Reduced diffusive limit equations for chaotic deterministic systems and their use in data assimilation

Abstract:

Whereas diffusion limits of stochastic multi-scale systems have a long and successful history, the case of constructing stochastic parametrizations of chaotic deterministic systems has been much less studied. We present rigorous results of convergence of a chaotic slow-fast system to a stochastic differential equation with multiplicative noise. Furthermore we present rigorous results for chaotic slow-fast maps, occurring as numerical discretizations of continuous time systems.

This raises the issue of how to interpret certain stochastic integrals; surprisingly the resulting integrals of the stochastic limit system are generically neither of Stratonovich nor of Ito type in the case of maps. It is shown that the limit system of a numerical discretisation is different to the associated continuous time system. This has important consequences when interpreting the statistics of long time simulations of multi-scale systems -- they may be very different to the one of the original continuous time system which we set out to study.

We then  consider a deterministic multi-scale toy model in which a chaotic fast subsystem triggers rare transitions between slow metastable regimes, akin to weather or climate regimes in the context of climate dynamics. Using homogenization techniques we derive a reduced stochastic model as a stochastic parametrization model for the slow dynamics only. We show that the stochastic reduced model can outperform the full deterministic model as forecast model in an ensemble data assimilation procedure, in particular in the realistic setting when observations are only available for the slow variables. We relate the observation intervals for which skill improvement can be obtained to the time scales of the system. We then set out to explain why stochastic climate models produce superior skill in an ensemble setting. The improvement in skill is due to the finite size of the ensemble, and we show that there is no skill improvement in very large ensembles or when the forecast variance is artificially and unreasonable inflated. We corroborate this with numerical simulations.