Most probable histories for nonlinear dynamics: Tracking climate transitions

The effective action provides an appropriate cost function to determine most probable (or optimal) histories for nonlinear dynamics with strong noise. In such strong-coupling problems, a nonperturbative technique is required to calculate the effective action. We have proposed a Rayleigh Ritz variational approximation, which employs simple moment-closures or intuitive guesses of the statistics to calculate the effective action. We consider here an application to climate dynamics, within a simple "bimodal" Langevin model similar to that proposed by C. Nicolis and G. Nicolis [Tellus 33:225 (1981)]. Capturing climate state transitions even in this simple model is known to present a serious problem for standard methods of data assimilation. In contrast, it is shown that the effective action for the climate history is already well-approximated by a one-moment closure and that the optimal, minimizing history robustly tracks climate change, even with large observation errors. Furthermore, the Hessian of the effective action provides the ensemble variance as a realistic measure of confidence level in the predicted optimal history.