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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 & 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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[34]

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

[35]

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

[36]

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

[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.  Google Scholar

[38]

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

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[34]

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

[35]

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

[36]

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

[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.  Google Scholar

[38]

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

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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