August  2014, 34(8): 3045-3060. doi: 10.3934/dcds.2014.34.3045

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

Received  July 2013 Revised  September 2013 Published  January 2014

We consider the two-dimensional equatorial water-wave problem with constant vorticity in the $f$-plane approximation. Within the framework of small-amplitude waves, we derive the dispersion relations and we find the analytic solutions of the nonlinear differential equation system describing the particle paths below such waves. We show that the solutions obtained are not closed curves. Some remarks on the stagnation points are also provided.
Citation: 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
References:
[1]

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

[2]

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

[3]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. 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, (2011). doi: 10.1137/1.9781611971873.

[5]

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

[6]

A. Constantin, An exact solution for equatorially trapped waves,, J. Geophys. Res., 117 (2012). doi: 10.1029/2012JC007879.

[7]

A. Constantin, On equatorial wind waves,, Differential and Integral Equations, 26 (2013), 237.

[8]

A. Constantin, Some three-dimensional nonlinear equatorial flows,, J. Phys. Ocean., 43 (2013), 165. 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. 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. 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. 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. 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. 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. 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. 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.

[17]

D. Henry, The trajectories of particles in deep-water Stokes waves,, Int. Math. Res. Not., (2006), 1. 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. doi: 10.2991/jnmp.2007.14.1.1.

[19]

D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., 15 (2008), 87. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00021-012-0102-5.

[27]

H. Lamb and H. Lamb, Hydrodynamics,, 6th ed., (1953).

[28]

J. Lighthill, Waves in Fluids,, Cambridge Univ. Press, (1978).

[29]

A. V. Matioc, On particle trajectories in linear deep-water waves,, Commun. Pure Appl. Anal., 11 (2012), 1537. 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). 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). 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., ().

[33]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Univ. Press, (1997). doi: 10.1017/CBO9780511624056.

[34]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer, (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. doi: 10.1098/rstl.1863.0006.

[36]

J. J. Stoker, Water Waves. The Mathematical Theory with Applications,, Interscience Publ. Inc., (1957).

[37]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.

[38]

E. Wahlen, Steady water waves with a critical layer,, J. Differential Eq., 246 (2009), 2468. 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. doi: 10.1088/0305-4470/34/7/313.

[2]

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

[3]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. 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, (2011). doi: 10.1137/1.9781611971873.

[5]

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

[6]

A. Constantin, An exact solution for equatorially trapped waves,, J. Geophys. Res., 117 (2012). doi: 10.1029/2012JC007879.

[7]

A. Constantin, On equatorial wind waves,, Differential and Integral Equations, 26 (2013), 237.

[8]

A. Constantin, Some three-dimensional nonlinear equatorial flows,, J. Phys. Ocean., 43 (2013), 165. 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. 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. 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. 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. 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. 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. 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. 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.

[17]

D. Henry, The trajectories of particles in deep-water Stokes waves,, Int. Math. Res. Not., (2006), 1. 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. doi: 10.2991/jnmp.2007.14.1.1.

[19]

D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., 15 (2008), 87. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00021-012-0102-5.

[27]

H. Lamb and H. Lamb, Hydrodynamics,, 6th ed., (1953).

[28]

J. Lighthill, Waves in Fluids,, Cambridge Univ. Press, (1978).

[29]

A. V. Matioc, On particle trajectories in linear deep-water waves,, Commun. Pure Appl. Anal., 11 (2012), 1537. 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). 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). 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., ().

[33]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Univ. Press, (1997). doi: 10.1017/CBO9780511624056.

[34]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer, (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. doi: 10.1098/rstl.1863.0006.

[36]

J. J. Stoker, Water Waves. The Mathematical Theory with Applications,, Interscience Publ. Inc., (1957).

[37]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.

[38]

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

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