September  2019, 39(9): 5149-5169. doi: 10.3934/dcds.2019209

Dispersion relations for steady periodic water waves of fixed mean-depth with two rotational layers

1. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

2. 

School of Mathematical Sciences, University College Cork, Western Road, T12 XF62, Cork, Ireland

* Corresponding author: Calin Iulian Martin

Received  July 2018 Revised  April 2019 Published  May 2019

Fund Project: C. I. Martin acknowledges the support of the Science Foundation Ireland (SFI)–research grant 13/CDA/2117 and the support of the Austrian Science Fund (FWF)–research grant P 30878-N32. A. Rodríguez-Sanjurjo acknowledges the support of the Science Foundation Ireland (SFI)–research grant 13/CDA/2117

The aim of this paper is to obtain the dispersion relation for small-amplitude periodic travelling water waves propagating over a flat bed with a specified mean depth under the presence of a discontinuous piecewise constant vorticity. An analysis of the dispersion relation for a model with two rotational layers each having a non-zero constant vorticity is presented. Moreover, we present a stability result for the bifurcation inducing laminar flow solutions.

Citation: Calin Iulian Martin, Adrián Rodríguez-Sanjurjo. Dispersion relations for steady periodic water waves of fixed mean-depth with two rotational layers. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5149-5169. doi: 10.3934/dcds.2019209
References:
[1]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), 755-768. doi: 10.1017/S0956792504005777.

[2]

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.

[3]

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

[4]

A. ConstantinD. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163. doi: 10.1017/S0022112005007469.

[5]

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

[6]

A. Constantin and W. A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.

[7]

A. ConstantinR. I. Ivanov and E. Prodanov, Nearly-Hamiltonian structure for water waves with constant vorticity, J. Math. Fluid Mech., 10 (2008), 224-237. doi: 10.1007/s00021-006-0230-x.

[8]

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.

[9]

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.

[10]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NFS Regional Conference Series in Applied Mathematics, 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[11]

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

[12]

A. Constantin and W. A. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[13]

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.

[14]

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.

[15]

A. Constantin and R. I. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids, Physics of Fluids, 27 (2015), 086603. doi: 10.1063/1.4929457.

[16]

A. ConstantinR. I. Ivanov and C. I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221 (2016), 1417-1447. doi: 10.1007/s00205-016-0990-2.

[17]

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

[18]

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

[19]

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.

[20]

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

[21]

J. Escher, Regularity of rotational travelling water waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1602-1615. doi: 10.1098/rsta.2011.0458.

[22]

J. Escher and B.-V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function, Differential Integral Equations, 27 (2014), 217-232.

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of the Second Order, Second edition. Springer-Verlag, Berlin, 2001.

[24]

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

[25]

D. Henry, Regularity for steady periodic capillary water waves with vorticity, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1616-1628. doi: 10.1098/rsta.2011.0449.

[26]

D. Henry, Dispersion relations for steady periodic water waves of fixed mean-depth with an isolated bottom vorticity layer, J. Nonlinear Math. Phys., 19 (2012), 1240007, 14 pp. doi: 10.1142/S1402925112400074.

[27]

D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves, Commun. Pure Appl. Anal., 11 (2012), 1453-1464. doi: 10.3934/cpaa.2012.11.1453.

[28]

D. Henry, Dispersion relations for steady periodic water waves of fixed mean-depth with an isolated layer of vorticity at the surface, Nonlinear Anal. Real World Appl., 14 (2013), 1034-1043. doi: 10.1016/j.nonrwa.2012.08.015.

[29]

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.

[30]

D. Henry and B.-V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 955–974.

[31]

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.

[32]

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

[33]

D. Henry and A.-V. Matioc, On the symmetry of steady equatorial wind waves, Nonlinear Anal. Real World Appl., 18 (2014), 50-56. doi: 10.1016/j.nonrwa.2014.01.009.

[34]

D. Henry and B.-V. Matioc, Aspects of the mathematical analysis of nonlinear stratified water waves, Elliptic and Parabolic Equations, 159–177, Springer Proc. Math. Stat., 119, Springer, Cham, 2015. doi: 10.1007/978-3-319-12547-3_7.

