Publications

2016
Ghil, Michael. 2016. “A Mathematical Theory of Climate Sensitivity: A Tale of Deterministic & Stochastic Dynamical Systems.” 11th AIMS Conf. on Dynamical Systems, Differential Equations & Applications, Honoring Peter Lax’s 90th Birthday, Orlando, FL, July 2016. Abstract

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Sella, Lisa, Gianna Vivaldo, Andreas Groth, and Michael Ghil. 2016. “Economic Cycles and Their Synchronization: A Comparison of Cyclic Modes in Three European Countries.” Journal of Business Cycle Research 12 (1). Springer: 25-48. Publisher's Version Abstract

The present work applies singular spectrum analysis (SSA) to the study of macroeconomic fluctuations in three European countries: Italy, The Netherlands, and the United Kingdom. This advanced spectral method provides valuable spatial and frequency information for multivariate data sets and goes far beyond the classical forms of time domain analysis. In particular, SSA enables us to identify dominant cycles that characterize the deterministic behavior of each time series separately, as well as their shared behavior. We demonstrate its usefulness by analyzing several fundamental indicators of the three countries' real aggregate economy in a univariate, as well as a multivariate setting. Since business cycles are international phenomena, which show common characteristics across countries, our aim is to uncover supranational behavior within the set of representative European economies selected herein. Finally, the analysis is extended to include several indicators from the U.S. economy, in order to examine its influence on the European economies under study and their interrelationships.

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Feliks, Yizhak, Andrew W. Robertson, and Michael Ghil. 2016. “Interannual Variability in North Atlantic Weather: Data Analysis and a Quasigeostrophic Model.” Journal of the Atmospheric Sciences 73 (8): 3227-3248. Abstract

This paper addresses the effect of interannual variability in jet stream orientation on weather systems over the North Atlantic basin (NAB). The observational analysis relies on 65 yr of NCEP–NCAR reanalysis (1948–2012). The total daily kinetic energy of the geostrophic wind (GTKE) is taken as a measure of storm activity over the North Atlantic. The NAB is partitioned into four rectangular regions, and the winter average of GTKE is calculated for each quadrant. The spatial GTKE average over all four quadrants shows striking year-to-year variability and is strongly correlated with the North Atlantic Oscillation (NAO).The GTKE strength in the northeast quadrant is closely related to the diffluence angle of the jet stream in the northwest quadrant. To gain insight into the relationship between the diffluence angle and its downstream impact, a quasigeostrophic baroclinic model is used. The results show that an initially zonal jet persists at its initial latitude over 30 days or longer, while a tilted jet propagates meridionally according to the Rossby wave group velocity, unless kept stationary by external forcing.A Gulf Stream–like narrow sea surface temperature (SST) front provides the requisite forcing for an analytical steady-state solution to this problem. This SST front influences the atmospheric jet in the northwest quadrant: it both strengthens the jet and tilts it northward at higher levels, while its effect is opposite at lower levels. Reanalysis data confirm these effects, which are consistent with thermal wind balance. The results suggest that the interannual variability found in the GTKE may be caused by intrinsic variability of the thermal Gulf Stream front.

Walwer, Damian, Eric Calais, and Michael Ghil. 2016. “Data-Adaptive Detection of Transient Deformation in Geodetic Networks.” Journal of Geophysical Research: Solid Earth 121 (3). Wiley Online Library: 2129-2152 . Abstract

The recent development of dense and continuously operating Global Navigation Satellite System (GNSS) networks worldwide has led to a significant increase in geodetic data sets that sometimes capture transient-deformation signals. It is challenging, however, to extract such transients of geophysical origin from the background noise inherent to GNSS time series and, even more so, to separate them from other signals, such as seasonal redistributions of geophysical fluid mass loads. In addition, because of the very large number of continuously recording GNSS stations now available, it has become impossible to systematically inspect each time series and visually compare them at all neighboring sites. Here we show that Multichannel Singular Spectrum Analysis (M-SSA), a method derived from the analysis of dynamical systems, can be used to extract transient deformations, seasonal oscillations, and background noise present in GNSS time series. M-SSA is a multivariate, nonparametric, statistical method that simultaneously exploits the spatial and temporal correlations of geophysical fields. The method allows for the extraction of common modes of variability, such as trends with nonconstant slopes and oscillations shared across time series, without a priori hypotheses about their spatiotemporal structure or their noise characteristics. We illustrate this method using synthetic examples and show applications to actual GPS data from Alaska to detect seasonal signals and microdeformation at the Akutan active volcano. The geophysically coherent spatiotemporal patterns of uplift and subsidence thus detected are compared to the results of an idealized model of such processes in the presence of a magma chamber source.

