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 and 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.

[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. 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.

[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. 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.

[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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