Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model

The theory of chaotic dynamical systems gives many tools that can be used in climate studies. The widely used ones are the Lyapunov exponents, the Kolmogorov entropy and the attractor dimension characterizing global quantities of a system. Another potentially useful tool from dynamical system theory arises from the fact that the local analysis of a system probability distribution function (PDF) can be accomplished by using a procedure that involves an expansion in terms of unstable periodic orbits (UPOs). The system measure (or its statistical characteristics) is approximated as a weighted sum over the orbits. The weights are inversely proportional to the orbit instability characteristics so that the least unstable orbits make larger contributions to the PDF. Consequently, one can expect some relationship between the least unstable orbits and the local maxima of the system PDF. As a result, the most probable system trajectories (or 'circulation regimes' in some sense) may be explained in terms of orbits. For the special classes of chaotic dynamical systems, there is a strict theory guaranteeing the accuracy of this approach. However, a typical atmospheric model may not qualify for these theorems. In our study, we will try to apply the idea of UPO expansion to the simple atmospheric system based on the barotropic vorticity equation of the sphere. We will check how well orbits approximate the system attractor, its statistical characteristics and PDF. The connection of the most probable states of the system with the least unstable periodic orbits will also be analysed.