Chen, C., M. A. Cane, N. Henderson, D. Eun Lee, D. Chapman, Dmitri Kondrashov, and Mickaël D. Chekroun. 2016. “Diversity, nonlinearity, seasonality and memory effect in ENSO simulation and prediction using empirical model reduction.” Journal of Climate 29 (5): 1809-1830. Abstract

A suite of empirical model experiments under the empirical model reduction framework are conducted to advance the understanding of ENSO diversity, nonlinearity, seasonality, and the memory effect in the simulation and prediction of tropical Pacific sea surface temperature (SST) anomalies. The model training and evaluation are carried out using 4000-yr preindustrial control simulation data from the coupled model GFDL CM2.1. The results show that multivariate models with tropical Pacific subsurface information and multilevel models with SST history information both improve the prediction skill dramatically. These two types of models represent the ENSO memory effect based on either the recharge oscillator or the time-delayed oscillator viewpoint. Multilevel SST models are a bit more efficient, requiring fewer model coefficients. Nonlinearity is found necessary to reproduce the ENSO diversity feature for extreme events. The nonlinear models reconstruct the skewed probability density function of SST anomalies and improve the prediction of the skewed amplitude, though the role of nonlinearity may be slightly overestimated given the strong nonlinear ENSO in GFDL CM2.1. The models with periodic terms reproduce the SST seasonal phase locking but do not improve the prediction appreciably. The models with multiple ingredients capture several ENSO characteristics simultaneously and exhibit overall better prediction skill for more diverse target patterns. In particular, they alleviate the spring/autumn prediction barrier and reduce the tendency for predicted values to lag the target month value.

Pierini, S., Michael Ghil, and Mickaël D. Chekroun. 2016. “Exploring the pullback attractors of a low-order quasigeostrophic ocean model: The deterministic case.” Journal of Climate 29 (11): 4185-4202. Abstract

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.

Chekroun, Mickaël D., Michael Ghil, Honghu Liu, and Shouhong Wang. 2016. “Low-dimensional Galerkin approximations of nonlinear delay differential equations.” Discrete and Continuous Dynamical Systems - Series S 36 (8): 4133-4177. Abstract

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

Edeline, Eric, Andreas Groth, Bernard Cazelles, David Claessen, Ian J. Winfield, Jan Ohlberger, L. Asbjørn Vøllestad, Nils C. Stenseth, and Michael Ghil. 2016. “Pathogens trigger top-down climate forcing on ecosystem dynamics.” Oecologia, 1–14. Abstract

Evaluating the effects of climate variation on ecosystems is of paramount importance for our ability to forecast and mitigate the consequences of global change. However, the ways in which complex food webs respond to climate variations remain poorly understood. Here, we use long-term time series to investigate the effects of temperature variation on the intraguild-predation (IGP) system of Windermere (UK), a lake where pike (Esox lucius, top predator) feed on small-sized perch (Perca fluviatilis) but compete with large-sized perch for the same food sources. Spectral analyses of time series reveal that pike recruitment dynamics are temperature controlled. In 1976, expansion of a size-truncating perch pathogen into the lake severely impacted large perch and favoured pike as the IGP-dominant species. This pathogen-induced regime shift to a pike-dominated IGP apparently triggered a temperature-controlled trophic cascade passing through pike down to dissolved nutrients. In simple food chains, warming is predicted to strengthen top–down control by accelerating metabolic rates in ectothermic consumers, while pathogens of top consumers are predicted to dampen this top–down control. In contrast, the local IGP structure in Windermere made warming and pathogens synergistic in their top–down effects on ecosystem functioning. More generally, our results point to top predators as major mediators of community response to global change, and show that size-selective agents (e.g. pathogens, fishers or hunters) may change the topological architecture of food webs and alter whole ecosystem sensitivity to climate variation.

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2015
Colon, Célian, David Claessen, and Michael Ghil. 2015. “Bifurcation analysis of an agent-based model for predator–prey interactions.” Ecological Modelling 317: 93 - 106. Publisher's Version Abstract

Abstract The Rosenzweig–MacArthur model is a set of ordinary differential equations (ODEs) that provides an aggregate description of the dynamics of a predator–prey system. When including an Allee effect on the prey, this model exhibits bistability and contains a pitchfork bifurcation, a Hopf bifurcation and a heteroclinic bifurcation. We develop an agent-based model (ABM) on a two-dimensional, square lattice that encompasses the key assumptions of the aggregate model. Although the two modelling approaches – \ODE\ and \ABM\ – differ, both models exhibit similar bifurcation patterns. The \ABM\ model's behaviour is richer and it is analysed using advanced statistical methods. In particular, singular spectrum analysis is used to robustly locate the transition between apparently random, small-amplitude fluctuations around a fixed point and stable, large-amplitude oscillations. Critical slowing down of model trajectories anticipates the heteroclinic bifurcation. Systematic comparison between the \ABM\ and the \ODE\ models’ behaviour helps one understand the predator–prey system better; it provides guidance in model exploration and allows one to draw more robust conclusions on the nature of predator–prey interactions.

