PHYSICAL INTERPRETATION OF THE ATTRACTOR DIMENSION FOR THE PRIMITIVE EQUATIONS OF ATMOSPHERIC CIRCULATION

In a series of recent papers, some of the authors have addressed with mathematical rigor Some aspects of the primitive equations governing t he large-scale atmospheric motion. Among other results, they derived w ithout evaluating it an expression for the dimension of the attractor for those equations. It is known that the long-term behavior of the mo tion and states of the atmosphere can be described by the global attra ctor. Namely, starting with a given initial value, the solution will t end to the attractor as t goes to infinity. The dimension estimate of the global attractor is evaluated in this article, showing that this g lobal attractor possesses a finite but large number of degrees of free dom. Using some arguments based on the known physical dissipation mech anisms, the bound on the dimension of the attractor in terms of the ob servable quantities governing the heating and energy dissipation accom panying the motion of the atmosphere is made immediately transparent. Consequently, to the extent that the resolution needed in numerical si mulations of the long-term. atmospheric motion is related to the dimen sion of the attractor, the result in this article suggests that the re quired resolution is guile sensitive to the magnitude of the effective (or eddy) viscosity, while it appears to be less sensitive to the det ails of the way that the atmosphere is heated.