Simple attractive sets of a viscous incompressible fluid on a sphere under quasi-periodic polynomial forcing are considered. Each set is the vorticity equation (VE) quasi-periodic solution of the complex (2n+ 1)-dimensional subspaceHnof homogeneous spherical polynomials of degreen. The Hausdorff dimension of its path being an open spiral densely wound around a2n-dimensional torus inHn, equals to 2n. As the generalized Grashof numbGbecomes small enough then the basin of attraction of such spiral solution is expanded fromHnto the entire VE phase space. It is shown that for givenG, there exists an integernGsuch that each spiral solution generated by a forcing ofHnwithn#x2265;nGis globally asymptotically stable. Thus, whereas the dimension of the fluid attractor under a stationary forcing is limited above byG, the dimension of the spiral attractive solution (equal to 2n) may, for a fixedG, become arbitrarily large as the degreenof the quasi-periodic polynomial forcing grows. Since the small scale quasi-periodic functions, unlike the stationary ones, more adequately depict the barotropic atmosphere forcing, this result is of meteorological interest and shows that the dimension of attractive sets depends not only on the forcing amplitude, but also on its spatial and temporal structure.
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