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Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories
1. | Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria, Austria |
References:
[1] |
A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[2] |
A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.
doi: 10.1088/0305-4470/34/45/311. |
[3] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[4] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, 81, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971873. |
[5] |
A. Constantin, On the modelling of Equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.
doi: 10.1029/2012GL051169. |
[6] |
A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029.
doi: 10.1029/2012JC007879. |
[7] |
A. Constantin, On equatorial wind waves, Differential and Integral Equations, 26 (2013), 237-252. |
[8] |
A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Ocean., 43 (2013), 165-175.
doi: 10.1175/JPO-D-12-062.1. |
[9] |
A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves}, Nonlinear Anal. Real World Appl., 9 (2008), 1336-1344.
doi: 10.1016/j.nonrwa.2007.03.003. |
[10] |
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. |
[11] |
A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2008), 1-18.
doi: 10.1007/s00021-005-0214-2. |
[12] |
A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.
doi: 10.1002/cpa.20299. |
[13] |
M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics, J. Math. Fluid Mech., 14 (2012), 407-419.
doi: 10.1007/s00021-011-0068-8. |
[14] |
M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909.
doi: 10.1016/j.jde.2008.01.012. |
[15] |
I. Gallagher and L. Saint-Raymond, On the influence of the Earth's rotation on geophysical flows, Handbook of Mathematical Fluid Mechanics, 4 (2007), 201-329.
doi: 10.1016/S1874-5792(07)80009-7. |
[16] |
F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. |
[17] |
D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not., Art., (2006), 1-13.
doi: 10.1155/IMRN/2006/23405. |
[18] |
D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlinear Math. Phys., 14 (2007), 1-7.
doi: 10.2991/jnmp.2007.14.1.1. |
[19] |
D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.S2.7. |
[20] |
D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21.
doi: 10.1016/j.euromechflu.2012.10.001. |
[21] |
D. Ionescu-Kruse, Particle trajectories in linearized irrotational shallow water flows, J. Nonlinear Math. Phys., 15 (2008), 13-27.
doi: 10.2991/jnmp.2008.15.s2.2. |
[22] |
D. Ionescu-Kruse, Particle trajectories beneath small amplitude shallow water waves in constant vorticity flows, Nonlinear Anal-Theor, 71 (2009), 3779-3793.
doi: 10.1016/j.na.2009.02.050. |
[23] |
D. Ionescu-Kruse, Small-amplitude capillary-gravity water waves: Exact solutions and particle motion beneath such waves, Nonlinear Anal. Real World Appl., 11 (2010), 2989-3000.
doi: 10.1016/j.nonrwa.2009.10.019. |
[24] |
D. Ionescu-Kruse, Peakons arising as particle paths beneath small-amplitude water waves in constant vorticity flows, J. Nonlinear Math. Phys., 17 (2010), 415-422.
doi: 10.1142/S140292511000101X. |
[25] |
D. Ionescu-Kruse, Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity, Commun. Pure Appl. Anal., 11 (2012), 1475-1496.
doi: 10.3934/cpaa.2012.11.1475. |
[26] |
D. Ionescu-Kruse, On the particle paths and the stagnation points in small-amplitude deep-water waves, J. Math. Fluid Mech., 15 (2013), 41-54.
doi: 10.1007/s00021-012-0102-5. |
[27] |
H. Lamb and H. Lamb, Hydrodynamics, 6th ed., Cambridge University Press, 1953. |
[28] |
J. Lighthill, Waves in Fluids, Cambridge Univ. Press, Cambridge, 1978. |
[29] |
A. V. Matioc, On particle trajectories in linear deep-water waves, Commun. Pure Appl. Anal., 11 (2012), 1537-1547.
doi: 10.3934/cpaa.2012.11.1537. |
[30] |
A. V. Matioc, An explicit solution for deep water waves with Coriolis effect, J. Nonlinear Math. Phys., 19 (2012), 15 pp.
doi: 10.1142/S1402925112400050. |
[31] |
A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 10 pp.
doi: 10.1088/1751-8113/45/36/365501. |
[32] |
A. V. Matioc, On particle motion in geophysical deep water waves traveling over uniform currents, Quart. Appl. Math., to appear. |
[33] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press, Cambridge, 1997.
doi: 10.1017/CBO9780511624056. |
[34] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979.
doi: 10.1115/1.3157711. |
[35] |
W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. A, 153 (1863), 127-138.
doi: 10.1098/rstl.1863.0006. |
[36] |
J. J. Stoker, Water Waves. The Mathematical Theory with Applications, Interscience Publ. Inc., New York, 1957. |
[37] |
J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. |
[38] |
E. Wahlen, Steady water waves with a critical layer, J. Differential Eq., 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
show all references
References:
[1] |
A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[2] |
A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.
