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July  2012, 11(4): 1475-1496. doi: 10.3934/cpaa.2012.11.1475

Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity

Delia Ionescu-Kruse 1,

1. 

Institute of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest

Received  April 2011 Revised  July 2011 Published  January 2012

We provide analytic solutions of the nonlinear differential equation system describing the particle paths below small-amplitude periodic gravity waves travelling on a constant vorticity current. We show that these paths are not closed curves. Some solutions can be expressed in terms of Jacobi elliptic functions, others in terms of hyperelliptic functions. We obtain new kinds of particle paths. We make some remarks on the stagnation points which could appear in the fluid due to the vorticity.
Keywords: Small-amplitude water waves, elliptic functions, hyperelliptic functions., vorticity, particle paths.
Mathematics Subject Classification: 76B15, 76F10, 33E05, 33E2.
Citation: Delia Ionescu-Kruse. Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1475-1496. doi: 10.3934/cpaa.2012.11.1475
References:
[1]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4.

[2]

P. F. Byrd and M. D. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists," Springer-Verlag, Berlin, Heidelberg, New York, 1971.

[3]

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

[4]

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

[5]

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.

[6]

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.

[7]

A. Constantin, M. Ehrnström M. 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.

[8]

A. Constantin and J. Escher, Symmetry of steady periodic water waves with vorticity, J. Fluid. Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.

[9]

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

[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 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.

[12]

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.

[13]

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

[14]

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

[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]

T. A. Da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid. Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.

[17]

L. Debnath, "Nonlinear Water Waves," Boston, MA: Academic Press Inc., 1994.

[18]

M.-L. Dubreil-Jacotin, Sur la determination rigoureuse des ondes permanentes periodiques dempleur finie, J. Math. Pures Appl., 13 (1934), 217-291.

[19]

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.

[20]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys. 2 (1809), 412-445. doi: 10.1002/andp.18090320808.

[21]

D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. (2006), Art. ID 23405, 13 pp. doi: 10.1155/IMRN/2006/23405.

[22]

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.

[23]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Phil. Trans. R. Soc. A, 365 (2007), 2241-2251. doi: 10.1098/rsta.2007.2005.

[24]

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

[25]

V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math Anal., 37 (2006), 1482-1521. doi: 10.1137/040621168.

[26]

V. M. Hur, Symmetry of steady periodic water waves with vorticity, Phil. Trans. R. Soc. A, 365 (2007), 2203-2214. doi: 10.1098/rsta.2007.2002.

[27]

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.

[28]

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.

[29]

D. Ionescu-Kruse, Exact solutions for small-amplitude capillary-gravity water waves, Wave Motion, 46 (2009), 379-388. doi: 10.1016/j.wavemoti.2009.06.003.

[30]

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.

[31]

D. Ionescu-Kruse, Peakons arising as particl epaths beneath small-amplitude water waves in cosntant vorticity flows, J. Nonlinear Math. Phys., 17 (2010), 415-422. doi: 10.1142/S140292511000101X.

[32]

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

[33]

H. Kalisch, Periodic traveling water waves with isobaric streamlines, J. Nonlinear Math. Phys., 11 (2004), 461-471. doi: 10.2991/jnmp.2004.11.4.3.

[34]

H. Lamb, "Hydrodynamics," 6th ed., Cambridge University Press, 1953.

[35]

J. Lighthill, "Waves in Fluids," Cambridge University Press, 2001.

[36]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts Appl. Math., vol. 27, Cambridge University Press, 2002.

[37]

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.

[38]

V. Smirnov, "Cours de Mathématiques supérieures, Tome III, deuxième partie," Mir, Moscou, 1972.

[39]

W. Strauss, Steady water waves, Bull. Amer. Math. Soc., 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1.

[40]

E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076. doi: 10.1016/j.jde.2008.12.018.

[41]

E. Wahlen, On rotational water waves with surface tension, Phil. Trans. R. Soc. A, 365 (2007), 2215-2225. doi: 10.1098/rsta.2007.2003.

[42]

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]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4.

[2]

P. F. Byrd and M. D. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists," Springer-Verlag, Berlin, Heidelberg, New York, 1971.

[3]

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

[4]

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

[5]

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.

[6]

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.

[7]

A. Constantin, M. Ehrnström M. 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.

[8]

A. Constantin and J. Escher, Symmetry of steady periodic water waves with vorticity, J. Fluid. Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.

[9]

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

[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 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.

[12]

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.

[13]

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

[14]

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

[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]

T. A. Da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid. Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.

[17]

L. Debnath, "Nonlinear Water Waves," Boston, MA: Academic Press Inc., 1994.

[18]

M.-L. Dubreil-Jacotin, Sur la determination rigoureuse des ondes permanentes periodiques dempleur finie, J. Math. Pures Appl., 13 (1934), 217-291.

[19]

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.

[20]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys. 2 (1809), 412-445. doi: 10.1002/andp.18090320808.

[21]

D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. (2006), Art. ID 23405, 13 pp. doi: 10.1155/IMRN/2006/23405.

[22]

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.

[23]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Phil. Trans. R. Soc. A, 365 (2007), 2241-2251. doi: 10.1098/rsta.2007.2005.

[24]

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

[25]

V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math Anal., 37 (2006), 1482-1521. doi: 10.1137/040621168.

[26]

V. M. Hur, Symmetry of steady periodic water waves with vorticity, Phil. Trans. R. Soc. A, 365 (2007), 2203-2214. doi: 10.1098/rsta.2007.2002.

[27]

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.

[28]

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.

[29]

D. Ionescu-Kruse, Exact solutions for small-amplitude capillary-gravity water waves, Wave Motion, 46 (2009), 379-388. doi: 10.1016/j.wavemoti.2009.06.003.

[30]

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.

[31]

D. Ionescu-Kruse, Peakons arising as particl epaths beneath small-amplitude water waves in cosntant vorticity flows, J. Nonlinear Math. Phys., 17 (2010), 415-422. doi: 10.1142/S140292511000101X.

[32]

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

[33]

H. Kalisch, Periodic traveling water waves with isobaric streamlines, J. Nonlinear Math. Phys., 11 (2004), 461-471. doi: 10.2991/jnmp.2004.11.4.3.

[34]

H. Lamb, "Hydrodynamics," 6th ed., Cambridge University Press, 1953.

[35]

J. Lighthill, "Waves in Fluids," Cambridge University Press, 2001.

[36]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts Appl. Math., vol. 27, Cambridge University Press, 2002.

[37]

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.

[38]

V. Smirnov, "Cours de Mathématiques supérieures, Tome III, deuxième partie," Mir, Moscou, 1972.

[39]

W. Strauss, Steady water waves, Bull. Amer. Math. Soc., 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1.

[40]

E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076. doi: 10.1016/j.jde.2008.12.018.

[41]

E. Wahlen, On rotational water waves with surface tension, Phil. Trans. R. Soc. A, 365 (2007), 2215-2225. doi: 10.1098/rsta.2007.2003.

[42]

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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