Abstract—The asymptotic behavior of solutions to vorticity equation that describes the dynamics of viscous nondivergent forced fluid on a rotating sphere is studied as t→∞. Special forms of forcing are given that guarantee the existence of a bounded set that eventually attracts all solutions. The asymptotic behavior of the BVE solutions depends on both the geometric structure and smoothness of forcing. Sufficient conditions for the global asymptotic stability of vorticity equation solutions are also obtained. It is well known that Hausdorff dimension of global attractor of vorticity equation is limited from above by the generalized Grashof number. An example given in this work shows that for a fixed bounded Grashof number, the Hausdorff dimension of spiral global attractor under a quasi-periodic polynomial forcing can become arbitrarily large. Since the small scale quasi-periodic forcing more adequately depicts the forcing in the barotropic vorticity equation, this result is of meteorological interest showing that the dimension of attractive sets depends not only on the forcing amplitude, but also on its spatial and temporal structure. It also shows that the search of finite-dimensional global attractor in the barotropic atmosphere is not well justified.
Index Terms—Viscous and forced nondivergent fluid, asymptotic behavior, global stability, attractor dimension.
Yuri N. Skiba is with the Centre for Atmospheric Sciences, Universidad Nacional Autónoma de México (e-mail: skiba@ unam.mx).
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Cite:Yuri N. Skiba, "Large-Time Dynamics of Incompressible Fluid on a Sphere," International Journal of Modeling and Optimization vol. 3, no. 4, pp. 344-348, 2013.
