American Institute of Mathematical Sciences

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

Steady periodic equatorial water waves with vorticity

Jifeng Chu 1,, and  Joachim Escher 2,

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.

Keywords: Steady periodic water waves, equatorial flows, vorticity, dispersion relation.
Mathematics Subject Classification: Primary: 76B15, 35J60, 47J15, 76B03.
Citation: Jifeng Chu, Joachim Escher. Steady periodic equatorial water waves with vorticity. Discrete & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[7]

A. Constantin, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

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

[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.  Google Scholar

[44]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[7]

A. Constantin, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

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

[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.  Google Scholar

[44]

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

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