August  2019, 39(8): 4713-4729. doi: 10.3934/dcds.2019191

Steady periodic equatorial water waves with vorticity

1. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

2. 

Institute for Applied Mathematics, Leibniz Universität Hannover, Hannover 30167, Germany

* Corresponding author: Jifeng Chu

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin.

Received  September 2018 Revised  January 2019 Published  May 2019

Fund Project: Jifeng Chu was supported by the Alexander von Humboldt-Stiftung of Germany, and the National Natural Science Foundation of China (Grants No. 11671118 and No. 11871273)

Of concern are steady two-dimensional periodic geophysical water waves of small amplitude near the equator. The analysis presented here is based on the bifurcation theory due to Crandall-Rabinowitz. Dispersion relations for various choices of the vorticity distribution, including constant, affine, and some nonlinear vorticities are obtained.

Citation: Jifeng Chu, Joachim Escher. Steady periodic equatorial water waves with vorticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4713-4729. doi: 10.3934/dcds.2019191
References:
[1]

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.

[2]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.

[3]

A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.

[4]

A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commun. Pure Appl. Anal., 11 (2012), 1397-1406. doi: 10.3934/cpaa.2012.11.1397.

[5]

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

[6]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.

[7]

A. ConstantinM. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.

[8]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., (2) 173 (2011), 559–568. doi: 10.4007/annals.2011.173.1.12.

[10]

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

[11]

A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358. doi: 10.1080/03091929.2015.1066785.

[12]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945.

[13]

A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Physics of Fluids, 29 (2017), 056604.

[14]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. Roy. Soc. London A., 473 (2017), 20170063, 17pp. doi: 10.1098/rspa.2017.0063.

[15]

A. Constantin and R. S. Johnson, Steady large-scale ocean flows in spherical coordinates, Oceanography, 31 (2018), 42-50. doi: 10.5670/oceanog.2018.308.

[16]

A. Constantin and E. Kartashova, Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves, Europhysics Letters, 86 (2009), 29001. doi: 10.1209/0295-5075/86/29001.

[17]

A. Constantin and S. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528. doi: 10.1017/jfm.2017.223.

[18]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[19]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.

[20]

M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[21]

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

[22]

M. L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, (French) 1934. 75 pp.

[23]

A. V. Fedorov and J. N. Brown, Equatorial waves, in: J. Steele (Ed.), Encyclopedia of Ocean Sciences, Academic, San Diego, Calif., 2009, 3679–3695.

[24]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16 (2014), 661-667. doi: 10.1007/s00021-014-0175-4.

[25]

D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal., 42 (2010), 3103-3111. doi: 10.1137/100801408.

[26]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487. doi: 10.1090/S0033-569X-2013-01293-8.

[27]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037. doi: 10.1080/03605302.2012.734889.

[28]

D. Henry, Exact equatorial water waves in the f-plane, Nonlinear Anal. Real World Appl., 28 (2016), 284-289. doi: 10.1016/j.nonrwa.2015.10.003.

[29]

D. Henry and H.-C. Hsu, Instability of Equatorial water waves in the f-plane, Discrete Contin. Dyn. Syst., 35 (2015), 909-916. doi: 10.3934/dcds.2015.35.909.

[30]

D. Henry and H.-C. Hsu, Instability of internal equatorial water waves, J. Differential Equations, 258 (2015), 1015-1024. doi: 10.1016/j.jde.2014.08.019.

[31]

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

[32]

D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Philos. Trans. Roy. Soc. A, 376 (2018), 20170090, 21 pp. doi: 10.1098/rsta.2017.0090.

[33]

D. Ionescu-Kruse and C. I. Martin, Local stability for an exact steady purely azimuthal equatorial flow, J. Math. Fluid Mech., 20 (2018), 27-34. doi: 10.1007/s00021-016-0311-4.

[34]

D. Ionescu-Kruse and A.-V. Matioc, Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories, Discrete Contin. Dyn. Syst., 34 (2014), 3045-3060. doi: 10.3934/dcds.2014.34.3045.

[35]

R. S. Johnson, Applications of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phil. Trans. R. Soc. A, 376 (2018), 20170092, 19pp. doi: 10.1098/rsta.2017.0092.

[36]

P. Karageorgis, Dispersion relation for water waves with non-constant vorticity, Eur. J. Mech. B Fluids, 34 (2012), 7-12. doi: 10.1016/j.euromechflu.2012.03.008.

[37]

V. Kozlov and N. Kuznetsov, Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents, Arch. Ration. Mech. Anal., 214 (2014), 971-1018. doi: 10.1007/s00205-014-0787-0.

[38]

V. Kozlov, N. Kuznetsov and E. Lokharu, Steady water waves with vorticity: An analysis of the dispersion equation, J. Fluid Mech., 751 (2014), R3, 13 pp. doi: 10.1017/jfm.2014.322.

[39]

C. I. Martin, Dispersion relations for gravity water flows with two rotational layer, Eur. J. Mech. B Fluids, 50 (2015), 9-18. doi: 10.1016/j.euromechflu.2014.10.005.

[40]

C. I. Martin, Dynamics of the thermocline in the equatorial region of the Pacific ocean, J. Nonlinear Math. Phys., 22 (2015), 516-522. doi: 10.1080/14029251.2015.1113049.

[41]

C. I. Martin, On periodic geophysical water flows with discontinuous vorticity in the equatorial f-plane approximation, Philos. Trans. Roy. Soc. A, 376 (2018), 20170096, 23 pp. doi: 10.1098/rsta.2017.0096.

