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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 |
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.
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. |
[1] |
Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397 |
[2] |
Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045 |
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