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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/s00222-007-0088-4. Google Scholar

[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. doi: 10.1098/rsta.1972.0032. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/090748500. Google Scholar

[5]

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

[6]

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

[7]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation. An Introduction,, Princeton University Press, (2003). Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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. doi: 10.1007/s10455-006-9042-8. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects,, Academic, (2011). Google Scholar

[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. doi: 10.1023/A:1021186408422. Google Scholar

[28]

A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and Perturbation Theory (eds. A. Degasperis and G. Gaeta), (1999), 23. Google Scholar

[29]

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

[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. doi: 10.1016/j.jfa.2006.03.022. Google Scholar

[31]

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

[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. Google Scholar

[33]

G. A. Gottwald, The Zakharov-Kuznetsov equation as a two-dimensional model for nonlinear Rossby waves,, preprint, (). Google Scholar

[34]

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. Google Scholar

[35]

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

[36]

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

[37]

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

[38]

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

[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. doi: 10.1006/aima.1998.1721. Google Scholar

[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. doi: 10.7546/jgsp-22-2011-13-49. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[46]

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

[47]

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. Google Scholar

[48]

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

[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). doi: 10.1029/2005JC003014. Google Scholar

[50]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1979). doi: 10.1115/1.3157711. Google Scholar

[51]

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

[52]

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

[53]

G. B. Whitham, Linear and Nonlinear Waves,, Wiley, (1974). Google Scholar

[54]

V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method,, Plenum, (1984). 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. doi: 10.1007/s00222-007-0088-4. Google Scholar

[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. doi: 10.1098/rsta.1972.0032. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/090748500. Google Scholar

[5]

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

[6]

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

[7]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation. An Introduction,, Princeton University Press, (2003). Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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. doi: 10.1007/s10455-006-9042-8. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects,, Academic, (2011). Google Scholar

[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. doi: 10.1023/A:1021186408422. Google Scholar

[28]

A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and Perturbation Theory (eds. A. Degasperis and G. Gaeta), (1999), 23. Google Scholar

[29]

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

[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. doi: 10.1016/j.jfa.2006.03.022. Google Scholar

[31]

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

[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. Google Scholar

[33]

G. A. Gottwald, The Zakharov-Kuznetsov equation as a two-dimensional model for nonlinear Rossby waves,, preprint, (). Google Scholar

[34]

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. Google Scholar

[35]

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

[36]

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

[37]

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

[38]

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

[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. doi: 10.1006/aima.1998.1721. Google Scholar

[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. doi: 10.7546/jgsp-22-2011-13-49. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[46]

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

[47]

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. Google Scholar

[48]

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

[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). doi: 10.1029/2005JC003014. Google Scholar

[50]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1979). doi: 10.1115/1.3157711. Google Scholar

[51]

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

[52]

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

[53]

G. B. Whitham, Linear and Nonlinear Waves,, Wiley, (1974). Google Scholar

[54]

V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method,, Plenum, (1984). Google Scholar

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