July  2022, 21(7): 2415-2431. doi: 10.3934/cpaa.2022053

On three-dimensional free surface water flows with constant vorticity

Oskar-Morgenstern Platz 1, 1090 Vienna, Austria

Received  October 2021 Revised  January 2022 Published  July 2022 Early access  March 2022

Fund Project: The author greatfully acknowledges the support of the Austrian Science Fund (FWF) through research grant P 33107-N. Comments and remarks from two anonymous referees, for which the author is thankful, have significantly improved the presentation. Many thanks to Prof. Robin S. Johnson (Newcastle University) for many interesting discussions on topics related to exact solutions concerning geophyical water flows.

We present a survey of recent results on gravity water flows satisfying the three-dimensional water wave problem with constant (non-vanishing) vorticity vector. The main focus is to show that a gravity water flow with constant non-vanishing vorticity has a two-dimensional character in spite of satisfying the three-dimensional water wave equations. More precisely, the flow does not change in one of the two horizontal directions. Passing to a rotating frame, and introducing thus geophysical effects (in the form of Coriolis acceleration) into the governing equations, the two-dimensional character of the flow remains in place. However, the two-dimensionality of the flow manifests now in a horizontal plane. Adding also centripetal terms into the equations further simplifies the flow (under the assumption of constant vorticity vector): the velocity field vanishes, but, however, the pressure function is a quadratic polynomial in the horizontal and vertical variables, and, surprisingly, the surface is non-flat.

Citation: Calin I. Martin. On three-dimensional free surface water flows with constant vorticity. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2415-2431. doi: 10.3934/cpaa.2022053
References:
[1]

A. Aleman and A. Constantin, On the decrease of kinetic energy with depth in wave-current interactions, Math. Ann., 378 (2020), 853-872.  doi: 10.1007/s00208-019-01910-8.

[2]

B. J. Bayly, Three-dimensional instabilities in quasi-two dimensional inviscid flows, in Nonlinear Wave Interactions in Fluids, ASME, New York, 1987.

[3]

D. Byrne, H. Xia and M. Shats, Robust inverse energy cascade and turbulence structure in three-dimensional layers of fluid, Phys. Fluids, 23 (2011), 8 pp.

[4]

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

[5]

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

[6]

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.

[7]

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

[8]

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

[9]

A. ConstantinM. Ehrnstr'̀om 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.

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A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7.

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A. Constantin and E. Kartashova, Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves, Euro. Lett., 86 (2009), 6 pp. doi: 10.1209/0295-5075/86/29001.

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

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A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, in CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

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

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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.

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A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), 4 pp. doi: 10.1029/2012GL051169.

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

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A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. Roy. Soc. A, 473 (2017), 18 pp. doi: 10.1098/rspa. 2017.0063.

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A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

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W. Craig, Non-existence of solitary water waves in three dimensions. Recent developments in the mathematical theory of water waves, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 360 (2002), 2127-2135.  doi: 10.1098/rsta.2002.1065.

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W. Craig and D. Nicholls, Traveling gravity water waves in two and three dimensions, Eur. J. Mech. B Fluids, 21 (2002), 615-641.  doi: 10.1016/S0997-7546(02)01207-4.

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M. EhrnströmJ. Escher and E. Wahlén, Steady water waves with multiple critical layers, SIAM J. Math. Anal., 43 (2011), 1436-1456.  doi: 10.1137/100792330.

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J. EscherA.-V. Matioc and B.-V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points, J. Differ. Equ., 251 (2011), 2932-2949.  doi: 10.1016/j.jde.2011.03.023.

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J. EscherP. KnopfC. Lienstromberg and B.-V. Matioc, Stratified periodic water waves with singular density gradients, Ann. Mat. Pura Appl., 199 (2020), 1923-1959.  doi: 10.1007/s10231-020-00950-1.

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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.

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

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M. GrovesM. Hărăgus and S. M. Sun, A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 360 (2002), 2189-2243.  doi: 10.1098/rsta.2002.1066.

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M. Groves and M. Hărăgus, A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves, J. Nonlinear Sci., 13 (2003), 397-447.  doi: 10.1007/s00332-003-0530-8.

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D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.  doi: 10.2991/jnmp.2008.15.s2.7.

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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.

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D. Henry and B.-V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 955-974. 

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D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity-stratified water waves, Proc. Roy. Soc. Edinb. Sect. A, 144 (2014), 775-786.  doi: 10.1017/S0308210512001990.

