1-6. doi: 10.1016/j.aml.2014.03.005.

[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), 110-123.

[32]

S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254. doi: 10.1017/S0022112004008444.

[33]

A. Lifschitz and E. Hameiri, Local stability conditions in fluid dynamics, Phys. Fluids, 3 (1991), 2644-2651. doi: 10.1063/1.858153.

[34]

A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 365501, 10pp. doi: 10.1088/1751-8113/45/36/365501.

[35]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2011), 127-137. doi: 10.1142/S1402925111001210.

show all references

References:
[1]

B. J. Bayly, Three-dimensional instabilities in quasi-two dimensional inviscid flows, in Nonlinear Wave Interactions in Fluids, edited by R. W. Miksad et al., ASME, New York, (1987), 71-77.

[2]

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

[3]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313.

[4]

A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311.

[5]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[6]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, Vol. 81, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[7]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879.

[8]

A. Constantin, On the modelling of Equatorial waves, Geophys. Res. Lett., 39 (2012), L05602. doi: 10.1029/2012GL051169.

[9]

A. Constantin, Some three-dimensional nonlinear Equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175. doi: 10.1175/JPO-D-12-062.1.

[10]

A. Constantin, On equatorial wind waves, Differential and Integral equations, 26 (2013), 237-252.

[11]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. doi: 10.1175/JPO-D-13-0174.1.

[12]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219.

[13]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.

[14]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011.

[15]

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

[16]

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

[17]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences, edited by J. Steele, Academic, San Diego, Calif., (2009), 271-287. doi: 10.1016/B978-012374473-9.00610-X.

[18]

S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett., 66 (1991), 2204-2206. doi: 10.1103/PhysRevLett.66.2204.

[19]

S. Friedlander and V. Yudovich, Instabilities in fluid motion, Not. Am. Math. Soc., 46 (1999), 1358-1367.

[20]

S. Friedlander, Lectures on stability and instability of an ideal fluid, in Hyperbolic Equations and Frequency Interactions, IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 5 (1999), 227-304.

[21]

I. Gallagher and L. Saint-Raymond, On the influence of the Earth's rotation on geophysical flows, in Handbook of Mathematical Fluid Mechanics, edited by S. Friedlander and D. Serre, North-Holland, Amsterdam, 4 (2007), 201-329.

[22]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., (2014), 1-2. doi: 10.1007/s00021-014-0175-4.

[23]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445.

[24]

D. Henry, On the deep-water Stokes flow, Int. Math. Res. Not. IMRN, (2008), 7 pp.

[25]

D. Henry, On Gerstner's water wave, J. Nonl. Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.S2.7.

[26]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21. doi: 10.1016/j.euromechflu.2012.10.001.

[27]

D. Henry and A. Matioc, On the existence of equatorial wind waves, Nonlinear Anal., 101 (2014), 113-123. doi: 10.1016/j.na.2014.01.018.

[28]

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

[29]

H.-C. Hsu, An exact solution for nonlinear internal equatorial waves in the $f-$plane approximation, J. Math. Fluid Mech., 16 (2014), 463-471. doi: 10.1007/s00021-014-0168-3.

[30]

H.-C. Hsu, Some nonlinear internal equatorial waves with a strong underlying current, Appl. Math. Letters, 34 (2014), 1-6. doi: 10.1016/j.aml.2014.03.005.

[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), 110-123.

[32]

S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254. doi: 10.1017/S0022112004008444.

[33]

A. Lifschitz and E. Hameiri, Local stability conditions in fluid dynamics, Phys. Fluids, 3 (1991), 2644-2651. doi: 10.1063/1.858153.

[34]

A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 365501, 10pp. doi: 10.1088/1751-8113/45/36/365501.

[35]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2011), 127-137. doi: 10.1142/S1402925111001210.

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