A Simple Stochastic Parameterization for Reduced Models of Multiscale Dynamics

Multiscale dynamics are frequently present in real-world processes, such as the atmosphere-ocean and climate science. Because of time scale separation between a small set of slowly evolving variables and much larger set of rapidly changing variables, direct numerical simulations of such systems are difficult to carry out due to many dynamical variables and the need for an extremely small time discretization step to resolve fast dynamics. One of the common remedies for that is to approximate a multiscale dynamical systems by a closed approximate model for slow variables alone, which reduces the total effective dimension of the phase space of dynamics, as well as allows for a longer time discretization step. Recently, we developed a new method for constructing a deterministic reduced model of multiscale dynamics where coupling terms were parameterized via the Fluctuation-Dissipation theorem. In this work we further improve this previously developed method for deterministic reduced models of multiscale dynamics by introducing a new method for parameterizing slow-fast interactions through additive stochastic noise in a systematic fashion. For the two-scale Lorenz 96 system with linear coupling, we demonstrate that the new method is able to recover additional features of multiscale dynamics in a stochastically forced reduced model, which the previously developed deterministic method could not reproduce.