Dynamical Systems Seminar

Unstable periodic orbits (UPOs) provide the so-called skeletal dynamics of a sufficiently well-behaved chaotic dynamical system and give a powerful tool for relating the response of the system to its variability. UPOs constitute natural modes of variability of the system and the resonant behavior of the system response to external forcing can be associated to good correspondence between the geometry of some UPOs and of the forcing term.

We have here analyzed a simple barotropic model of the atmosphere and constructed a large number of UPOs approximating the system attractor. We have then studied the change in the model climate resulting from changes in the forcing, in the orography, and in the Eckman friction. The most interesting result is the presence of a strong resonance in the orographic response on time scales of the order of about 3 days, corresponding to forced waves. Interestingly, this phenomenon is entirely absent from the natural variability of the system and correspond to the excitation of a specific group of UPOs.

This clarifies the fact that, as opposed to the case of quasi-equilibrium systems, it is far from obvious to associate forced and free variability in the spirit of the fluctuation-dissipation theorem (FDT). Reassuringly, using the complementary point of view of covariant Lyapunov vectors, we discover that the forcing projects substantially in the stable direction of the flow, which is exactly the mathematical setting under which the FDT cannot be applied.

References:

1. A.Gritsun,V.Lucarini, Fluctuations, response, and resonances in a simple atmospheric model, Physica D, (2017), 10.1016/j.physd.2017.02.015
2. D. Ruelle, General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium, Phys. Letters A 245, 220-224 (1998)
3. Cvitanovic, P., and B. Eckhardt, Periodic orbit expansions for classical smooth flows, Journal of Physics A 24 L237 (1991)
4. Gritsun, A., Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model. Phil. Trans. R. Phil. Soc. A 371, 20120336-20120336 (2012)