Climate dynamics and fluid mechanics: Natural variability and related uncertainties

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 wind-driven circulation of the oceans. In doing so, we concentrate on the large-scale, wind-driven 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

Noise-driven Topological Changes in Chaotic Dynamics

Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be "strange" but it is frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use Branched Manifold Analysis through Homologies (BraMAH) -- a technique originally introduced to characterize the topological structure of deterministically chaotic flows -- which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA's evolution includes sharp transitions that appear as topological tipping points.Comment: 12 pages and 4 figure

Collaborative Research: Robust Climate Projections and Stochastic Stability of Dynamical Systems

The project was completed along the lines of the original proposal, with additional elements arising as new results were obtained. The originally proposed three thrusts were expanded to include an additional, fourth one. (i) The e#11;ffects of stochastic perturbations on climate models have been examined at the fundamental level by using the theory of deterministic and random dynamical systems, in both #12;nite and in#12;nite dimensions. (ii) The theoretical results have been implemented #12;first on a delay-diff#11;erential equation (DDE) model of the El-Nino/Southern-Oscillation (ENSO) phenomenon. (iii) More detailed, physical aspects of model robustness have been considered, as proposed, within the stripped-down ICTP-AGCM (formerly SPEEDY) climate model. This aspect of the research has been complemented by both observational and intermediate-model aspects of mid-latitude and tropical climate. (iv) An additional thrust of the research relied on new and unexpected results of (i) and involved reduced-modeling strategies and associated prediction aspects have been tested within the team's empirical model reduction (EMR) framework. Finally, more detailed, physical aspects have been considered within the stripped-down SPEEDY climate model. The results of each of these four complementary e#11;fforts are presented in the next four sections, organized by topic and by the team members concentrating on the topic under discussion

Reduced-order models for coupled dynamical systems: data-driven methods and the Koopman operator

Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations. Parameterizations aim to reduce the complexity of high-dimensional dynamical systems. Here, a theory-based and a data-driven approach for the parameterization of coupled systems are compared, showing that both yield the same stochastic multilevel structure. The results provide very strong support to the use of empirical methods in model reduction and clarify the practical relevance of the proposed theoretical framework

Inverse stochastic-dynamic models for high-resolution Greenland ice core records

Proxy records from Greenland ice cores have been studied for several decades, yet many open questions remain regarding the climate variability encoded therein. Here, we use a Bayesian framework for inferring inverse, stochastic–dynamic models from δ¹⁸O and dust records of unprecedented, subdecadal temporal resolution. The records stem from the North Greenland Ice Core Project (NGRIP), and we focus on the time interval 59–22 ka b2k. Our model reproduces the dynamical characteristics of both the δ¹⁸O and dust proxy records, including the millennial-scale Dansgaard–Oeschger variability, as well as statistical properties such as probability density functions, waiting times and power spectra, with no need for any external forcing. The crucial ingredients for capturing these properties are (i) high-resolution training data, (ii) cubic drift terms, (iii) nonlinear coupling terms between the δ¹⁸O and dust time series, and (iv) non-Markovian contributions that represent short-term memory effects

Comment on “Nonparametric forecasting of low-dimensional dynamical systems ”

The comparison performed in Berry et al. [Phys. Rev. E 91, 032915 (2015)] between the skill in predicting the El Niño-Southern Oscillation climate phenomenon by the prediction method of Berry et al. and the "past-noise" forecasting method of Chekroun et al. [Proc. Natl. Acad. Sci. USA 108, 11766 (2011)] is flawed. Three specific misunderstandings in Berry et al. are pointed out and corrected

Climate dynamics and fluid mechanics: Natural variability and related uncertainties

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 focus on the large-scale, wind-driven flow of the mid-latitude oceans which contribute in a crucial way to Earth's climate, and to changes therein. We study the low-frequency variability (LFV) 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 LFV of the ocean circulation is but one of the causes of uncertainties in climate projections. Another major cause of such uncertainties could reside in the structural instability in the 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. As a very first step, we study the effect of noise on the topological classes of the Arnol'd family of circle maps, a paradigmatic model of frequency locking as occurring in the nonlinear interactions between the El Nino-Southern Oscillations (ENSO) and the seasonal cycle. It is shown that the maps' fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification. This result is consistent with stabilizing effects of stochastic parametrization obtained in modeling of ENSO phenomenon via some general circulation models.Comment: Invited survey paper for Special Issue on The Euler Equations: 250 Years On, in Physica D: Nonlinear phenomen

Global error analysis and inertial manifold reduction

Four types of global error for initial value problems are considered in a common framework. They include classical forward error analysis and shadowing error analysis together with extensions of both to include rescaling of time. To determine the amplificatioh of the local error that bounds the global error we present a linear analysis similar in spirit to condition number estimation for linear systems of equations. We combine these ideas with techniques for dimension reduction of differential equations via a boundary value formulation of numerical inertial manifold reduction. These global error concepts are exercised to illustrate their utility on the Lorenz equations and inertial manifold reductions of the Kuramoto-Sivashinsky equation. (C) 2016 Elsevier B.V. All rights reserved

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

International audienceThis 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 NioSouthern 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 SinaRuelleBowen (SRB) measures. © 2011 Elsevier B.V. All rights reserved