Stochastic climate dynamics: Random attractors and time-dependent invariant measures

This article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s: the nonlinear deterministic and the linear stochastic one. This unification, via the theory of random dynamical systems (RDS), allows one to consider the detailed geometric structure of the random attractors associated with nonlinear, stochastically perturbed systems. We report on high-resolution numerical studies of two idealized models of fundamental interest for climate dynamics. The first of the two is a stochastically forced version of the classical Lorenz model. The second one is a low-dimensional, nonlinear stochastic model of the El Niño–Southern Oscillation (ENSO). These studies provide a good approximation of the two models’ global random attractors, as well as of the time-dependent invariant measures supported by these attractors; the latter are shown to have an intuitive physical interpretation as random versions of Sinaï–Ruelle–Bowen (SRB) measures. ► Measure-theoretic concepts of random dynamical systems theory are introduced. ► Random SRB measures of nonlinear stochastic systems are then considered. ► Two idealized models of fundamental interest for climate dynamics serve as examples. ► The geometry of the corresponding random SRB measures is numerically explored. ► The physical insights provided by these mathematical objects are enlightened.