ON DIMENSIONS OF ATTRACTIVE SETS OF VISCOUS FLUIDS ON A SPHERE UNDER QUASI-PERIODIC FORCING

Simple attractive sets of a viscous incompressible fluid on a sphere u nder quasi-periodic polynomial forcing are considered. Each set is the vorticity equation (VE) quasi-periodic solution of the complex (2n 1)-dimensional subspace H-n of homogeneous spherical polynomials of de gree n. The Hausdorff dimension of its path being an open spiral dense ly wound around a 2n-dimensional torus in H-n, equals to 2n. As the ge neralized Grashof numb G becomes small enough then the basin of attrac tion of such spiral solution is expanded from H-n to the entire VE pha se space. It is shown that for given G, there exists an integer n(G) s uch that each spiral solution generated by a forcing of H-n with n gre ater than or equal to n(G) is globally asymptotically stable. Thus, wh ereas 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 deg ree n of the quasi-periodic polynomial forcing grows. Since the small scale quasi-periodic functions, unlike the stationary ones, more adequ ately depict the barotropic atmosphere forcing, this result is of mete orological interest and shows that the dimension of attractive sets de pends not only on the forcing amplitude, but also on its spatial and t emporal structure.