Nonlinear analysis of perturbed rotating whirlpools in the Ocean and atmosphere. (English) Zbl 1445.76033

Summary: A nonlinear, uniformly stratified ocean model is used to investigate the stability and existence of perturbed rotating whirlpools that represent a certain class of exact solutions for the governing nonlinear equations describing the propagation of internal gravity waves in stratified media. A cut-off frequency is identified, beyond which the flow is completely stable to linear disturbances. An analysis close to cut-offs reveals the latitude-dependent distribution of eigenfrequencies of oscillatory and evanescent perturbations of whirlpools.


76B55 Internal waves for incompressible inviscid fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76U60 Geophysical flows
86A05 Hydrology, hydrography, oceanography
76E20 Stability and instability of geophysical and astrophysical flows
76E30 Nonlinear effects in hydrodynamic stability


Full Text: DOI Link


[1] H. Anwar. A study of the turbulent structure in a tidal flow. Estuarine, Coastal and Shelf Sci., 13(4), (1981), 373-387.
[2] J. Apel. A new analytical model for internal solitons in the ocean. J Phys Oceanogr, 33, (2003), 2247-2269.
[3] L. Arm. Effects of variation in eddy diffusivity on property distributions in the oceans. J. Mar. Res., 37, (1997), 515-530.
[4] S. Balasuriya. Vanishing viscosity in the barotropic ;-plane. J. Math. Anal. Appl. 214, (1997), 128-150. · Zbl 0906.35072
[5] G. Castelão, W. Johns. Sea surface structure of North Brazil Current rings derived from shipboard and moored acoustic Doppler current profiler observations. J. Geophys. Res., 116, (2011), C01010, doi:.
[6] D. Chillymanjaro. Giant ocean whirlpools puzzle scientists, Earth changes. Featured stories, Ocean, (2011),
[7] E. Dewan, R. Picard, R. O’Neil, H. Gardiner, J. Gibson. MSX satellite observations of thunderstorm-generated gravity waves in mid-wave infrared images of the upper stratosphere. Geophys. Res. Lett. 25, (1998), 939-942.
[8] N. Fraser. Surfing an oil rig. Energy Rev. 20 (4), (1999), 4-8.
[9] S. Friedlander, W. Siegmann. Internal waves in a contained rotating stratified fluid. J. Fluid Mech. 114, (1982), 123-156. · Zbl 0477.76027
[10] L. Funkquist, L. Gidhagen, U. Svensson. The mathematical modelling of baroclinic waves and fronts in the ocean. Appl. Math. Modelling, 11, (1987), 11-18. · Zbl 0611.76120
[11] J. Gibbon, A. Fokas, C. Doering. Dynamically Streched Vortices as Solutions of the Navier-Stokes Equations. Physica D 132, (1999), 497-510. · Zbl 0956.76018
[12] A. Gill. Atmosphere-Ocean Dynamics. New York, (1983), Academic Press.
[13] S. Hamdi, W. Enright, W. Schiesser, J. Gottlieb. Exact solutions and invariants of motion for eneral types of regularized long wave equations. Math. Comput. Simul., 65, (2004), 535-545. · Zbl 1059.35118
[14] S. Hamdi, B. Morse, B. Halphen, W. Schiesser. Analytical solutions of long nonlinear internal waves, Nat Hazards, 57, (2011), 597-607.
[15] G, Haltiner, R. Williams. Numerical prediction and dynamic meteorology. (1980), Wiley.
[16] K. Helfric, W. Melville. Long nonlinear internal waves. Annu Rev Fluid Mech 38, (2006), 395-425. · Zbl 1098.76018
[17] P. Hsieh. Application of modflow for oil reservoir simulation during the Deepwater Horizon crisis. Ground Water.. 49 (3), (2011), 319-323.
[18] R. Ibragimov, G. Jefferson, J. Carminati. Invariant and approximately invariant solutions on non-linear internal gravity waves forming a column of stratified fluid affected by the Earth’s rotation. Int. J. Non-Linear Mech. 51, (2013), 28-44.
[19] R. Ibragimov. Oscillatory nature and dissipation of the internal waves energy spectrum in the deep ocean. Eur. Phys. J. Appl. Phys., 40, (2007), 315-334.
[20] R. Ibragimov. Generation of internal tides by an oscillating background flow along a corrugated slope. Phys. Scr. 78, (2008), 065801.
[21] R. Ibragimov, D. Pelinovsky. Effects of rotation on stability of viscous stationary flows on a spherical surface. Phys. Fluids, 22, (2010), 126602. · Zbl 1308.76298
[22] R. Ibragimov. Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications. Phys. Fluids, 23, (2011), 123102. · Zbl 1308.76297
[23] R. Ibragimov, N. Yilmaz, A. Bakhtiyarov. Experimental mixing parameterization due to multiphase fluid-structure interaction. Mech. Re. Commun. 38, (2011), 261-266.
