On dimensions of attractive sets of viscous fluids on a sphere under quasi-periodic forcing

Abstract

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 subspace Hn of homogeneous spherical polynomials of degree n. The Hausdorff dimension of its path being an open spiral densely wound around a 2n-dimensional torus in Hn , equals to 2n. As the generalized Grashof numb G becomes small enough then the basin of attraction of such spiral solution is expanded from Hn to the entire VE phase space. It is shown that for given G, there exists an integer nG such that each spiral solution generated by a forcing of Hn with n ≥ nG is globally asymptotically stable. Thus, whereas the dimension of the fluid attractor under a stationary forcing is limited above by G, the dimension of the spiral attractive solution (equal to 2n) may, for a fixed G, become arbitrarily large as the degree n of 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.