Asymptotic Analysis of Conserved Densities Evaluated on Invariant Solutions Associated with Large Scale Nonlinear Zonal Flows Around the Rotating Sphere

  • Ranis N. Ibragimov
  • Michael Dameron
  • Chamath Dannangoda


We study the asymptotic behavior of the conserved densities deduced form the Lagrangian corresponding to the nonlinear two-dimensional Euler equations describing nonviscous incompressible fluid flows on a three-dimensional rotating spherical surface superimposed by a particular stationary latitude dependent flow. Under the assumption of no friction and a distribution of temperature dependent only upon latitude, the equations in question can be used to model zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. The conserved densities were analyzed and visualized by using the exact invariant solutions associated with the given model for the particular form of finite disturbances for which the invariant solutions are also exact solutions of Navier-Stokes equations.


Navier Stokes equations Conservation laws Lagrangian Euler equations Atmospheric modeling 

Mathematics Subject Classifications (2010)

35Q30 35Q86 35L65 


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  1. 1.
    Anderson, R.F., Ali, S., Brandtmiller, L.L., Nielsen, S.H.H., Fleisher, M.Q.: Wind-driven upwelling in the Southern Ocean and the deglacial rise in atmospheric CO2. Sci. 323, 1443–1448 (2006)ADSCrossRefGoogle Scholar
  2. 2.
    Bachelor, G.K.: An introduction to fluid dynamics. Cambridge University Press, Cambridge (1967)Google Scholar
  3. 3.
    Balasuriya, S.: Vanishing viscosity in the barotropic β−plane. J. Math. Anal. Appl. 214, 128–150 (1997)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Belotserkovskii, O.M., Mingalev, I.V., Mingalev O.V.: Formation of large-scale vortices in shear flows of the lower atmosphere of the earth in the region of tropical latitudes. Cosm. Res. 47(6), 466–479 (2009)ADSCrossRefGoogle Scholar
  5. 5.
    Ben-Yu, G.: Spectral method for vorticity equations on spherical surface. Math. Comput. 64, 1067–1079 (1995)CrossRefGoogle Scholar
  6. 6.
    Blinova, E.N.: A hydrodynamical theory of pressure and temperature waves and of centres of atmospheric action. C.R. (Doklady) Acad. Sci. USSR 39, 257–260 (1943)MATHMathSciNetGoogle Scholar
  7. 7.
    Blinova, E.N.: A method of solution of the nonlinear problem of atmospheric motions on a planetary scale. Dokl. Acad. Nauk USSR 110, 975–977 (1956)MATHMathSciNetGoogle Scholar
  8. 8.
    Cenedese, C., Linden, P.F.: Cyclone and anticyclone formation in a rotating stratified fluid over a sloping bottom. J. Fluid Mech. 381, 199–223 (1999)ADSCrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Furnier, A., Bunger, H., Hollerbach, R., Vilotte, I.: Application of the spectral-element method to the axisymetric Navier-Stokes equations. Geophys. J. Int. 156, 682–700 (2004)ADSCrossRefGoogle Scholar
  10. 10.
    Golovkin, H.: Vanishing viscosity in Cauchy’s problem for hydromechanics equation. Proc. Steklov Inst. Math. 92, 33–53 (1966)Google Scholar
  11. 11.
    Herrmann, E.: The motions of the atmosphere and especially its waves. Bull. Amer. Math. Soc. 2(9), 285–296 (1896)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Hsieh, P.A.: Application of modflow for oil reservoir simulation during the Deepwater Horizon crisis. Ground Water 49(3), 319–323 (2011)CrossRefGoogle Scholar
  13. 13.
    Ibragimov, R.N.: Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications. Phys. Fluids 23, 123102 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    Ibragimov, R.N., Dameron, M.: Spinning phenomena and energetics of spherically pulsating patterns in stratified fluids. Physica Scripta 84, 015402 (2011)Google Scholar
  15. 15.
    Ibragimov, N.H., Ibragimov, R.N.: Intergarion by quadratures of the nonlinear Euler equations modeling atmoaspheric flows in a thin rotating spherical shell. Phys. Lett. A 375, 3858 (2011)ADSCrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Ibragimov, N.H., Ibragimov, R.N.: Applications of lie group analysis in geophysical fluid dynamics. Series on Complexity, Nonlinearity and Chaos, vol. 2. World Scientific Publishers, ISBN: 978-981-4340-46-5 (2011)Google Scholar
