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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

Delia Ionescu-Kruse 1, and  Anca-Voichita Matioc 1,

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.
Keywords: constant vorticity, equatorial f-plane approximation, small-amplitude waves, particle paths, Geophysical water waves, disperion relations..
Mathematics Subject Classification: Primary: 35Q86, 37N10; Secondary: 76B15, 76F1.
Citation: Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete & Continuous Dynamical Systems, 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), 1405-1417. doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

A. Constantin, On the modelling of Equatorial waves, Geophys. Res. Lett., 39 (2012), L05602. doi: 10.1029/2012GL051169.  Google Scholar

[6]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879.  Google Scholar

[7]

A. Constantin, On equatorial wind waves, Differential and Integral Equations, 26 (2013), 237-252.  Google Scholar

[8]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Ocean., 43 (2013), 165-175. doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Lamb and H. Lamb, Hydrodynamics, 6th ed., Cambridge University Press, 1953.  Google Scholar

[28]

J. Lighthill, Waves in Fluids, Cambridge Univ. Press, Cambridge, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

A. V. Matioc, On particle motion in geophysical deep water waves traveling over uniform currents,, Quart. Appl. Math., ().   Google Scholar

[33]

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

[34]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. doi: 10.1115/1.3157711.  Google Scholar

[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.  Google Scholar

[36]

J. J. Stoker, Water Waves. The Mathematical Theory with Applications, Interscience Publ. Inc., New York, 1957.  Google Scholar

[37]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

A. Constantin, On the modelling of Equatorial waves, Geophys. Res. Lett., 39 (2012), L05602. doi: 10.1029/2012GL051169.  Google Scholar

[6]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879.  Google Scholar

[7]

A. Constantin, On equatorial wind waves, Differential and Integral Equations, 26 (2013), 237-252.  Google Scholar

[8]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Ocean., 43 (2013), 165-175. doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Lamb and H. Lamb, Hydrodynamics, 6th ed., Cambridge University Press, 1953.  Google Scholar

[28]

J. Lighthill, Waves in Fluids, Cambridge Univ. Press, Cambridge, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

A. V. Matioc, On particle motion in geophysical deep water waves traveling over uniform currents,, Quart. Appl. Math., ().   Google Scholar

[33]

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

[34]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. doi: 10.1115/1.3157711.  Google Scholar

[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.  Google Scholar

[36]

J. J. Stoker, Water Waves. The Mathematical Theory with Applications, Interscience Publ. Inc., New York, 1957.  Google Scholar

[37]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar

[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.  Google Scholar

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