[35]

D. Henry and S. Sastre-Gómez, Steady periodic water waves bifurcating for fixed-depth rotational flows with discontinuous vorticity, Differential Integral Equations, 31 (2018), 1-26.

[36]

D. Henry, C.-I. Martin and S. Sastre-Gómez, Large amplitude steady periodic water waves for fixed-depth flows with discontinuous vorticity, submitted.

[37]

D. Ionescu-Kruse and C. I. Martin, Periodic Equatorial water flows from a Hamiltonian perspective, J. Differential Equations, 262 (2017), 4451-4474. doi: 10.1016/j.jde.2017.01.001.

[38]

I. G. Jonsson, Wave-current interactions. In: Le Mehaut, M., Hanes, D.M., eds., The Sea, Ocean Engineering Science, Vol. 9. New York: Wiley, (1990), 65–120.

[39]

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.

[40]

M. Kluczek and C. I. Martin, Dispersion relations for fixed mean-depth flows with two discontinuities in vorticity, Nonlinear Anal., 181 (2019), 62-86. doi: 10.1016/j.na.2018.11.007.

[41]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 2008,197–215. doi: 10.1017/S0022112008002371.

[42]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids, 27 (2008), 96-109. doi: 10.1016/j.euromechflu.2007.04.004.

[43]

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.

[44]

C. I. Martin, Dispersion relations for periodic water waves with surface tension and discontinuous vorticity, Discrete Cont. Dyn. Syst. - Ser. A, 34 (2013), 3109-3123. doi: 10.3934/dcds.2014.34.3109.

[45]

C. I. Martin, Dispersion relations for rotational gravity water flows having two jumps in the vorticity distribution, J. Math. Anal. Appl., 418 (2014), 595-611. doi: 10.1016/j.jmaa.2014.04.014.

[46]

C. I. Martin and B.-V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, J. Differential Equations, 256 (2014), 3086-3114. doi: 10.1016/j.jde.2014.01.036.

[47]

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

[48]

C. I. Martin and B.-V. Matioc, Gravity water flows with discontinuous vorticity and stagnation points, Commun. Math. Sci., 14 (2016), 415-441. doi: 10.4310/CMS.2016.v14.n2.a5.

[49]

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.

[50]

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

[51]

A.-V. Matioc and B.-V. Matioc, Capillary-gravity water waves with discontinuous vorticity: Existence and regularity results, Comm. Math. Phys., 330 (2014), 859-886. doi: 10.1007/s00220-014-1918-z.

[52]

A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity, Differential Integral Equations, 26 (2013), 129-140.

[53]

A. Nachbin and R. Ribeiro-Junior, Capturing the flow beneath water waves, Phil. Trans. R. Soc. A, 376 (2018), 20170098, 17pp. doi: 10.1098/rsta.2017.0098.

[54]

K. Okuda, Internal flow structure of short wind waves, J. Oceanogr. Soc. Jpn., 38 (1982), 28-42.

[55]

D. H. Peregrine, Interactions of water waves and currents, Adv. Appl. Mech., 16 (1976), 9-117.

[56]

O. M. Phillips and M. L. Banner, Wave breaking in the presence of wind drift and swell, J. Fluid Mech., 66 (1974), 625-640. doi: 10.1017/S0022112074000413.

[57]

G. P. Thomas and G. Klopman, Wave-current interactions in the nearshore region, in: J. N. Hunt (Ed.), Gravity Waves in Water of Finite Depth, Computational Mechanics Publications, WIT, Southampton, United Kingdom, (1997), 255–319.

[58]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

[59]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[60]

S. Walsh, Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete Contin, Dyn. Syst. Ser. A, 34 (2014), 3241-3285. doi: 10.3934/dcds.2014.34.3241.

show all references

References:
[1]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), 755-768. doi: 10.1017/S0956792504005777.

[2]

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.

[3]

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

[4]

A. ConstantinD. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163. doi: 10.1017/S0022112005007469.

[5]

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

[6]

A. Constantin and W. A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.

[7]

A. ConstantinR. I. Ivanov and E. Prodanov, Nearly-Hamiltonian structure for water waves with constant vorticity, J. Math. Fluid Mech., 10 (2008), 224-237. doi: 10.1007/s00021-006-0230-x.

