Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T19:19:46.877Z Has data issue: false hasContentIssue false
  • English
  • Français

Conserved Vectors for a Model of Nonlinear Atmospheric FlowsAround The Rotating Spherical Surface

Published online by Cambridge University Press:  28 January 2013

A.M. Araslanov ,
L.R. Galiakberova ,
N.H. Ibragimov  and
R. N. Ibragimov
A.M. Araslanov
Affiliation:
Laboratory “Group analysis of mathematical models in natural and engineering sciences” Ufa State Aviation Technical University 12, K. Marx, Str., 450000 Ufa, Russia
L.R. Galiakberova
Affiliation:
Laboratory “Group analysis of mathematical models in natural and engineering sciences” Ufa State Aviation Technical University 12, K. Marx, Str., 450000 Ufa, Russia
N.H. Ibragimov
Affiliation:
Laboratory “Group analysis of mathematical models in natural and engineering sciences” Ufa State Aviation Technical University 12, K. Marx, Str., 450000 Ufa, Russia
R. N. Ibragimov*
Affiliation:
Department of Mathematics University of Texas at Brownsville Brownsville, TX 78520, USA
*
∗ Corresponding author. E-mail: [email protected]
Get access

Abstract

We derive the conserved vectors for the nonlinear two-dimensional Euler equationsdescribing nonviscous incompressible fluid flows on a three-dimensional rotating sphericalsurface superimposed by a particular stationary latitude dependent flow. Under theassumption 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 upperatmosphere between the Ferrel and Polar cells. As a particualr example, the conserveddensities are analyzed by visualizing the exact invariant solutions associated with thegiven model for the particular form of finite disturbances for which the invariantsolutions are also exact solutions of Navier-Stokes equations.

Keywords

nonlinear Navier-Stokes equationsatmospheric wavesexact solutuonsnonlinear conservation laws
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 1 - 17
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

