Abstract
The nonlinear Euler equations are used to model two-dimensional atmosphere dynamics in a thin rotating spherical shell. The energy balance is deduced on the basis of two classes of functorially independent invariant solutions associated with the model. It it shown that the energy balance is exactly the conservation law for one class of the solutions whereas the second class of invariant solutions provides and asymptotic convergence of the energy balance to the conservation law.
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Aguiar, A.C., Read, P.L., Wordsworth, R.D., Salter, T., Yamazaki, Y.H.: A laboratory model of Saturn’s North Polar Hexagon. Icarus 206(2), 755–763 (2010)
Ali, A., Kalisch, H.: On the formulation of mass, momentum and energy conservation in the KdV equation. Aca Appl. Math. 133, 113–131 (2014)
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 323, 1443–1448 (2006)
Bachelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)
Balasuriya, S.: Vanishing viscosity in the barotropic \( \beta \)-plane. J. Math. Anal. Appl. 214, 128–150 (1997)
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)
Ben-Yu, G.: Spectral method for vorticity equations on spherical surface. Math. Comput. 64, 1067–1079 (1995)
Blinova, E.N.: A hydrodynamical theory of pressure and temperature waves and of centres of atmospheric action. C.R. (Dokl.) Acad. Sci. USSR 39, 257–260 (1943)
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)
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)
Flertcher, L.N., Irwin, P.G., Orton, G.S.: Temperature and composition of Saturn’s polar hot spots and hexagon. Science 319, 79–82 (2008)
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)
Golovkin, H.: Vanishing viscosity in Cauchy’s problem for hydromechanics equation. Proc. Steklov Inst. Math. 92, 33–53 (1966)
Herrmann, E.: The motions of the atmosphere and especially its waves. Bull. Am. Math. Soc. 2(9), 285–296 (1896)
Hsieh, P.A.: Application of modflow for oil reservoir simulation during the Deepwater Horizon crisis. Ground Water 49(3), 319–323 (2011)
Ibragimov, R.N.: Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications. Phys. Fluids 23, 123102 (2011)
Ibragimov, R.N., Dameron, M.: Spinning phenomena and energetics of spherically pulsating patterns in stratified fluids. Phys. Scr. 84, 015402 (2011)
Ibragimov, N.H., Ibragimov, R.N.: Intergarion by quadratures of the nonlinear Euler equations modeling atmospheric flows in a thin rotating spherical shell. Phys. Lett. A 375, 3858 (2011)
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 (2011) (ISBN: 978-981-4340-46-5)
Ibragimov, R.N., Pelinovsky, D.E.: Effects of rotation on stability of viscous stationary flows on a spherical surface. Phys. Fluids 22, 126602 (2010)
Ibragimov, R.N., Jefferson, G., Carminati, J.: Invariant and approximately invariant solutions associated with nonlinear zonally averaged atmospheric motion in a thin rotating atmospheric shell. Anal. Math. Phys. 3, 375–391 (2013)
Ibragimov, R.N., Pelinovsky, D.E.: Incompressible viscous fluid flows in a thin spherical shell. J. Math. Fluid Mech. 11, 60–90 (2009)
Ibragimov, R.N.: Shallow water theory and solutions of the free boundary problem on the atmospheric motion around the Earth. Phys. Scr. 61, 391–395 (2000)
Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333(1), 311–328 (2007)
Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Nauka, Moscow (1983) (English. transl, Reidel, Dordrecht (1985))
Iftimie, D., Raugel, G.: Some results on the Navier–Stokes equations in thin 3D domains. J. Differ. Equ. 169, 281–331 (2001)
Karczewska, A., Rozmej, P., Infeld, E.: Energy invariant for shallow-water waves and the Korteweg–de Vries equation: doubts about the invariance of energy. Phys. Rev. E 92, 053202 (2015)
Lamb, H.: Hydrodynamics, 5th edn. Cambridge University Press, Cambridge (1924)
Lions, J.L., Teman, R., Wang, S.: On the equations of the large-scale ocean. Nonlinearity 5, 1007–1053 (1992)
Lions, J.L., Teman, R., Wang, S.: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5, 237–288 (1992)
Noether, E.: Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 1918, 235–257 (1918)
Ovsyannikov, L.V.: Group Analysis of Differential Equations. Nauka, Moscow (1978) (English transl., ed. W.F. Ames, Academic Press, New York (1982))
Shindell, D.T., Schmidt, G.A.: Southern Hemisphere climate response to ozone changes and greenhouse gas increases. Res. Lett. 31, L18209 (2004)
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)
Summerhayes, C.P., Thorpe, S.A.: Oceanography. An Illustrative Guide. Willey, New York (1996)
Swarztrauber, P.N.: Shallow water flow on the sphere. Mon. Weather Rev. 132, 3010–3018 (2004)
Swarztrauber, P.N.: The approximation of vector functions and their derivatives on the sphere. SIAM J. Numer. Anal. 18, 181–210 (1981)
Temam, R., Ziane, M.: Navier–Stokes equations in thin spherical domains. Contemp. Math. 209, 281–314 (1997)
Toggweiler, J.R., Russel, J.L.: Ocean circulation on a warming climate. Nature 451, 286–288 (2008)
Vassada, A.R., Horst, S.M., Kennedy, M.R., Ingersoll, A.P.: Cassini imaging of Saturn: Southern hemisphere winds and vorticities. J. Geophys. Res. 111, 5004–5017 (2006)
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)
Williamson, D.: A standard test for numerical approximation to the shallow water equations in spherical geometry. J. Comput. Phys. 102, 211–224 (1992)
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Ibragimov, R.N. Effects of zonal flows on correlation between energy balance and energy conservation associated with nonlinear nonviscous atmospheric dynamics in a thin rotating spherical shell . Anal.Math.Phys. 8, 11–24 (2018). https://doi.org/10.1007/s13324-016-0158-0
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DOI: https://doi.org/10.1007/s13324-016-0158-0