Dynamical systems

Zaliapin, Ilya, and Michael Ghil. “A delay differential model of ENSO variability, Part 2: Phase locking, multiple solutions, and dynamics of extrema.” Nonlinear Processes in Geophysics 17, no. 2 (2010): 123–135.
Chekroun, Mickaël D., Michael Ghil, Jean Roux, and Ferenc Varadi. “Averaging of time-periodic systems without a small parameter.” Discrete and Continuous Dynamical Systems 14, no. 4 (2010): 753–782.
Roques, Lionel, and Mickaël D. Chekroun. “Does reaction-diffusion support the duality of fragmentation effect?Ecological Complexity 7, no. 1 (2010): 100–106.
Ghil, Michael, Peter Read, and Leonard Smith. “Geophysical flows as dynamical systems: the influence of Hide's experiments.” Astronomy & Geophysics 51, no. 4 (2010): 4–28.
Zaliapin, Ilya, Efi Foufoula-Georgiou, and Michael Ghil. “Transport on river networks: A dynamic tree approach.” Journal of Geophysical Research: Earth Surface 115, no. F2 (2010).
Kravtsov, Sergey, Dmitri Kondrashov, and Michael Ghil. “Empirical model reduction and the modelling hierarchy in climate dynamics and the geosciences.” Stochastic physics and climate modelling. Cambridge University Press, Cambridge (2009): 35–72. Abstract
Modern climate dynamics uses a two-fisted approach in attacking and solving the problems of atmospheric and oceanic flows. The two fists are: (i) observational analyses; and (ii) simulations of the geofluids, including the coupled atmosphere–ocean system, using a hierarchy of dynamical models. These models represent interactions between many processes that act on a broad range of spatial and time scales, from a few to tens of thousands of kilometers, and from diurnal to multidecadal, respectively. The evolution of virtual climates simulated by the most detailed and realistic models in the hierarchy is typically as difficult to interpret as that of the actual climate system, based on the available observations thereof. Highly simplified models of weather and climate, though, help gain a deeper understanding of a few isolated processes, as well as giving clues on how the interaction between these processes and the rest of the climate system may participate in shaping climate variability. Finally, models of intermediate complexity, which resolve well a subset of the climate system and parameterise the remainder of the processes or scales of motion, serve as a conduit between the models at the two ends of the hierarchy. We present here a methodology for constructing intermediate mod- els based almost entirely on the observed evolution of selected climate fields, without reference to dynamical equations that may govern this evolution; these models parameterise unresolved processes as multi- variate stochastic forcing. This methodology may be applied with equal success to actual observational data sets, as well as to data sets resulting from a high-end model simulation. We illustrate this methodology by its applications to: (i) observed and simulated low-frequency variability of atmospheric flows in the Northern Hemisphere; (ii) observed evo- lution of tropical sea-surface temperatures; and (iii) observed air–sea interaction in the Southern Ocean. Similar results have been obtained for (iv) radial-diffusion model simulations of Earth’s radiation belts, but are not included here because of space restrictions. In each case, the reduced stochastic model represents surprisingly well a variety of linear and nonlinear statistical properties of the resolved fields. Our methodology thus provides an efficient means of constructing reduced, numerically inexpensive climate models. These models can be thought of as stochastic–dynamic prototypes of more complex deterministic models, as in examples (i) and (iv), but work just as well in the situation when the actual governing equations are poorly known, as in (ii) and (iii). These models can serve as competitive prediction tools, as in (ii), or be included as stochastic parameterisations of certain processes within more complex climate models, as in (iii). Finally, the methodology can be applied, with some modifications, to geophysical problems outside climate dynamics, as illustrated by (iv).
Simonnet, Eric, Henk A. Dijkstra, and Michael Ghil. “Bifurcation analysis of ocean, atmosphere, and climate models.” In Handbook of numerical analysis, edited by R. Temam and J. Tribbia, 14:187–229. Elsevier, 2009.
Bordi, Isabella, Klaus Fraedrich, Michael Ghil, and Alfonso Sutera. “Zonal flow regime changes in a GCM and in a simple quasigeostrophic model: The role of stratospheric dynamics.” Journal of the Atmospheric Sciences 66, no. 5 (2009): 1366–1383.
Ghil, Michael, Ilya Zaliapin, and Barbara Coluzzi. “Boolean delay equations: A simple way of looking at complex systems.” Physica D: Nonlinear Phenomena 237, no. 23 (2008): 2967–2986.
Hallegatte, Stéphane, Michael Ghil, Patrice Dumas, and Jean-Charles Hourcade. “Business cycles, bifurcations and chaos in a neo-classical model with investment dynamics.” Journal of Economic Behavior & Organization 67, no. 1 (2008): 57–77. Abstract

This paper presents a non-equilibrium dynamic model (NEDyM) that introduces investment dynamics and non-equilibrium effects into a Solow growth model. NEDyM can reproduce several typical economic regimes and, for certain ranges of parameter values, exhibits endogenous business cycles with realistic characteristics. The cycles arise from the investment-profit instability and are constrained by the increase in labor costs and the inertia of production capacity. For other parameter ranges, the model exhibits chaotic behavior. These results show that complex variability in the economic system may be due to deterministic, intrinsic factors, even if the long-term equilibrium is neo-classical in nature.