Groth, Andreas. 2015. “Business cycle analysis and forecasting using advanced spectral methods and data-based low-order models.” 35th International Symposium on Forecasting Riverside, California, June 2015. Abstract

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Mukhin, Dmitry, Evgeny Loskutov, Anna Mukhina, Alexander Feigin, Ilia Zaliapin, and Michael Ghil. 2015. “Predicting critical transitions in ENSO models. Part I: Methodology and simple models with memory.” Journal of Climate 28 (5): 1940–1961. Abstract
A new empirical approach is proposed for predicting critical transitions in the climate system based on a time series alone. This approach relies on nonlinear stochastic modeling of the system’s time-dependent evolution operator by the analysis of observed behavior. Empirical models that take the form of a discrete random dynamical system are constructed using artificial neural networks; these models include state-dependent stochastic components. To demonstrate the usefulness of such models in predicting critical climate transitions, they are applied here to time series generated by a number of delay-differential equation (DDE) models of sea surface temperature anomalies. These DDE models take into account the main conceptual elements responsible for the El Niño–Southern Oscillation phenomenon. The DDE models used here have been modified to include slow trends in the control parameters in such a way that critical transitions occur beyond the learning interval in the time series. Numerical results suggest that the empirical models proposed herein are able to forecast sequences of critical transitions that manifest themselves in future abrupt changes of the climate system’s statistics.
Mukhin, Dmitry, Dmitri Kondrashov, Evgeny Loskutov, Andrey Gavrilov, Alexander Feigin, and Michael Ghil. 2015. “Predicting critical transitions in ENSO models. Part II: Spatially dependent models.” Journal of Climate 28 (5): 1962–1976. Abstract
The present paper is the second part of a two-part study on empirical modeling and prediction of climate variability. This paper deals with spatially distributed data, as opposed to the univariate data of Part I. The choice of a basis for effective data compression becomes of the essence. In many applications, it is the set of spatial empirical orthogonal functions that provides the uncorrelated time series of principal components (PCs) used in the learning set. In this paper, the basis of the learning set is obtained instead by applying multichannel singular-spectrum analysis to climatic time series and using the leading spatiotemporal PCs to construct a reduced stochastic model. The effectiveness of this approach is illustrated by predicting the behavior of the Jin–Neelin–Ghil (JNG) hybrid seasonally forced coupled ocean–atmosphere model of El Niño–Southern Oscillation. The JNG model produces spatially distributed and weakly nonstationary time series to which the model reduction and prediction methodology is applied. Critical transitions in the hybrid periodically forced coupled model are successfully predicted on time scales that are substantially longer than the duration of the learning sample.
Vannitsem, Stéphane, Jonathan Demaeyer, Lesley De Cruz, and Michael Ghil. 2015. “Low-frequency variability and heat transport in a low-order nonlinear coupled ocean–atmosphere model.” Physica D: Nonlinear Phenomena 309. Elsevier: 71–85. Abstract
We formulate and study a low-order nonlinear coupled ocean–atmosphere model with an emphasis on the impact of radiative and heat fluxes and of the frictional coupling between the two components. This model version extends a previous 24-variable version by adding a dynamical equation for the passive advection of temperature in the ocean, together with an energy balance model. The bifurcation analysis and the numerical integration of the model reveal the presence of low-frequency variability (LFV) concentrated on and near a long-periodic, attracting orbit. This orbit combines atmospheric and oceanic modes, and it arises for large values of the meridional gradient of radiative input and of frictional coupling. Chaotic behavior develops around this orbit as it loses its stability; this behavior is still dominated by the LFV on decadal and multi-decadal time scales that is typical of oceanic processes. Atmospheric diagnostics also reveals the presence of predominant low- and high-pressure zones, as well as of a subtropical jet; these features recall realistic climatological properties of the oceanic atmosphere. Finally, a predictability analysis is performed. Once the decadal-scale periodic orbits develop, the coupled system’s short-term instabilities–as measured by its Lyapunov exponents–are drastically reduced, indicating the ocean’s stabilizing role on the atmospheric dynamics. On decadal time scales, the recurrence of the solution in a certain region of the invariant subspace associated with slow modes displays some extended predictability, as reflected by the oscillatory behavior of the error for the atmospheric variables at long lead times.
Chekroun, Mickaël D., Honghu Liu, and S. Wang. 2015. Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. New York: Springer Briefs in Mathematics, Springer. Publisher's Version Abstract

This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.

