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

Abstract

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. A high-resolution numerical study of two highly idealized models of fundamental interest for climate dynamics allows one to obtain a good approximation of their global random attractors, as well as of the time-dependent invariant measures supported by these attractors; the latter are shown to be random Sinai-Ruelle-Bowen (SRB) measures. The first of the two models 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). Keywords: Climate Dynamics, Dynamical Systems, El Niño, Random Dynamical Systems, Stochastic ForcingThe geometric [1] and ergodic [2] theory of dynamical systems represents a significant achievement of the last century. In the meantime, the foundations of the stochastic calculus also led to the birth of a rigorous theory of time-dependent random phenomena. Historically, theoretical developments in climate dynamics have been largely motivated by these two complementary approaches, based on the work of E. N. Lorenz [3] and that of K. Hasselmann [4], respectively.It now seems clear that these two approaches complement, rather than exclude each other. Incomplete knowledge of small-, subgrid-scale processes, as well as computational limitations will always require one to account for these processes in a stochastic way. As a result of sensitive dependence on initial data and on parameters, numerical weather forecasts [5] as well as climate projections [6] are both expressed these days in probabilistic terms. In addition to the intrinsic challenge of addressing the nonlinearity along with the stochasticity of climatic processes, it is thus more convenient -and becoming more and more necessaryto rely on a model's (or set of models') probability density function (PDF) rather than on its individual, pointwise simulations or predictions.We show in this paper that finer, highly relevant and still computable statistics exist for stochastic nonlinear systems, which provide meaningful physical information not described by the PDF alone. These statistics are supported by a random attractor that extends the concept of a strange attractor [3,7] and of its invariant measures [2] from deterministic to stochastic dynamics.The attractor of a deterministic dynamical system provides crucial geometric information about its asymptotic regime as t → ∞, while the Sinaï-Ruelle-Bowen (SRB) measure provides, when it exists, the * Corresponding author.

Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

“…1, respectively. The gray dots in both panels arise from 15 000 IDs evenly distributed in G and are associated with the filamentary patterns that characterize chaotic attractors (e.g., Chekroun et al 2011;Drótos et al 2015).…”

“…We have studied this double-gyre model's PBAs (i.e., those timedependent invariant sets that attract all trajectories initialized in the remote past) along with the invariant sample measures that live on these sets. PBAs (Arnold 1998;Ghil et al 2008;Chekroun et al 2011) and their close cousins, snapshot attractors (Romeiras et al 1990;Bódai et al 2011Bódai et al , 2013Bódai and Tél 2012;Drótos et al 2015), have only recently been recognized as the natural tools for investigating basic features of a changing nonequilibrium climate, and the theory of nonautonomous dynamical systems as the proper mathematical framework for such investigations.…”

“…The limit t / 2' is the pullback limit and it provides the natural generalization of the forward asymptotics associated with autonomous systems (Arnold 1998;Rasmussen 2007;Kloeden and Rasmussen 2011;Carvalho et al 2012;Ghil et al 2008;Chekroun et al 2011); see the appendix herein. In practical terms, though, a question arises: How large should jtj be for A(t, t) to be virtually independent of t?…”

“…The concepts and methods of nonautonomous and random dynamical systems, including pullback attraction, were introduced into climate dynamics by Ghil et al (2008) and Chekroun et al (2011), and they were also pursued vigorously by T. Tél and his group (Bódai et al 2011;Bódai and Tél 2012;Bódai et al 2013). Several recent studies have explored the properties of PBAs for low-order climate models using ensemble simulations (Bódai et al 2011(Bódai et al , 2013Bódai and Tél 2012;De Saedeleer et al 2013;Pierini 2014a;Drótos et al 2015).…”

Extremes and Recurrence in Dynamical Systems

“…In [89] attention is focussed on the trajectories of the individual ensemble members driven by the same sequence of random variables T t and constructed according to the time-dependent measure. While mathematically sound, this approach seems a bit unnatural with respect to describing reasonable experimental conditions because we are assuming to be able to reproduce the same sequence of random events for all initial conditions considered.…”