[35]

D. Henry and C. I. Martin, Free-surface, purely azimuthal equatorial flows in spherical coordinates with stratification, J. Differ. Equ., 266 (2019), 6788-6808.  doi: 10.1016/j.jde.2018.11.017.

[36]

D. Henry and C. I. Martin, Azimuthal equatorial flows with variable density in spherical coordinate, Arch. Ration. Mech. Anal., 233 (2019), 497-512.  doi: 10.1007/s00205-019-01362-z.

[37]

D. Henry and C. I. Martin, Stratified equatorial flows in cylindrical coordinates, Nonlinearity, 33 (2020), 3889-3904.  doi: 10.1088/1361-6544/ab801f.

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M. H. Holmes, Introduction to Perturbation Methods, Texts in Applied Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-5477-9.

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

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D. Ionescu-Kruse, Local Stability for an Exact Steady Purely Azimuthal Flow which Models the Antarctic Circumpolar Current, J. Math Fluid Mech., 20 (2018), 569-579.  doi: 10.1007/s00021-017-0335-4.

[41]

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.

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R. S. Johnson, Singular Perturbation Theory. Mathematical and Analytical Techniques with Applications to Engineering, Springer, 2005.

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

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

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E. LokharuD. S. Seth and E. Wahlén, An existence theory for small amplitude doubly periodic water waves with vorticity, Arch. Rational Mech. Anal., 238 (2020), 607-637.  doi: 10.1007/s00205-020-01550-2.

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S. A. Maslowe, Critical layers in shear flows, Annu. Rev. Fluid Mech., 18 (1986), 405-432. 

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C. I. Martin and B.-V. Matioc, Existence of wilton ripples for water waves with constant vorticity and capillary effects, SIAM J. Appl. Math., 73 (2013), 1582-1595.  doi: 10.1137/120900290.

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C. I. Martin, Resonant interactions of capillary-gravity water waves, J. Math. Fluid Mech., 19 (2017), 807-817.  doi: 10.1007/s00021-016-0306-1.

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C. I. Martin, Two-dimensionality of gravity water flows governed by the equatorial f-plane approximation, Ann. Mat. Pura Appl., 196 (2017), 2253-2260.  doi: 10.1007/s10231-017-0663-2.

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C. I. Martin, Non-existence of time-dependent three-dimensional gravity water flows with constant non-zero vorticity, Phys. Fluids, 30 (2018), 7 pp. doi: 10.1063/1.5048580.

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C. I. Martin, Constant vorticity water flows with full Coriolis term, Nonlinearity, 32 (2019), 2327-2336.  doi: 10.1088/1361-6544/ab1c76.

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show all references

References:
[1]

A. Aleman and A. Constantin, On the decrease of kinetic energy with depth in wave-current interactions, Math. Ann., 378 (2020), 853-872.  doi: 10.1007/s00208-019-01910-8.

[2]

B. J. Bayly, Three-dimensional instabilities in quasi-two dimensional inviscid flows, in Nonlinear Wave Interactions in Fluids, ASME, New York, 1987.

[3]

D. Byrne, H. Xia and M. Shats, Robust inverse energy cascade and turbulence structure in three-dimensional layers of fluid, Phys. Fluids, 23 (2011), 8 pp.

[4]

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

[5]

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

[6]

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.

[7]

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

[8]

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

[9]

A. ConstantinM. Ehrnstr'̀om 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.

[10]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7.

[11]

A. Constantin and E. Kartashova, Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves, Euro. Lett., 86 (2009), 6 pp. doi: 10.1209/0295-5075/86/29001.

[12]

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

[13]

A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, in CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[14]

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

[15]

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.

[16]

A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), 4 pp. doi: 10.1029/2012GL051169.

[17]

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

[18]

A. Constantin and R. I. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids, 27 (2015), 8 pp. doi: 10.1063/1.4929457.

[19]

A. ConstantinW. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers, Acta Math., 217 (2016), 195-262.  doi: 10.1007/s11511-017-0144-x.

[20]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. Roy. Soc. A, 473 (2017), 18 pp. doi: 10.1098/rspa. 2017.0063.

[21]

A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

[22]

W. Craig, Non-existence of solitary water waves in three dimensions. Recent developments in the mathematical theory of water waves, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 360 (2002), 2127-2135.  doi: 10.1098/rsta.2002.1065.

[23]

W. Craig and D. Nicholls, Traveling gravity water waves in two and three dimensions, Eur. J. Mech. B Fluids, 21 (2002), 615-641.  doi: 10.1016/S0997-7546(02)01207-4.