Chang, C. P., Michael Ghil, M. Latif, and J. M. Wallace, ed. 2015. Climate Change: Multidecadal and Beyond. World Scientific Publ. Co./Imperial College Press.
Ghil, Michael, Mickaël D. Chekroun, and Gabor Stepan. 2015. “A collection on `Climate Dynamics: Multiple Scales and Memory Effects'.” Proceedings of the Royal Society A 471 (20150097). Royal Society London. Publisher's Version
Kondrashov, Dmitri, Mickaël D. Chekroun, and Michael Ghil. 2015. “Data-driven non-Markovian closure models.” Physica D: Nonlinear Phenomena 297. Elsevier: 33–55. Abstract

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 broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The EMR–MSM methodology is first applied to a conceptual, nonlinear, stochastic climate model of coupled slow and fast variables, in which only slow variables are observed. It is shown that the resulting closure model with energy-conserving nonlinearities efficiently captures the main statistical features of the slow variables, even when there is no formal scale separation and the fast variables are quite energetic. Second, an MSM is shown to successfully reproduce the statistics of a partially observed, generalized Lotka–Volterra model of population dynamics in its chaotic regime. The challenges here include the rarity of strange attractors in the model’s parameter space and the existence of multiple attractor basins with fractal boundaries. The positivity constraint on the solutions’ components replaces here the quadratic-energy–preserving constraint of fluid-flow problems and it successfully prevents blow-up.

Chavez, M., Michael Ghil, and J. Urrutia-Fucugauchi, ed. 2015. Extreme Events: Observations, Modeling and Economics. Geophysical Monographs. Vol. 214. Washington, DC: American Geophysical Union & Wiley, 214. Publisher's Version Abstract

The monograph covers the fundamentals and the consequences of extreme geophysical phenomena like asteroid impacts, climatic change, earthquakes, tsunamis, hurricanes, landslides, volcanic eruptions, flooding, and space weather. This monograph also addresses their associated, local and worldwide socio-economic impacts. The understanding and modeling of these phenomena is critical to the development of timely worldwide strategies for the prediction of natural and anthropogenic extreme events, in order to mitigate their adverse consequences. This monograph is unique in as much as it is dedicated to recent theoretical, numerical and empirical developments that aim to improve: (i) the understanding, modeling and prediction of extreme events in the geosciences, and, (ii) the quantitative evaluation of their economic consequences. The emphasis is on coupled, integrative assessment of the physical phenomena and their socio-economic impacts. With its overarching theme, Extreme Events: Observations, Modeling and Economics will be relevant to and become an important tool for researchers and practitioners in the fields of hazard and risk analysis in general, as well as to those with a special interest in climate change, atmospheric and oceanic sciences, seismo-tectonics, hydrology, and space weather.

Chekroun, Mickaël D., and Honghu Liu. 2015. “Finite-horizon parameterizing manifolds, and applications to suboptimal control of nonlinear parabolic PDEs.” Acta Applicandae Mathematicae 135 (1): 81–144.
Groth, Andreas, Patrice Dumas, Michael Ghil, and Stéphane Hallegatte. 2015. “Impacts of natural disasters on a dynamic economy.” Extreme Events : Observations, Modeling, and Economics, edited by Eric Chavez, Michael Ghil, and Jaime Urrutia-Fucugauchi, 343–360. American Geophysical Union and Wiley-Blackwell. Abstract

This chapter presents a modeling framework for macroeconomic growth dynamics; it is motivated by recent attempts to formulate and study “integrated models” of the coupling between natural and socioeconomic phe­ nomena. The challenge is to describe the interfaces between human activities and the functioning of the earth system. We examine the way in which this interface works in the presence of endogenous business cycle dynam­ ics, based on a nonequilibrium dynamic model. Recent findings about the macroeconomic response to natural disasters in such a nonequilibrium setting have shown a more severe response to natural disasters during expan­ sions than during recessions. These findings raise questions about the assessment of climate change damages or natural disaster losses that are based purely on long-term growth models. In order to compare the theoretical findings with observational data, we analyze cyclic behavior in the U.S. economy, based on multivariate singular spectrum analysis. We analyze a total of nine aggregate indicators in a 52 year interval (1954–2005) and demon­ strate that the behavior of the U.S. economy changes significantly between intervals of growth and recession, with higher volatility during expansions.

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