Qualitative properties of solutions to vorticity equation for a viscous incompressible fluid on a rotating sphere

A nonlinear vorticity equation describing the behavior of a viscous incompressible fluid on a rotating sphere is considered. The viscosity term is modeled by a real degree of the Laplace operator. The smoothness of external forcing is established that guarantee the existence of a limited attractive set in the space of solutions. Theorems on the existence and uniqueness of non-stationary and stationary weak solutions are given. Sufficient conditions for the global asymptotic stability of solutions are obtained. An example is constructed that shows that, in contrast to the stationary forcing, the Hausdorff dimension of the global attractor generated by a quasiperiodic (in time) and polynomial (in space) forcing can be arbitrarily large, even if the generalized Grashof number is limited.