[24] R. Ibragimov, M. Dameron. Spinning phenomena and energetics of spherically pulsating patterns in stratified fluids. Phys. Scr. 84, (2011), 015402. · Zbl 1219.86005
[25] G. Jefferson, J. Carminati. ASP: Approximate Symbolic Computation of Approximate Symmetries of Differential Equations. Computer Physics Communications, 184(3), (2013), 1045-1063. · Zbl 1306.65267
[26] V. Kamenkovich. Fundamentals of Ocean Dynamics. (1977), Elsevier.
[27] W. Krauss. Methoden und Ergebnisse der Theoretischen Oceanographie II: Interne Wellen. (1966), Berlin : Gebrueder-Borntraeger.
[28] M. Kudlick. On transient motions in a contained rotating fluid. Ph.D. dissertation, (1966), Massachusetts Institute of Technology.
[29] P. Kundu, I. Cohen, D. Dowling. Fluid Mechanics. 5th Ed.. (1990), Academic Press.
[30] C. Lee, H. Peng, H. Yuan, J. Wu, M. Zhou, F. Hussain. Experimental studies of surface waves inside a cylindrical container. J. Fluid Mech., 677, (2011), 39-62. · Zbl 1241.76035
[31] J. Lions, R. Teman, S. Wang. On the equations of the large-scale ocean. Nonlinearity, 5, (1992), 1007-1053. · Zbl 0766.35039
[32] J. Lions, R. Teman, S. Wang. New formulations of the primitive equations of atmosphere and applications. Nonlinearity, 5, (1992), 237-288. · Zbl 0746.76019
[33] J. McCreary. Easern tropical ocean response to changing wind systems with applications to El Niño. J. Phys, Oceanogr.. 6, (1976) : ,632-645.
[34] J. McCreary. A linear stratified ocean model of the equatorial undercurrent. Phil. Trans. Roy. Soc. London.. 302, (1981), 385-413.
[35] J. McCreary. Equatorial beams. J. Mar. Res. 42, (1984), 395-430.
[36] L. Musak. Topographically trapped waves. Ann. Rev. Fluid. Mech.. 12, (1980), 45-76.
[37] P. Müller, G. Holloway, F. Henyey, N. Pomphrey. Nonlinear interactions among internal gravity waves. Rev. Geophys.. 24, 3, (1986), 493-536.
[38] G. Needler, Dispersion in the ocean by physical, geochemical and biological processes. Phil. Trans. R. Soc. London, 319, (1986), 177-187.
[39] D. Nethery. Vertical propagation of baroclinic Kelvin waves along the west coast of India. J. Earth. Syst. Sci.. 116 (4), (2007), 331-339.
[40] S. Philanders. Variability of the tropical oceans. Dynamics Oceans & Atmos.. 3, (1978), 191.
[41] E. Rasmussen, J, Turner. Polar lows: mesoscale weather systems in the polar regions. Cambridge University Press. (2003), 224. ISBN 978-0-521-62430-5.
[42] P. Reid, A. Fischer, E. Lewis-Brown, et al.. Chapter 1: Impacts of the Oceans on Climate Change. Advances in Marine Biology, 56, (2009), 1-1500.
[43] R. Romea, J. Allen. On vertically propagating coastal Kelvin waves at low latitudes. J. Phys. Oceanogr.. 13 (1), (1983), 241-254.
[44] D. Shindell, G. Schmidt. Southern Hemisphere climate response to ozone changes and greenhouse gas increases. Res. Lett., 31, (2004), L18209.
[45] R. Smith. Poleward propagating perturbations in currents and sea levels along the Peru coast. J. Geophys. Res.. 83, (1978), 6083.
[46] C. Staquet, J. Sommeria. Internal Gravity Waves: From instabilities to turbulence. Annu. Rev. Fluid Mech.. 34, (2002), 559-593. · Zbl 1047.76014
[47] C. Summerhayes, S. Thorpe. Oceanography. An Illustrative Guide. New York : (1996), John Willey & Sons.
[48] R. Szoeke, R. Samelson. The duality between the Boussinesq and non-Boussinesq hydrostatic equations of motion. J. Phys. Oceanogr.. 32, (2002), 2194-2203.
[49] A. Timmermann, M. Latif, A. Grotzner, R. Voss. Modes of climate variability as simulated by a copled general circulation model. Part I: ENSO-like climate variability and its low-frequency modulation. Climate Dynamics.. 15 (8), (1999), 605-618.
[50] K. Vu, G. Jefferson, J. Carminati. Finding generalised symmetries of differential equations using the MAPLE package DESOLVII. Computer Physics Communications, 183, (2012, 1044-1054. · Zbl 1308.35002
[51] G. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, (1966), 2nd ed.. · Zbl 0174.36202
[52] Z. Zhang, W. Wang, B. Qiu. Science, 345 (6194), (2014), 322-324, doi: .
[53] J. Zimmerman. The tidal whirlpool: A review of horizontal dispersion by tidal and residual currents. Netherlands J. of See Res., 20 (2-3), (1986), 133-154.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.