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Explicit invariant solutions associated with nonlinear atmospheric flows in a thin rotating spherical shell with and without west-to-east jets perturbations

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Abstract

A class of non-stationary exact solutions of two-dimensional nonlinear Navier–Stokes (NS) equations within a thin rotating spherical shell were found as invariant and approximately invariant solutions. The model is used to describe a simple zonally averaged atmospheric circulation caused by the difference in temperature between the equator and the poles. Coriolis effects are generated by pseudoforces, which support the stable west-to-east flows providing the achievable meteorological flows. The model is superimposed by a stationary latitude dependent flow. Under the assumption of no friction, the perturbed model describes zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. In terms of nonlinear modeling for the NS equations, two small parameters are chosen for the viscosity and the rate of the earth’s rotation and exact solutions in terms of elementary functions are found using approximate symmetry analysis. It is shown that approximately invariant solutions are also valid in the absence of the flow perturbation to a zonally averaged mean flow.

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Notes

  1. We use here the term “exact” when referring to the classical Lie point symmetries of a given system. This is in order to make the distinction clear between such symmetries and their approximate symmetry counterparts.

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Acknowledgments

This research was supported in part by an appointment to the U.S. Department of Energy’s Visiting Faculty Program.

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Authors and Affiliations

  1. Pacific Northwest National Laboratory, Richland, WA , 99352, USA

    Ranis Ibragimov

  2. School of Information Technology, Deakin University, Waurn Ponds, Australia

    Grace Jefferson & John Carminati

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  1. Ranis Ibragimov
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Correspondence to Ranis Ibragimov.

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Ibragimov, R., 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. Anal.Math.Phys. 3, 375–391 (2013). https://doi.org/10.1007/s13324-013-0062-9

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