On the long-time behavior of solutions to the barotropic atmosphere model

Abstract

Abstract Long-term large scale behavior and location of the attractors of the barotropic atmosphere model described by the dissipative and forced vorticity equation (VE) on a rotating sphere are studied analytically. Size of a bounded invariant set B that eventually attracts the trajectories of all the VE solutions is estimated depending on the linear drag, turbulence and spectral composition and smoothness of the forcing. If the VE forcing belongs to the set Hn of the homogeneous spherical polynomials of degree n, the solutions show quite different behavior for ideal fluid (a); nonturbulent fluid with linear drag (b), and turbulent fluid (c). For n ≫ 1, the whole space of the VE solutions is divided into sets M n + and M n − of the small and large scale fields defined by χ> n(n + 1) and χ n(n + 1) respectively (χ is the Fjortoft average spectral number of field on a sphere), and the interface M n 0:χ = n(n + 1) that includes Hn . In cases (a) and (b), M n +, M n 0, M n − and H n are invariant sets of the...