Connection of unstable periodic orbits with statistical characteristics and predictability of barotropic atmospheric model

Unstable periodic orbits (UPOs) are an important feature of chaotic dissipative systems(i.e. systems having positive Lyapunov exponents and contracting the phase space). For some chaotic systems like Anosov or Axiom A systems UPOs are dense on the system attractor so that any trajectory of the system can be approximated by some orbit with any given accuracy. As a result, predictability of trajectories for such a system can be quantified in terms of stability characteristics of nearby UPOs. In addition, all system statistical characteristics can be calculated by set of UPOs.

Atmospheric systems are dissipative and chaotic but likely do not have Axiom A property. In this study we try to understand to what extent UPOs of simple atmospheric system approximate its statistical and predictability characteristics. The system under consideration is a Galerkin approximation for barotropic vorticity equation on a rotating sphere with T12 truncation.

With the help of damped Newton and Gauss-Newton methods we were able to find more than 500 of the system UPOs. It is shown that average state of the system as well as its second order statistical characteristics (variance and leading EOFs) can be calculated by UPOs with very high accuracy. Other system properties like Kaplan-Yorke attractor dimension and number of positive Lyapunov exponents can also be reconstructed using UPOs. These results suggest that UPOs form a skeleton of the system attractor and may be important in understanding atmospheric dynamics. The predictability of the system trajectory is also controlled by the behavior of nearby orbits. Namely, the 3-5 member ensemble of adjacent orbits is shadowing the target trajectory for up to 3 weeks and the divergence of the ensemble reflects the loss of predictability.