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

Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them-as filtered through an observable of the system-is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap-defined as the distance between the subdominant RP resonance and the unit circle-plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño-Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.climate dynamics | Markov operators | parametric dependence | sensitivity bounds | uncertainty quantification S ensitive behavior of long-term general circulation model (GCM) statistics is attracting increased attention (1-3), but its origin and fundamental mechanisms remain unclear. These sensitive-behavior issues are of practical, as well as theoretical, importance in climate dynamics and elsewhere (4). For some GCMs, involving millions of variables, circumstances have been found where certain climate observables vary smoothly through a plausible parameter range (5) or where linear response theory applies over some range (6). On the other hand, this may not hold for every observable or parameter, and concerns arise regarding the role of some type of "structural instability" in sensitive parameter dependence (1, 2, 4).The low-order Lorenz (L63) model (7) illustrates some of the relevant issues. Various statistics exhibit linear dependence over a broad range of parameters for which the dynamics is chaotic (e.g., figure 2 of ref. 8). The statistics' linear dependence coexists here with structural instability of this model's global attractor, as small variations in the parameters cause a plethora of topological changes (9). In particular, the unstable periodic orbits that appear or disappear as a parameter chan...

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A low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states whereas the trajectories in the latter converge to small portions of the global PBA. This complex scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.

Interannual and interdecadal prediction are major challenges of climate dynamics. In this article we develop a prediction method for climate processes that exhibit low-frequency variability (LFV). The method constructs a nonlinear stochastic model from past observations and estimates a path of the "weather" noise that drives this model over previous finite-time windows. The method has two steps: (i) select noise samples-or "snippets"-from the past noise, which have forced the system during short-time intervals that resemble the LFV phase just preceding the currently observed state; and (ii) use these snippets to drive the system from the current state into the future. The method is placed in the framework of pathwise linear-response theory and is then applied to an El Niño-Southern Oscillation (ENSO) model derived by the empirical model reduction (EMR) methodology; this nonlinear model has 40 coupled, slow, and fast variables. The domain of validity of this forecasting procedure depends on the nature of the system's pathwise response; it is shown numerically that the ENSO model's response is linear on interannual time scales. As a result, the method's skill at a 6-to 16-month lead is highly competitive when compared with currently used dynamic and statistic prediction methods for the Niño-3 index and the global sea surface temperature field. E NSO forecasting has a decade-long history and relies mainly on two classes of models: dynamical and statistical (1, 2). Still, a further distinction has to be made within the latter class: Some of the statistical models do not make any use of dynamical information, like Lorenz's method of analogues (3) and its followers (4-6), while others do use a dynamical model-previously fitted to the observations from the past-to drive the statistics in the future (2,7,8). Empirical stochastic models belong to this hybrid category, and linear versions of such models have been used in ENSO forecasting for two decades; see ref. 9 for a survey. More recently, Kravtsov et al. (10) have extended this approach to nonlinear models by developing an EMR methodology that can include quadratic nonlinearities as well as state-dependent noise that parameterizes small-scale effects, without assuming a priori scale separation (11).The purpose of this paper is to show that, under suitable circumstances, a better understanding of the role of the fast processes, weather or noise, can help predict the slow ones-namely, the climate. To achieve this purpose, we proceed in two steps: (i) develop a special prediction methodology, called past noise forecasting (PNF), using EMR models; and (ii) provide a theoretical framework for applying the PNF method-or any other forecasting method based on perturbations of the noise-to other empirical stochastic models. Of late, probabilistic forecasts in weather and climate prediction have become fairly widespread: They are grounded in an estimation of the probability density function (PDF: 12-14).We take here a distinct, pathwise approach instead, and will show that th...

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