[8]

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.

[9]

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.

[10]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NFS Regional Conference Series in Applied Mathematics, 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[11]

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

[12]

A. Constantin and W. A. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[13]

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.

[14]

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.

[15]

A. Constantin and R. I. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids, Physics of Fluids, 27 (2015), 086603. doi: 10.1063/1.4929457.

[16]

A. ConstantinR. I. Ivanov and C. I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221 (2016), 1417-1447. doi: 10.1007/s00205-016-0990-2.

[17]

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

[18]

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

[19]

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.

[20]

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

[21]

J. Escher, Regularity of rotational travelling water waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1602-1615. doi: 10.1098/rsta.2011.0458.

[22]

J. Escher and B.-V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function, Differential Integral Equations, 27 (2014), 217-232.

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of the Second Order, Second edition. Springer-Verlag, Berlin, 2001.

[24]

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

[25]

D. Henry, Regularity for steady periodic capillary water waves with vorticity, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1616-1628. doi: 10.1098/rsta.2011.0449.

[26]

D. Henry, Dispersion relations for steady periodic water waves of fixed mean-depth with an isolated bottom vorticity layer, J. Nonlinear Math. Phys., 19 (2012), 1240007, 14 pp. doi: 10.1142/S1402925112400074.

[27]

D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves, Commun. Pure Appl. Anal., 11 (2012), 1453-1464. doi: 10.3934/cpaa.2012.11.1453.

[28]

D. Henry, Dispersion relations for steady periodic water waves of fixed mean-depth with an isolated layer of vorticity at the surface, Nonlinear Anal. Real World Appl., 14 (2013), 1034-1043. doi: 10.1016/j.nonrwa.2012.08.015.

[29]

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.

[30]

D. Henry and B.-V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 955–974.

[31]

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.

[32]

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

[33]

D. Henry and A.-V. Matioc, On the symmetry of steady equatorial wind waves, Nonlinear Anal. Real World Appl., 18 (2014), 50-56. doi: 10.1016/j.nonrwa.2014.01.009.

[34]

D. Henry and B.-V. Matioc, Aspects of the mathematical analysis of nonlinear stratified water waves, Elliptic and Parabolic Equations, 159–177, Springer Proc. Math. Stat., 119, Springer, Cham, 2015. doi: 10.1007/978-3-319-12547-3_7.

[35]

D. Henry and S. Sastre-Gómez, Steady periodic water waves bifurcating for fixed-depth rotational flows with discontinuous vorticity, Differential Integral Equations, 31 (2018), 1-26.

[36]

D. Henry, C.-I. Martin and S. Sastre-Gómez, Large amplitude steady periodic water waves for fixed-depth flows with discontinuous vorticity, submitted.

[37]

D. Ionescu-Kruse and C. I. Martin, Periodic Equatorial water flows from a Hamiltonian perspective, J. Differential Equations, 262 (2017), 4451-4474. doi: 10.1016/j.jde.2017.01.001.

[38]

I. G. Jonsson, Wave-current interactions. In: Le Mehaut, M., Hanes, D.M., eds., The Sea, Ocean Engineering Science, Vol. 9. New York: Wiley, (1990), 65–120.

[39]

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.

[40]

M. Kluczek and C. I. Martin, Dispersion relations for fixed mean-depth flows with two discontinuities in vorticity, Nonlinear Anal., 181 (2019), 62-86. doi: 10.1016/j.na.2018.11.007.

[41]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 2008,197–215. doi: 10.1017/S0022112008002371.

[42]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids, 27 (2008), 96-109. doi: 10.1016/j.euromechflu.2007.04.004.

[43]

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.

[44]

C. I. Martin, Dispersion relations for periodic water waves with surface tension and discontinuous vorticity, Discrete Cont. Dyn. Syst. - Ser. A, 34 (2013), 3109-3123. doi: 10.3934/dcds.2014.34.3109.

[45]

C. I. Martin, Dispersion relations for rotational gravity water flows having two jumps in the vorticity distribution, J. Math. Anal. Appl., 418 (2014), 595-611. doi: 10.1016/j.jmaa.2014.04.014.