Carrassi, Alberto, Michael Ghil, Anna Trevisan, and Francesco Uboldi. “Data assimilation as a nonlinear dynamical systems problem: Stability and convergence of the prediction-assimilation system.” Chaos 18, no. 2 (2008): 023112. Abstract

We study prediction-assimilation systems, which have become routine in meteorology and oceanography and are rapidly spreading to other areas of the geosciences and of continuum physics. The long-term, nonlinear stability of such a system leads to the uniqueness of its sequentially estimated solutions and is required for the convergence of these solutions to the system's true, chaotic evolution. The key ideas of our approach are illustrated for a linearized Lorenz system. Stability of two nonlinear prediction-assimilation systems from dynamic meteorology is studied next via the complete spectrum of their Lyapunov exponents; these two systems are governed by a large set of ordinary and of partial differential equations, respectively. The degree of data-induced stabilization is crucial for the performance of such a system. This degree, in turn, depends on two key ingredients: (i) the observational network, either fixed or data-adaptive, and (ii) the assimilation method.

Hallegatte, Stéphane, and Michael Ghil. “Natural disasters impacting a macroeconomic model with endogenous dynamics.” Ecological Economics 68, no. 1-2 (2008): 582–592. Abstract

We investigate the macroeconomic response to natural disasters by using an endogenous business cycle (EnBC) model in which cyclical behavior arises from the investment-profit instability. Our model exhibits a larger response to natural disasters during expansions than during recessions. This apparently paradoxical result can be traced to the disasters amplifying pre-existing disequilibria during expansions, while the existence of unused resources during recessions damps the exogenous shocks. It thus appears that high-growth periods are also highly vulnerable to supply-side shocks. In our EnBC model, the average production loss due to a set of disasters distributed at random in time is highly sensitive to the dynamical characteristics of the impacted economy. Larger economic flexibility allows for a more efficient and rapid response to supply-side shocks and reduces production losses. On the other hand, too high a flexibility can lead to vulnerability phases that cause average production losses to soar. These results raise questions about the assessment of climate change damages or natural disaster losses that are based purely on long-term growth models.

Spyratos, V., P. S. Bourgeron, and Michael Ghil. “Development at the wildland urban interface and the mitigation of forest-fire risk.” Proceedings of the National Academy of Sciences 104, no. 36 (2007): 14272–14276. Abstract

This work addresses the impacts of development at the wildland-urban interface on forest fires that spread to human habitats. Catastrophic fires in the western United States and elsewhere make these impacts a matter of urgency for decision makers, scientists, and the general public. Using a simple fire-spread model, along with housing and vegetation data, we show that fire size probability distributions can be strongly modified by the density and flammability of houses. We highlight a sharp transition zone in the parameter space of vegetation flammability and house density. Many actual fire landscapes in the United States appear to have spreading properties close to this transition. Thus, the density and flammability of buildings should be taken into account when assessing fire risk at the wildland-urban interface. Moreover, our results highlight ways for regulation at this interface to help mitigate fire risk.

Kravtsov, S, P Berloff, William K. Dewar, M Ghil, and James C. McWilliams. “Dynamical origin of low-frequency variability in a highly nonlinear midlatitude coupled model.Journal of Climate 19, no. 24 (2006). Abstract

A novel mechanism of decadal midlatitude coupled variability, which crucially depends on the nonlinear dynamics of both the atmosphere and the ocean, is presented. The coupled model studied involves quasigeostrophic atmospheric and oceanic components, which communicate with each other via a constant-depth oceanic mixed layer. A series of coupled and uncoupled experiments show that the decadal coupled mode is active across parameter ranges that allow the bimodality of the atmospheric zonal flow to coexist with oceanic turbulence. The latter is most intense in the regions of inertial recirculation (IR). Bimodality is associated with the existence of two distinct anomalously persistent zonal-flow modes, which are characterized by different latitudes of the atmospheric jet stream. The IR reorganizations caused by transitions of the atmosphere from its high- to low-latitude state and vice versa create sea surface temperature anomalies that tend to induce transition to the opposite atmospheric state. The decadal–interdecadal time scale of the resulting oscillation is set by the IR adjustment; the latter depends most sensitively on the oceanic bottom drag. The period T of the nonlinear oscillation is 7–25 yr for the range of parameters explored, with the most realistic parameter values yielding T \approx 20 yr. Aside from this nonlinear oscillation, an interannual Rossby wave mode is present in all coupled experiments. This coupled mode depends neither on atmospheric bimodality, nor on ocean eddy dynamics; it is analogous to the mode found previously in a channel configuration. Its time scale in the model with a closed ocean basin is set by cross-basin wave propagation and equals 3–5 yr for a basin width comparable with the North Atlantic.

