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

Turbulence closure with small, local neural networks: Forced two-dimensional and β\beta-plane flows

We parameterize sub-grid scale (SGS) fluxes in sinusoidally forced two-dimensional turbulence on the $\beta$-plane at high Reynolds numbers (Re$\sim$25000) using simple 2-layer Convolutional Neural Networks (CNN) having only O(1000)parameters, two orders of magnitude smaller than recent studies employing deeper CNNs with 8-10 layers; we obtain stable, accurate, and long-term online or a posteriori solutions at 16X downscaling factors. Our methodology significantly improves training efficiency and speed of online Large Eddy Simulations (LES) runs, while offering insights into the physics of closure in such turbulent flows. Our approach benefits from extensive hyperparameter searching in learning rate and weight decay coefficient space, as well as the use of cyclical learning rate annealing, which leads to more robust and accurate online solutions compared to fixed learning rates. Our CNNs use either the coarse velocity or the vorticity and strain fields as inputs, and output the two components of the deviatoric stress tensor. We minimize a loss between the SGS vorticity flux divergence (computed from the high-resolution solver) and that obtained from the CNN-modeled deviatoric stress tensor, without requiring energy or enstrophy preserving constraints. The success of shallow CNNs in accurately parameterizing this class of turbulent flows implies that the SGS stresses have a weak non-local dependence on coarse fields; it also aligns with our physical conception that small-scales are locally controlled by larger scales such as vortices and their strained filaments. Furthermore, 2-layer CNN-parameterizations are more likely to be interpretable and generalizable because of their intrinsic low dimensionality.Comment: 27 pages, 13 figure

Equations discovery of organized cloud fields: Stochastic generator and dynamical insights

The emergence of organized multiscale patterns resulting from convection is ubiquitous, observed throughout different cloud types. The reproduction of such patterns by general circulation models remains a challenge due to the complex nature of clouds, characterized by processes interacting over a wide range of spatio-temporal scales. The new advances in data-driven modeling techniques have raised a lot of promises to discover dynamical equations from partial observations of complex systems. This study presents such a discovery from high-resolution satellite datasets of continental cloud fields. The model is made of stochastic differential equations able to simulate with high fidelity the spatio-temporal coherence and variability of the cloud patterns such as the characteristic lifetime of individual clouds or global organizational features governed by convective inertia gravity waves. This feat is achieved through the model's lagged effects associated with convection recirculation times, and hidden variables parameterizing the unobserved processes and variables.Comment: 11 pages, 9 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

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

Exploring the pullback attractors of a low-order quasigeostrophic ocean model: The deterministic case

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