August  2014, 34(8): 3025-3034. doi: 10.3934/dcds.2014.34.3025

One-dimensional weakly nonlinear model equations for Rossby waves

1. 

School of Mathematical Sciences, University College Cork, Cork, Ireland

2. 

School of Mathematical Science, Dublin Institute of Technology, Kevin Street, Dublin 8

Received  July 2013 Revised  September 2013 Published  January 2014

In this study we explore several possibilities for modelling weakly nonlinear Rossby waves in fluid of constant depth, which propagate predominantly in one direction. The model equations obtained include the BBM equation, as well as the integrable KdV and Degasperis-Procesi equations.
Citation: David Henry, Rossen Ivanov. One-dimensional weakly nonlinear model equations for Rossby waves. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3025-3034. doi: 10.3934/dcds.2014.34.3025
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]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.

[3]

J. L. Bona, W. G. Pritchard and L. R. Scott, A comparison of solutions of two model equations for lonf waves, Lectures in Applied Mathematics, 20 (1983), 235-267.

[4]

A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588. doi: 10.1137/090748500.

[5]

A. Boutet de Monvel and D. Shepelsky, A Riemann-Hilbert approach for the Degasperis-Procesi equation, Nonlinearity, 26 (2013), 2081-2107. doi: 10.1088/0951-7715/26/7/2081.

[6]

J. P. Boyd, Equatorial solitary waves. Part I: Rossby solitons, Journal of Physical Oceanography, 10 (1980), 1699-1717. doi: 10.1175/1520-0485(1980)010<1699:ESWPIR>2.0.CO;2.

[7]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation. An Introduction, Princeton University Press, Princeton, NJ, 2003.

[8]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[14]

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

[15]

A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506, 4 pp. doi: 10.1063/1.1845603.

[16]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.

[17]

A. Constantin and J. Escher, Particle trajectories in solitary water wave, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.

[18]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586.

[19]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017.

[20]

A. Constantin, R. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012.

[21]

A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180. doi: 10.1007/s10455-006-9042-8.

[22]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.

[23]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[24]

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

[25]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[26]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011.

[27]

A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoretical and Mathematical Physics, 133 (2002), 1461-1472. doi: 10.1023/A:1021186408422.

[28]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory (eds. A. Degasperis and G. Gaeta), World Scientific, 1999, 23-37.

[29]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153 doi: 10.1007/s00209-010-0778-2.

[30]

J. Escher, Y. Liu and Z. Yin, Global weak solutions blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022.

[31]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences (ed. J. Steele), Academic, San Diego, Calif., 2009, 3679-3695. doi: 10.1016/B978-012374473-9.00610-X.

[32]

C. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Letters, 19 (1967), 1095-1097.

[33]

G. A. Gottwald, The Zakharov-Kuznetsov equation as a two-dimensional model for nonlinear Rossby waves, preprint, arXiv:nlin/0312009.

[34]

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.

[35]

D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565-1573. doi: 10.1016/j.na.2008.02.104.

[36]

D. Henry, Compactly supported solutions of a family of nonlinear partial differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 15 (2008), 145-150.

[37]

D. Henry, Infinite propagation speed for the Degasperis-Procesi equation, J. Math. Anal. Appl., 311 (2005), 755-759. doi: 10.1016/j.jmaa.2005.03.001.

[38]

D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. doi: 10.2991/jnmp.2005.12.3.3.

[39]

D. Holm, J. Marsden and T. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.

[40]

D. Holm and R. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications, J. of Geometry and Symmetry in Physics, 22 (2011), 13-49. doi: 10.7546/jgsp-22-2011-13-49.

[41]

R. Ivanov, Water waves and integrability, Philos. Trans. Roy. Soc.: Ser. A., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.

[42]

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

[43]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.

[44]

B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation, Wave Motion, 46 (2009), 412-419. doi: 10.1016/j.wavemoti.2009.06.005.

[45]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangularchannel, an on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.

[46]

Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146. doi: 10.1002/cpa.20239.

[47]

A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 365501. doi: 10.1088/1751-8113/45/36/365501.

[48]

O. Mustafa, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14. doi: 10.2991/jnmp.2005.12.1.2.

[49]

G. W. Owen, A. J. Willmott and I. D. Abrahams, Scattering of barotropic Rossby waves by the Antarctic Circumpolar Current, J. Geophys. Res., 111 (2006), C12024, 14 pp. doi: 10.1029/2005JC003014.