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 CO 2. Science. (2006) 323, 1443-1448. CrossRefGoogle ScholarPubMed
G.K. Bachelor. An Introduction to Fluid Dynamics. Cambridge University Press, (1967) Cambridge.
Balasuriya, S.. Vanishing viscosity in the barotropic β–plane J. Math.Anal. Appl., (1997) 214, 128-150. CrossRefGoogle Scholar
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. Cosmic Research, (2009) 47, (6), 466-479. CrossRefGoogle Scholar
Ben-Yu, G.. Spectral method for vorticity equations on spherical surface. Math. Comput. (1995) 64, 1067-1079. CrossRefGoogle Scholar
Blinova, E.N.. A hydrodynamical theory of pressure and temperature waves and of centres of atmospheric action. C.R. (Doklady) Acad. Sci USSR, (1943) 39, 257-260. Google Scholar
Blinova, E.N.. A method of solution of the nonlinear problem of atmospheric motions on a planetary scale. Dokl. Acad. Nauk USSR, (1956) 110, 975-977. Google Scholar
Cenedese, C., Linden, P.F.. Cyclone and anticyclone formation in a rotating stratified fluid over a sloping bottom. J. Fluid Mech., (1999) 381, 199-223. CrossRefGoogle Scholar
Furnier, A., Bunger, H., Hollerbach, R., Vilotte, I.. Application of the spectral-element method to the axisymetric Navier-Stokes equations. Geophys. J. Int. 156, (2004) 682-700. CrossRefGoogle Scholar
Golovkin, H.. Vanishing viscosity in Cauchy’s problem for hydromechanics equation. Proc. Steklov Inst. Math. (1966) 92, 33-53. Google Scholar
Herrmann, E.. The motions of the atmosphere and especially its waves. Bull. Amer. Math. Soc. 2 (9), 285-296. CrossRef
Hsieh, P.A.. Application of modflow for oil reservoir simulation during the Deepwater Horizon crisis. Ground Water. (2011) 49 (3), 319-323. CrossRefGoogle ScholarPubMed
Ibragimov, R.N.. Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications. Phys. Fluids. (2011) 23, 123102. CrossRefGoogle Scholar
Ibragimov, R.N., Dameron, M.. Spinning phenomena and energetics of spherically pulsating patterns in stratified fluids. Physica Scripta. (2011) 84, 015402. CrossRefGoogle Scholar
Ibragimov, N.H., Ibragimov, R.N.. Intergarion by quadratures of the nonlinear Euler equations modeling atmoaspheric flows in a thin rotating spherical shell. Phys. Letters A. (2011) 375, 3858. CrossRefGoogle Scholar
N.H. Ibragimov, R.N. Ibragimov. Applications of Lie Group Analysis in Geophysical Fluid Dynamics. (2011) Series on Complexity, Nonlinearity and Chaos, Vol 2, World Scientific Publishers, ISBN : 978-981-4340-46-5.
N.H. Ibragimov, R.N. Ibragimov. Conservation laws and invariant solutions for a model of nonlinear atmospheric zonal flows in a thin rotatimng spherical shell. (2012) Archives of ALGA, vol. 9, pp.27-38.
Ibragimov, R.N., Pelinovsky, D.E.. Effects of rotation on stability of viscous stationary flows on a spherical surface. Phys. Fluids. (2010) 22, 126602. CrossRefGoogle Scholar
Ibragimov, R.N.. Mechanism of energy transfers to smaller scales within the rotational internal wave field . Springer. Mathematical Physics, Analysis and Geometry, (2010) 13 (4), 331-355. CrossRefGoogle Scholar
Ibragimov, R.N., Pelinovsky, D.E.. Incompressible viscous fluid flows in a thin spherical shell. J. Math. Fluid. Mech. (2009) 11, 60-90. CrossRefGoogle Scholar
Ibragimov, R.N.. Shallow water theory and solutions of the free boundary problem on the atmospheric motion around the Earth. Physica Scripta. (2000) 61, 391-395. CrossRefGoogle Scholar
Ibragimov, N.H.. A new conservation theorem. Journal of Mathematical Analysis and Applications, (2007) 333 (1), 311–328. CrossRefGoogle Scholar
Iftimie, D., Raugel, G.. Some results on the Navier-Stokes equations in thin 3D domains. J. Diff. Eqs. (2001) 169, 281-331. CrossRefGoogle Scholar
H. Lamb. Hydrodynamics. Cambridge University Press, 5th edition (1924) .
Lions, J.L., Teman, R., Wang, S.. On the equations of the large-scale ocean. Nonlinearity. (1992) 5, 1007-1053. CrossRefGoogle Scholar
Lions, J.L., Teman, R., Wang, S.. New formulations of the primitive equations of atmosphere and applications. Nonlinearity, (1992) 5, 237-288. CrossRefGoogle Scholar
E. Noether. Invariante Variationsprobleme. Konigliche Gessellschaft der Wissenschaften, Gottingen Math. Phys. K1., (1918) English transl. : Transport Theory and Statistical Physics 1(3) (1971) 186-207.
Shindell, D.T., Schmidt, G.A.. Southern Hemisphere climate response to ozone changes and greenhouse gas increases. Res. Lett., (2004) 31, L18209. CrossRefGoogle Scholar
J. Shen. On pressure stabilization method and projection method for unsteady Navier-Stokes equations, in : Advances in Computer Methods for Partial Differential Equations, (1992) 658-662, IMACS, New Brunswick, NJ.
C.P. Summerhayes, S.A. Thorpe. Oceanography, An Illustrative Guide. (1996) New York : John Willey & Sons.
Swarztrauber, P.N.. Shallow water flow on the sphere. Mon. Wea. Rev. (2004) 132, 3010-3018. CrossRefGoogle Scholar
Swarztrauber, P.N.. The approximation of vector functions and their derivatives on the sphere. SIAM J. Numer. Anal. (1981) 18, 181-210. CrossRefGoogle Scholar
Temam, R., Ziane, M.. Navier-Stokes equations in thin spherical domains. Contemp. Math. (1997) 209, 281-314. CrossRefGoogle Scholar
Toggweiler, J.R., Russel, J.L.. Ocean circulation on a warming climate. Nature. (2008) 451, 286-288. CrossRefGoogle ScholarPubMed
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., (2007) 37, 2869-2881. CrossRefGoogle Scholar
Williamson, D.. A standard test for numerical approximation to the shallow water equations in spherical geometry. J. Comput. Physics., (1992) 102, 211-224.CrossRefGoogle Scholar