  17. 17.
    Ibragimov, R.N., Pelinovsky, D.E.: Effects of rotation on stability of viscous stationary flows on a spherical surface. Phys. Fluids 22, 126602 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Ibragimov, R.N.: Mechanism of energy transfers to smaller scales within the rotational internal wave field. Springer Math. Phys. Anal. Geom. 13(4), 331–355 (2010)CrossRefMATHGoogle Scholar
  19. 19.
    Ibragimov, R.N., Pelinovsky, D.E.: Incompressible viscous fluid flows in a thin spherical shell. J. Math. Fluid. Mech. 11, 60–90 (2009)ADSCrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Ibragimov, R.N.: Shallow water theory and solutions of the free boundary problem on the atmospheric motion around the Earth. Physica Scr. 61, 391–395 (2000)ADSCrossRefMATHGoogle Scholar
  21. 21.
    Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333(1), 311–328 (2007)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Ibragimov, R.N., Jefferson, G., Carminati, J.: Invariant and approximately invariant solutions of nonlinear internal gravity waves forming a column of stratified fluid affected the Earth’s rotation. Int. J. Non-Linear Mech. 51, 28–44 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    Ibragimov, R.N., Jefferson, G., Carminati, J.: Explicit invariant solutions associated with nonlinear atmospheric flows in a thin rotating spherical shell with and without west-to-east jets perturbations. Spinger Math. Phys. Anal. Geom. 3(3), 201–294 (2013)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Iftimie, D., Raugel, G.: Some results on the Navier-Stokes equations in thin 3D domains. J. Diff. Eqs. 169, 281–331 (2001)ADSCrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Lamb, H.: Hydrodynamics, 5th edn. Cambridge University, Cambridge (1924)Google Scholar
  26. 26.
    Lions, J.L., Teman, R., Wang, S.: On the equations of the large-scale ocean. Nonlinearity 5, 1007–1053 (1992)Google Scholar
  27. 27.
    Lions, J.L., Teman, R., Wang, S.: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5, 237–288 (1992)ADSCrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Noether, E.: Invariante Variationsprobleme. Konigliche Gessellschaft der Wissenschaften, Gottingen Math. Phys. K1., English transl.: Transport Theory and Statistical Physics 1(3), 186–207 (1918/1971)Google Scholar
  29. 29.
    Shindell, D.T., Schmidt, G.A.: Southern Hemisphere climate response to ozone changes and greenhouse gas increases. Res. Lett. 31, L18209 (2004)ADSCrossRefGoogle Scholar
  30. 30.
    Shen, J.: On pressure stabilization method and projection method for unsteady Navier-Stokes equations. In: Advances in Computer Methods for Partial Differential Equations, pp. 658–662. IMACS, New Brunswick (1992)Google Scholar
  31. 31.
    Summerhayes, C.P., Thorpe, S.A.: Oceanography: An illustrative guide. Willey, New York (1996)Google Scholar
  32. 32.
    Swarztrauber, P.N.: Shallow water flow on the sphere. Mon. Weather Rev. 132, 3010–3018 (2004)ADSCrossRefGoogle Scholar
  33. 33.
    The approximation of vector functions and their derivatives on the sphere. SIAM J. Numer. Anal. 18, 181–210 (1981)Google Scholar
  34. 34.
    Temam, R., Ziane, M.: Navier-Stokes equations in thin spherical domains. Contemp. Math. 209, 281–314 (1997)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Toggweiler, J.R., Russel, J.L.: Ocean circulation on a warming climate. Nature 451, 286–288 (2008)ADSCrossRefGoogle Scholar
  36. 36.
    Weijer, W., Vivier, F., Gille, S.T., Dijkstra, H.: Multiple oscillatory modes of the Argentine basin. Part II: the spectral origin of basin modes. J. Phys. Oceanogr. 37, 2869–2881 (2007)ADSCrossRefGoogle Scholar
  37. 37.
    Williamson, D.: A standard test for numerical approximation to the shallow water equations in spherical geometry. J. Comput. Physics. 102, 211–224 (1992)ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Ranis N. Ibragimov
    • 1
  • Michael Dameron
    • 1
  • Chamath Dannangoda
    • 2
  1. 1.Department of MathematicsUniversity of Texas at BrownsvilleBrownsvilleUSA
  2. 2.Department of Physics and AstronomyUniversity of Texas at BrownsvilleBrownsvilleUSA

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