[46]

C. I. Martin and B.-V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, J. Differential Equations, 256 (2014), 3086-3114. doi: 10.1016/j.jde.2014.01.036.

[47]

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

[48]

C. I. Martin and B.-V. Matioc, Gravity water flows with discontinuous vorticity and stagnation points, Commun. Math. Sci., 14 (2016), 415-441. doi: 10.4310/CMS.2016.v14.n2.a5.

[49]

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.

[50]

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

[51]

A.-V. Matioc and B.-V. Matioc, Capillary-gravity water waves with discontinuous vorticity: Existence and regularity results, Comm. Math. Phys., 330 (2014), 859-886. doi: 10.1007/s00220-014-1918-z.

[52]

A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity, Differential Integral Equations, 26 (2013), 129-140.

[53]

A. Nachbin and R. Ribeiro-Junior, Capturing the flow beneath water waves, Phil. Trans. R. Soc. A, 376 (2018), 20170098, 17pp. doi: 10.1098/rsta.2017.0098.

[54]

K. Okuda, Internal flow structure of short wind waves, J. Oceanogr. Soc. Jpn., 38 (1982), 28-42.

[55]

D. H. Peregrine, Interactions of water waves and currents, Adv. Appl. Mech., 16 (1976), 9-117.

[56]

O. M. Phillips and M. L. Banner, Wave breaking in the presence of wind drift and swell, J. Fluid Mech., 66 (1974), 625-640. doi: 10.1017/S0022112074000413.

[57]

G. P. Thomas and G. Klopman, Wave-current interactions in the nearshore region, in: J. N. Hunt (Ed.), Gravity Waves in Water of Finite Depth, Computational Mechanics Publications, WIT, Southampton, United Kingdom, (1997), 255–319.

[58]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

[59]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[60]

S. Walsh, Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete Contin, Dyn. Syst. Ser. A, 34 (2014), 3241-3285. doi: 10.3934/dcds.2014.34.3241.

Figure 1.  Sketch of the graph of the polynomial providing the dispersion relation for two negative vorticities under the condition (44). The root x01 does not satisfy the non-stagnation condition while the root x02 verifies it if and only if (47) is assumed
[1]

Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114

[2]

Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397

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Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109

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Walter A. Strauss. Vorticity jumps in steady water waves. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101

[5]

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

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G. A. Athanassoulis, K. A. Belibassakis. New evolution equations for non-linear water waves in general bathymetry with application to steady travelling solutions in constant, but arbitrary, depth. Conference Publications, 2007, 2007 (Special) : 75-84. doi: 10.3934/proc.2007.2007.75

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Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045

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Shu-Ming Sun. Existence theory of capillary-gravity waves on water of finite depth. Mathematical Control & Related Fields, 2014, 4 (3) : 315-363. doi: 10.3934/mcrf.2014.4.315

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Mats Ehrnström, Gabriele Villari. Recent progress on particle trajectories in steady water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 539-559. doi: 10.3934/dcdsb.2009.12.539

[10]

David Henry, Bogdan--Vasile Matioc. On the regularity of steady periodic stratified water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1453-1464. doi: 10.3934/cpaa.2012.11.1453

[11]

Gerhard Tulzer. On the symmetry of steady periodic water waves with stagnation points. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1577-1586. doi: 10.3934/cpaa.2012.11.1577

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Mats Ehrnström. Deep-water waves with vorticity: symmetry and rotational behaviour. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 483-491. doi: 10.3934/dcds.2007.19.483

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Bogdan-Vasile Matioc. A characterization of the symmetric steady water waves in terms of the underlying flow. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3125-3133. doi: 10.3934/dcds.2014.34.3125

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Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997

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Delia Ionescu-Kruse. Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1475-1496. doi: 10.3934/cpaa.2012.11.1475

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Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5521-5541. doi: 10.3934/dcds.2019225

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Shunlian Liu, David M. Ambrose. Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3123-3147. doi: 10.3934/dcds.2019129

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Cheng Wang. The primitive equations formulated in mean vorticity. Conference Publications, 2003, 2003 (Special) : 880-887. doi: 10.3934/proc.2003.2003.880

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Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3357-3385. doi: 10.3934/dcds.2018144

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Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331

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