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.

This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori-Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of MSMs can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the MZ formalism. A simple correlation-based stopping criterion for an EMR-MSM model is derived to assess how well it approximates the GLE solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given MSM to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a very broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The approach is applied to conceptual nonlinear models borrowed from climate dynamics and population dynamics. In both cases, it is shown that the resulting closure models are able to capture the main statistical features of the dynamics, even in presence of weak time-scale separation.Comment: 47 pages, 7 figure

The purpose of this review-and-research paper is twofold: (i) to review the role played in climate dynamics by fluid-dynamical models; and (ii) to contribute to the understanding and reduction of the uncertainties in future climate-change projections. To illustrate the first point, we review recent theoretical advances in studying the winddriven circulation of the oceans. In doing so, we concentrate on the large-scale, winddriven flow of the mid-latitude oceans, which is dominated by the presence of a larger, anticyclonic and a smaller, cyclonic gyre. The two gyres share the eastward extension of western boundary currents, such as the Gulf Stream or Kuroshio, and are induced by the shear in the winds that cross the respective ocean basins. The boundary currents and eastward jets carry substantial amounts of heat and momentum, and thus contribute in a crucial way to Earth's climate, and to changes therein.Changes in this double-gyre circulation occur from year to year and decade to decade. We study this low-frequency variability of the wind-driven, double-gyre circulation in mid-latitude ocean basins, via the bifurcation sequence that leads from steady states through periodic solutions and on to the chaotic, irregular flows documented in the observations. This sequence involves local, pitchfork and Hopf bifurcations, as well as global, homoclinic ones.The natural climate variability induced by the low-frequency variability of the ocean circulation is but one of the causes of uncertainties in climate projections. The range of these uncertainties has barely decreased, or even increased, over the last three decades. Another major cause of such uncertainties could reside in the structural instability-in the classical, topological sense-of the equations governing climate dynamics, including but not restricted to those of atmospheric and ocean dynamics.We propose a novel approach to understand, and possibly reduce, these uncertainties, based on the concepts and methods of random dynamical systems theory. The idea is to compare the climate simulations of distinct general circulation models (GCMs) used in climate projections, by applying stochastic-conjugacy methods and thus perform a stochastic classification of GCM families. This approach is particularly appropriate given recent interest in stochastic parametrization of subgrid-scale processes in GCMs.As a very first step in this direction, we study the behavior of the Arnol'd family of circle maps in the presence of noise. The maps' fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification.1