doi: 10.1088/0305-4470/34/45/311. |
[3] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[4] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, 81, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971873. |
[5] |
A. Constantin, On the modelling of Equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.
doi: 10.1029/2012GL051169. |
[6] |
A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029.
doi: 10.1029/2012JC007879. |
[7] |
A. Constantin, On equatorial wind waves, Differential and Integral Equations, 26 (2013), 237-252. |
[8] |
A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Ocean., 43 (2013), 165-175.
doi: 10.1175/JPO-D-12-062.1. |
[9] |
A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves}, Nonlinear Anal. Real World Appl., 9 (2008), 1336-1344.
doi: 10.1016/j.nonrwa.2007.03.003. |
[10] |
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. |
[11] |
A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2008), 1-18.
doi: 10.1007/s00021-005-0214-2. |
[12] |
A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.
doi: 10.1002/cpa.20299. |
[13] |
M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics, J. Math. Fluid Mech., 14 (2012), 407-419.
doi: 10.1007/s00021-011-0068-8. |
[14] |
M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909.
doi: 10.1016/j.jde.2008.01.012. |
[15] |
I. Gallagher and L. Saint-Raymond, On the influence of the Earth's rotation on geophysical flows, Handbook of Mathematical Fluid Mechanics, 4 (2007), 201-329.
doi: 10.1016/S1874-5792(07)80009-7. |
[16] |
F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. |
[17] |
D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not., Art., (2006), 1-13.
doi: 10.1155/IMRN/2006/23405. |
[18] |
D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlinear Math. Phys., 14 (2007), 1-7.
doi: 10.2991/jnmp.2007.14.1.1. |
[19] |
D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.S2.7. |
[20] |
D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21.
doi: 10.1016/j.euromechflu.2012.10.001. |
[21] |
D. Ionescu-Kruse, Particle trajectories in linearized irrotational shallow water flows, J. Nonlinear Math. Phys., 15 (2008), 13-27.
doi: 10.2991/jnmp.2008.15.s2.2. |
[22] |
D. Ionescu-Kruse, Particle trajectories beneath small amplitude shallow water waves in constant vorticity flows, Nonlinear Anal-Theor, 71 (2009), 3779-3793.
doi: 10.1016/j.na.2009.02.050. |
[23] |
D. Ionescu-Kruse, Small-amplitude capillary-gravity water waves: Exact solutions and particle motion beneath such waves, Nonlinear Anal. Real World Appl., 11 (2010), 2989-3000.
doi: 10.1016/j.nonrwa.2009.10.019. |
[24] |
D. Ionescu-Kruse, Peakons arising as particle paths beneath small-amplitude water waves in constant vorticity flows, J. Nonlinear Math. Phys., 17 (2010), 415-422.
doi: 10.1142/S140292511000101X. |
[25] |
D. Ionescu-Kruse, Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity, Commun. Pure Appl. Anal., 11 (2012), 1475-1496.
doi: 10.3934/cpaa.2012.11.1475. |
[26] |
D. Ionescu-Kruse, On the particle paths and the stagnation points in small-amplitude deep-water waves, J. Math. Fluid Mech., 15 (2013), 41-54.
doi: 10.1007/s00021-012-0102-5. |
[27] |
H. Lamb and H. Lamb, Hydrodynamics, 6th ed., Cambridge University Press, 1953. |
[28] |
J. Lighthill, Waves in Fluids, Cambridge Univ. Press, Cambridge, 1978. |
[29] |
A. V. Matioc, On particle trajectories in linear deep-water waves, Commun. Pure Appl. Anal., 11 (2012), 1537-1547.
doi: 10.3934/cpaa.2012.11.1537. |
[30] |
A. V. Matioc, An explicit solution for deep water waves with Coriolis effect, J. Nonlinear Math. Phys., 19 (2012), 15 pp.
doi: 10.1142/S1402925112400050. |
[31] |
A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 10 pp.
doi: 10.1088/1751-8113/45/36/365501. |
[32] |
A. V. Matioc, On particle motion in geophysical deep water waves traveling over uniform currents, Quart. Appl. Math., to appear. |
[33] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press, Cambridge, 1997.
doi: 10.1017/CBO9780511624056. |
[34] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979.
doi: 10.1115/1.3157711. |
[35] |
W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. A, 153 (1863), 127-138.
doi: 10.1098/rstl.1863.0006. |
[36] |
J. J. Stoker, Water Waves. The Mathematical Theory with Applications, Interscience Publ. Inc., New York, 1957. |
[37] |
J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. |
[38] |
E. Wahlen, Steady water waves with a critical layer, J. Differential Eq., 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
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