[24]

M. EhrnströmJ. Escher and E. Wahlén, Steady water waves with multiple critical layers, SIAM J. Math. Anal., 43 (2011), 1436-1456.  doi: 10.1137/100792330.

[25]

J. EscherA.-V. Matioc and B.-V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points, J. Differ. Equ., 251 (2011), 2932-2949.  doi: 10.1016/j.jde.2011.03.023.

[26]

J. EscherP. KnopfC. Lienstromberg and B.-V. Matioc, Stratified periodic water waves with singular density gradients, Ann. Mat. Pura Appl., 199 (2020), 1923-1959.  doi: 10.1007/s10231-020-00950-1.

[27]

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.

[28]

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

[29]

M. GrovesM. Hărăgus and S. M. Sun, A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 360 (2002), 2189-2243.  doi: 10.1098/rsta.2002.1066.

[30]

M. Groves and M. Hărăgus, A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves, J. Nonlinear Sci., 13 (2003), 397-447.  doi: 10.1007/s00332-003-0530-8.

[31]

D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.  doi: 10.2991/jnmp.2008.15.s2.7.

[32]

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.

[33]

D. Henry and B.-V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 955-974. 

[34]

D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity-stratified water waves, Proc. Roy. Soc. Edinb. Sect. A, 144 (2014), 775-786.  doi: 10.1017/S0308210512001990.

[35]

D. Henry and C. I. Martin, Free-surface, purely azimuthal equatorial flows in spherical coordinates with stratification, J. Differ. Equ., 266 (2019), 6788-6808.  doi: 10.1016/j.jde.2018.11.017.

[36]

D. Henry and C. I. Martin, Azimuthal equatorial flows with variable density in spherical coordinate, Arch. Ration. Mech. Anal., 233 (2019), 497-512.  doi: 10.1007/s00205-019-01362-z.

[37]

D. Henry and C. I. Martin, Stratified equatorial flows in cylindrical coordinates, Nonlinearity, 33 (2020), 3889-3904.  doi: 10.1088/1361-6544/ab801f.

[38]

M. H. Holmes, Introduction to Perturbation Methods, Texts in Applied Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-5477-9.

[39]

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

[40]

D. Ionescu-Kruse, Local Stability for an Exact Steady Purely Azimuthal Flow which Models the Antarctic Circumpolar Current, J. Math Fluid Mech., 20 (2018), 569-579.  doi: 10.1007/s00021-017-0335-4.

[41]

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.

[42]

G. Iooss and P. Plotnikov, Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves, Memoirs of the American Mathematical Society, 2009. doi: 10.1090/memo/0940.

[43] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, 1997.  doi: 10.1017/CBO9780511624056.
[44]

R. S. Johnson, Singular Perturbation Theory. Mathematical and Analytical Techniques with Applications to Engineering, Springer, 2005.

[45]

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

[46]

I. G. Jonsson, Wave-current interactions, Wiley, 9 (1989), 65-120. 

[47] P. K. KunduI. M. Cohen and D. R. Dowling, Fluid Mechanics, Academic Press, 2016. 
[48]

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

[49]

E. LokharuD. S. Seth and E. Wahlén, An existence theory for small amplitude doubly periodic water waves with vorticity, Arch. Rational Mech. Anal., 238 (2020), 607-637.  doi: 10.1007/s00205-020-01550-2.

[50] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. 
[51]

S. A. Maslowe, Critical layers in shear flows, Annu. Rev. Fluid Mech., 18 (1986), 405-432. 

[52]

C. I. Martin and B.-V. Matioc, Existence of wilton ripples for water waves with constant vorticity and capillary effects, SIAM J. Appl. Math., 73 (2013), 1582-1595.  doi: 10.1137/120900290.

[53]

C. I. Martin, Resonant interactions of capillary-gravity water waves, J. Math. Fluid Mech., 19 (2017), 807-817.  doi: 10.1007/s00021-016-0306-1.

[54]

C. I. Martin, Two-dimensionality of gravity water flows governed by the equatorial f-plane approximation, Ann. Mat. Pura Appl., 196 (2017), 2253-2260.  doi: 10.1007/s10231-017-0663-2.

[55]

C. I. Martin, Non-existence of time-dependent three-dimensional gravity water flows with constant non-zero vorticity, Phys. Fluids, 30 (2018), 7 pp. doi: 10.1063/1.5048580.

[56]

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