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 the large-scale atmospheric motion. Among other results, they derived without evaluating it an expression for the dimension of the attractor for those equations. It is known that the long-term behavior of the motion and states of the atmosphere can be described by the global attractor. Namely, starting with a given initial value, the solution will tend to the attractor as t goes to infinity. The dimension estimate of the global attractor is evaluated in this article, showing that this global attractor possesses a finite but large number of degrees of freedom. Using some arguments based on the known physical dissipation mechanisms, the bound on the dimension of the attractor in terms of the observable quantities governing the heating and energy dissipation accompanying the motion of the atmosphere is made immediately transparent. Consequently, to the extent that the resolution needed in numerical simulations of the long-term atmospheric motion is related to the dimension of the attractor, the result in this article suggests that the required resolution is quite sensitive to the magnitude of the effective (or eddy) viscosity, while it appears to be less sensitive to the details of the way that the atmosphere is heated.