Discontinuity, Nonlinearity, and Complexity
Sources and Sinks of Energy Balance for Nonlinear Atmospheric Motion Perturbed by Westtoeast Winds Progressing on a Surface of a Rotating Spherical Shell
Discontinuity, Nonlinearity, and Complexity 1(1) (2012) 4155  DOI:10.5890/DNC.2012.02.002
Ranis N. Ibragimov$^{1}$; Michael Dameron$^{1}$; Chamath Dannangoda$^{2}$
$^{1}$ Department of Mathematics, University of Texas at Brownsville, Brownsville, TX 78520, USA.
$^{2}$ Department of Physics and Astronomy, University of Texas at Brownsville, Brownsville, TX 78520, USA.
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Abstract
We study the asymptotic behavior of sources and sinks associated with the effects of rotation and nonlinearity of the energy balance of atmospheric motion perturbed by westtoeast winds progressing on the surface of a rotating spherical shell. The model uses nonlinear viscous and nonviscous incompressible fluid flows on a rotating spherical domain with infinitely small thickness. The energy density and associated sources and sinks were determined and visualized by means of elementary functions that provide the exact solutions of the nonviscous baratropic vorticity equation on the rotating sphere. It is shown that there exists a particular form of westto east flows for which the source and sink terms associated with the energy balance vanishes providing thus the energy conservation law. Moreover, this particular form of atmospheric perturbation preserves the exact solutions for the case of viscous flows.
References

[1]  Anderson, R.F., Ali, S., Brandtmiller, L.L., Nielsen, S.H.H, Fleisher, M.Q. (2006), Winddriven upwelling in the Southern Ocean and the deglacial rise in atmosphericCO2, Science , 323, 14431448. 

[2]  Bachelor, G.K. (1967), An Introduction to Fluid Dynamics,Cambridge University Press, Cambridge. 

[3]  Balasuriya, S. (1997), Vanishing viscosity in the barotropic β—plane, J. Math.Anal. Appl., 214, 128150. 

[4]  Belotserkovskii, O.M., Mingalev, I.V., Mingalev O.V.( 2009), Formation of largescale vortices in shear flows of the lower atmosphere of the earth in the region of tropical latitudes, Cosmic Research, 47 (6), 466479. 

[5]  BenYu, G. (1995), Spectral method for vorticity equations on spherical surface, Math. Comput., 64, 10671079. 

[6]  Blinova, E.N. ( 1943), A hydrodynamical theory of pressure and temperature waves and of centres of atmospheric action, C.R. (Doklady) Acad. Sci USSR, 39, 257260. 

[7]  Blinova, E.N. (1956), A method of solution of the nonlinear problem of atmospheric motions on a planetary scale, Dokl. Acad. Nauk USSR, 110, 975977. 

[8]  Callaghan, T.G., Forbes, L.K. (2006), Computing largescale progressive Rossby waves on a sphere, J. Comput. Phys., 217, 845. 

[9]  Cenedese, C., Linden, P.F. (1999), Cyclone and anticyclone formation in a rotating stratified fluid over a sloping bottom, J. Fluid Mech., 381, 199223. 

[10]  Chang, H.H., Zhou, J., Fuentes, M. (2010), Impact of climate change on ambient ozone level and mortality in southeastern United States, Int. J. Environ. Res. Public Health, 7, 28662880. 

[11]  Herrmann, E. (1896), The motions of the atmosphere and especially its waves, Bull. Amer. Math. Soc., 2 (9), 285296. 

[12]  Hsieh, P.A. (2011), Application of modflow for oil reservoir simulation during the Deepwater Horizon crisis, Ground Water, 49 (3), 319323. 

[13]  Ibragimov, R.N. (2000), Shallow water theory and solutions of the free boundary problem on the atmospheric motion around the Earth, Physica Scripta, 61, 391395. 

[14]  Ibragimov, R.N. ( 2011), Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications, Phys. Fluids, 23, 1223102. 

[15]  Ibragimov, R.N., Pelinovsky, D.E. (2009), Incompressible viscous fluid flows in a thin spherical shell, J. Math. Fluid. Mech., 11, 6090. 

[16]  Ibragimov, R.N., Pelinovsky, D.E. (2010), Effects of rotation on stability of viscous stationary flows on a spherical surface, Phys. Fluids, 22, 126602. 

[17]  Ibragimov, N.H., Ibragimov, R.N. (2011), Integration by quadratures of the nonlinear Euler equations modeling atmospheric flows in a thin rotating spherical shell, Ocean Modelling, To appear. 

[18]  Ibragimov, N.H., Ibragimov, R.N. (2011), Applications of Lie Group Analysis in Geophysical Fluid Dynamics, Series of Complexity, Nonlinearity and Chaos, Vol. 2, World Scientific Publishers, Singapore. 

[19]  Iftimie, D., Raugel, G. (2001), Some results on the NavierStokes equations in thin 3D domains, J. Diff. Eqs., 169, 281331. 

[20]  Lamb, H.(1924), Hydrodynamics, Cambridge University Press, 5th edition,Cambridge. 

[21]  Lions, J.L., Teman, R., Wang, S. (1992), On the equations of the largescale ocean, Nonlinearity, 5, 10071053. 

[22]  Lions, J.L., Teman, R., Wang, S. (1992), New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5, 237288. 

[23]  Shindell, D.T., Schmidt, G.A. (2004), Southern Hemisphere climate response to ozone changes and greenhouse gas increases, Res. Lett., 31, L18209. 

[24]  Sleijpen, G.L.G., Van der Vorst, H.A. (1996), A JacobiDavidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17, 410425. 

[25]  Shen, J. (1992), On pressure stabilization method and projection method for unsteady NavierStokes equations, in: Advances in Computer Methods for Partial Differential Equations, 658662, IMACS, New Brunswick, NJ, 1992. 

[26]  Summerhayes, C.P., Thorpe, S.A. (1996), Oceanography, An Illustrative Guide, John Willey & Sons, New York. 

[27]  Swarztrauber, P.N. (2004), Shallow water flow on the sphere, Mon. Weather Rev., 132, 3010. 

[28]  Temam, R., Ziane, M. (1997), NavierStokes equations in thin spherical domains, Contemp. Math., 209, 281314. 

[29]  Toggweiler, J.R, Russel, J.L. (2008), Ocean circulation on a warming climate, Nature, 451, 286288. 

[30]  Vallis, G.K. (2006), Atmospheric anf Ocean Fluid Dynamics, Cambridge University Press, Cambridge. 

[31]  Weijer, W., Vivier, F., Gille, S.T., Dijkstra, H.( 2007), Multiple oscillatory modes of the Argentine Basin. Part II: The spectral origin of basin modes, J. Phys. Oceanogr., 37, 28692881. 

[32]  Williamson, D. (1992), A standard test for numerical approximation to the shallow water equations in spherical geometry, J. Comput. Physics., 102, 211224. 