ON THE LONG-TIME BEHAVIOR OF SOLUTIONS TO THE BAROTROPIC ATMOSPHERE MODEL

Long-term large scale behavior and location of the attractors of the b arotropic atmosphere model described by the dissipative and forced vor ticity equation (VE) on a rotating sphere are studied analytically. Si ze of a bounded invariant set B that eventually attracts the trajector ies 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 H-n of the homogeneous spherical po lynomials of degree n, the solutions show quite different behavior for ideal fluid (a); nonturbulent fluid with linear drag (b), and turbule nt fluid (c). For n greater than or equal to 1, the whole space of the VE solutions is divided into sets M(+)(n) and M(-)(n) of the small an d large scale fields defined by chi > n(n + 1) and chi < n(n + 1) resp ectively (chi is the Fjorthoft average spectral number of field on a s phere), and the interface M(n0):chi = n(n + 1) that includes H-n. In c ases (a) and (b), M(+)(n), M(0)(n), M(-)(n) and H-n are invariant sets of the VE solutions, also, in case (b), any solution of M(+)(n) or M( -)(n) tends to the intersection of M(0)(n) and B. In the case (c), the sets M(-)(n), M(+)(n) and M(0)(n) are no longer invariant, however, H -n, and the union of M(+)(n) and M(0)(n) remain invariant, and all sta tionary attractors are in the intersection of B and the set (49). In t he cases (b) and (c), H-n is the basin of attraction of a steady state , limit cycle or quasi-periodic VE solution according to whether the f orcing is stationary, periodic or quasi-periodic. Conditions providing for such an attractor to be global are also given. As examples, it is shown that (1) the global attractor in H-n can cyclically change its structure from a zonal to a blocking-like flow; (2) bifurcation of the global attractor from one set H-n to another can take place.