[42]

C. I. Martin, On the vorticity of mesoscale ocean currents, Oceanography, 31 (2018), 28-35. doi: 10.5670/oceanog.2018.306.

[43]

C. I. Martin, Non-existence of time-dependent three-dimensional gravity water flows with constant non-zero vorticity, Physics of Fluids, 30 (2018), 107102. doi: 10.1063/1.5048580.

[44]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979.

show all references

References:
[1]

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.

[2]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.

[3]

A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.

[4]

A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commun. Pure Appl. Anal., 11 (2012), 1397-1406. doi: 10.3934/cpaa.2012.11.1397.

[5]

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

[6]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.

[7]

A. ConstantinM. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.

[8]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., (2) 173 (2011), 559–568. doi: 10.4007/annals.2011.173.1.12.

[10]

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

[11]

A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358. doi: 10.1080/03091929.2015.1066785.

[12]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945.

[13]

A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Physics of Fluids, 29 (2017), 056604.

[14]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. Roy. Soc. London A., 473 (2017), 20170063, 17pp. doi: 10.1098/rspa.2017.0063.

[15]

A. Constantin and R. S. Johnson, Steady large-scale ocean flows in spherical coordinates, Oceanography, 31 (2018), 42-50. doi: 10.5670/oceanog.2018.308.

[16]

A. Constantin and E. Kartashova, Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves, Europhysics Letters, 86 (2009), 29001. doi: 10.1209/0295-5075/86/29001.

[17]

A. Constantin and S. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528. doi: 10.1017/jfm.2017.223.

[18]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[19]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.

[20]

M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[21]

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

[22]

M. L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, (French) 1934. 75 pp.

[23]

A. V. Fedorov and J. N. Brown, Equatorial waves, in: J. Steele (Ed.), Encyclopedia of Ocean Sciences, Academic, San Diego, Calif., 2009, 3679–3695.

[24]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16 (2014), 661-667. doi: 10.1007/s00021-014-0175-4.

[25]

D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal., 42 (2010), 3103-3111. doi: 10.1137/100801408.

[26]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487. doi: 10.1090/S0033-569X-2013-01293-8.

[27]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037. doi: 10.1080/03605302.2012.734889.

[28]

D. Henry, Exact equatorial water waves in the f-plane, Nonlinear Anal. Real World Appl., 28 (2016), 284-289. doi: 10.1016/j.nonrwa.2015.10.003.

[29]

D. Henry and H.-C. Hsu, Instability of Equatorial water waves in the f-plane, Discrete Contin. Dyn. Syst., 35 (2015), 909-916. doi: 10.3934/dcds.2015.35.909.

[30]

D. Henry and H.-C. Hsu, Instability of internal equatorial water waves, J. Differential Equations, 258 (2015), 1015-1024. doi: 10.1016/j.jde.2014.08.019.

[31]

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

[32]

D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Philos. Trans. Roy. Soc. A, 376 (2018), 20170090, 21 pp. doi: 10.1098/rsta.2017.0090.

[33]

D. Ionescu-Kruse and C. I. Martin, Local stability for an exact steady purely azimuthal equatorial flow, J. Math. Fluid Mech., 20 (2018), 27-34. doi: 10.1007/s00021-016-0311-4.

[34]

D. Ionescu-Kruse and A.-V. Matioc, Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories, Discrete Contin. Dyn. Syst., 34 (2014), 3045-3060. doi: 10.3934/dcds.2014.34.3045.

[35]

R. S. Johnson, Applications of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phil. Trans. R. Soc. A, 376 (2018), 20170092, 19pp. doi: 10.1098/rsta.2017.0092.

[36]

P. Karageorgis, Dispersion relation for water waves with non-constant vorticity, Eur. J. Mech. B Fluids, 34 (2012), 7-12. doi: 10.1016/j.euromechflu.2012.03.008.

[37]

V. Kozlov and N. Kuznetsov, Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents, Arch. Ration. Mech. Anal., 214 (2014), 971-1018. doi: 10.1007/s00205-014-0787-0.

[38]

V. Kozlov, N. Kuznetsov and E. Lokharu, Steady water waves with vorticity: An analysis of the dispersion equation, J. Fluid Mech., 751 (2014), R3, 13 pp. doi: 10.1017/jfm.2014.322.

[39]

C. I. Martin, Dispersion relations for gravity water flows with two rotational layer, Eur. J. Mech. B Fluids, 50 (2015), 9-18. doi: 10.1016/j.euromechflu.2014.10.005.

[40]

C. I. Martin, Dynamics of the thermocline in the equatorial region of the Pacific ocean, J. Nonlinear Math. Phys., 22 (2015), 516-522. doi: 10.1080/14029251.2015.1113049.

[41]

C. I. Martin, On periodic geophysical water flows with discontinuous vorticity in the equatorial f-plane approximation, Philos. Trans. Roy. Soc. A, 376 (2018), 20170096, 23 pp. doi: 10.1098/rsta.2017.0096.

[42]

C. I. Martin, On the vorticity of mesoscale ocean currents, Oceanography, 31 (2018), 28-35. doi: 10.5670/oceanog.2018.306.

[43]

C. I. Martin, Non-existence of time-dependent three-dimensional gravity water flows with constant non-zero vorticity, Physics of Fluids, 30 (2018), 107102. doi: 10.1063/1.5048580.

[44]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979.

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