[50]

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

[51]

P. B. Rhines, Lectures in Geophysical Fluid Dynamics, Lectures in Applied Mathematics, 20 (1983), 3-58.

[52]

G. B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. Lond. A, 299 (1967), 6-25.

[53]

G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.

[54]

V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method, Plenum, New York, 1984.

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]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.

[3]

J. L. Bona, W. G. Pritchard and L. R. Scott, A comparison of solutions of two model equations for lonf waves, Lectures in Applied Mathematics, 20 (1983), 235-267.

[4]

A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588. doi: 10.1137/090748500.

[5]

A. Boutet de Monvel and D. Shepelsky, A Riemann-Hilbert approach for the Degasperis-Procesi equation, Nonlinearity, 26 (2013), 2081-2107. doi: 10.1088/0951-7715/26/7/2081.

[6]

J. P. Boyd, Equatorial solitary waves. Part I: Rossby solitons, Journal of Physical Oceanography, 10 (1980), 1699-1717. doi: 10.1175/1520-0485(1980)010<1699:ESWPIR>2.0.CO;2.

[7]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation. An Introduction, Princeton University Press, Princeton, NJ, 2003.

[8]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[14]

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

[15]

A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506, 4 pp. doi: 10.1063/1.1845603.

[16]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.

[17]

A. Constantin and J. Escher, Particle trajectories in solitary water wave, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.

[18]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586.

[19]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017.

[20]

A. Constantin, R. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012.

[21]

A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180. doi: 10.1007/s10455-006-9042-8.

[22]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.

[23]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[24]

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

[25]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[26]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011.

[27]

A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoretical and Mathematical Physics, 133 (2002), 1461-1472. doi: 10.1023/A:1021186408422.

[28]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory (eds. A. Degasperis and G. Gaeta), World Scientific, 1999, 23-37.

[29]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153 doi: 10.1007/s00209-010-0778-2.

[30]

J. Escher, Y. Liu and Z. Yin, Global weak solutions blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022.

[31]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences (ed. J. Steele), Academic, San Diego, Calif., 2009, 3679-3695. doi: 10.1016/B978-012374473-9.00610-X.

[32]

C. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Letters, 19 (1967), 1095-1097.

[33]

G. A. Gottwald, The Zakharov-Kuznetsov equation as a two-dimensional model for nonlinear Rossby waves, preprint, arXiv:nlin/0312009.

[34]

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.

[35]

D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565-1573. doi: 10.1016/j.na.2008.02.104.

[36]

D. Henry, Compactly supported solutions of a family of nonlinear partial differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 15 (2008), 145-150.

[37]

D. Henry, Infinite propagation speed for the Degasperis-Procesi equation, J. Math. Anal. Appl., 311 (2005), 755-759. doi: 10.1016/j.jmaa.2005.03.001.

[38]

D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. doi: 10.2991/jnmp.2005.12.3.3.

[39]

D. Holm, J. Marsden and T. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.

[40]

D. Holm and R. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications, J. of Geometry and Symmetry in Physics, 22 (2011), 13-49. doi: 10.7546/jgsp-22-2011-13-49.

[41]

R. Ivanov, Water waves and integrability, Philos. Trans. Roy. Soc.: Ser. A., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.

[42]

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

[43]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.

[44]

B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation, Wave Motion, 46 (2009), 412-419. doi: 10.1016/j.wavemoti.2009.06.005.

[45]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangularchannel, an on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.

[46]

Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146. doi: 10.1002/cpa.20239.

[47]

A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 365501. doi: 10.1088/1751-8113/45/36/365501.

[48]

O. Mustafa, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14. doi: 10.2991/jnmp.2005.12.1.2.

[49]

G. W. Owen, A. J. Willmott and I. D. Abrahams, Scattering of barotropic Rossby waves by the Antarctic Circumpolar Current, J. Geophys. Res., 111 (2006), C12024, 14 pp. doi: 10.1029/2005JC003014.

[50]

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

[51]

P. B. Rhines, Lectures in Geophysical Fluid Dynamics, Lectures in Applied Mathematics, 20 (1983), 3-58.

[52]

G. B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. Lond. A, 299 (1967), 6-25.

[53]

G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.

[54]

V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method, Plenum, New York, 1984.

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