Ghil, Michael, and Ilya Zaliapin. “Une Nouvelle source de Fractales: Les Equations Booléennes avec Retard, et leurs Applications aux Sciences de la Planete.” In L'irruption Des Géométries Fractales Dans Les Sciences,une Apologie De L'oeuvre De Benoît Mandelbrot, 161–187. Paris: Editions de l'Académie Européenne Interdisciplinaire des Sciences, 2006.
PDF PDF (front matter)
Simonnet, Eric, Michael Ghil, and Henk Dijkstra. “Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation.” Journal of Marine Research 63, no. 5 (2005): 931–956. Abstract

The wind-driven double-gyre circulation in a rectangular basin goes through several dynamical regimes as the amount of lateral friction is decreased. This paper studies the transition to irregular flow in the double-gyre circulation by applying dynamical systems methodology to a quasi-geostrophic, equivalent-barotropic model with a 10-km resolution. The origin of the irregularities, in space and time, is the occurrence of homoclinic bifurcations that involve phase-space behavior far from stationary solutions. The connection between these homoclinic bifurcations and earlier transitions, which occur at larger lateral friction, is explained. The earlier transitions, such as pitchfork and asymmetric Hopf bifurcation, only involve the nonlinear saturation of linear instabilities, while the homoclinic bifurcations are associated with genuinely nonlinear behavior. The sequence of bifurcations—pitchfork, Hopf, and homoclinic—is independent of the lateral friction and may be described as the unfolding of a singularity that occurs in the frictionless, Hamiltonian limit of the governing equations. Two distinct chaotic regimes are identified: Lorenz chaos at relatively large lateral friction versus Shilnikov chaos at relatively small lateral friction. Both types of homoclinic bifurcations induce chaotic behavior of the recirculation gyres that is dominated by relaxation oscillations with a well-defined period. The relevance of these results to the mid-latitude oceans' observed low-frequency variations is discussed. A previously documented 7-year peak in observed North-Atlantic variability is shown to exist across a hierarchy of models that share the gyre modes and homoclinic bifurcations discussed herein.

Dijkstra, Henk A., and Michael Ghil. “Low-frequency variability of the large-scale ocean circulation: a dynamical systems approach.” Reviews of Geophysics 43 (2005). Abstract

Oceanic variability on interannual, interdecadal, and longer timescales plays a key role in climate variability and climate change. Paleoclimatic records suggest major changes in the location and rate of deepwater formation in the Atlantic and Southern oceans on timescales from millennia to millions of years. Instrumental records of increasing duration and spatial coverage document substantial variability in the path and intensity of ocean surface currents on timescales of months to decades. We review recent theoretical and numerical results that help explain the physical processes governing the large-scale ocean circulation and its intrinsic variability. To do so, we apply systematically the methods of dynamical systems theory. The dynamical systems approach is proving successful for more and more detailed and realistic models, up to and including oceanic and coupled ocean-atmosphere general circulation models. In this approach one follows the road from simple, highly symmetric model solutions, through a “bifurcation tree,” toward the observed, complex behavior of the system under investigation. The observed variability can be shown to have its roots in simple transitions from a circulation with high symmetry in space and regularity in time to circulations with successively lower symmetry in space and less regularity in time. This road of successive bifurcations leads through multiple equilibria to oscillatory and eventually chaotic solutions. Key features of this approach are illustrated in detail for simplified models of two basic problems of the ocean circulation. First, a barotropic model is used to capture major features of the wind-driven ocean circulation and of the changes in its behavior as wind stress increases. Second, a zonally averaged model is used to show how the thermohaline ocean circulation changes as buoyancy fluxes at the surface increase. For the wind-driven circulation, multiple separation patterns of a “Gulf-Stream like” eastward jet are obtained. These multiple equilibria are followed by subannual and interannual oscillations of the jet and of the entire basin's circulation. The multiple equilibria of the thermohaline circulation include deepwater formation near the equator, near either pole or both, as well as intermediate possibilities that bear some degree of resemblance to the currently observed Atlantic overturning pattern. Some of these multiple equilibria are subject, in turn, to oscillatory instabilities with timescales of decades, centuries, and millennia. Interdecadal and centennial oscillations are the ones of greatest interest in the current debate on global warming and on the relative roles of natural and anthropogenic variability in it. They involve the physics of the truly three-dimensional coupling between the wind-driven and thermohaline circulation. To arrive at this three-dimensional picture, the bifurcation tree is sketched out for increasingly complex models for both the wind-driven and the thermohaline circulation.

Ghil, Michael, Tian Ma, and Shouhong Wang. “Structural Bifurcation of 2-D Nondivergent Flows with Dirichlet Boundary Conditions: Applications to Boundary-Layer Separation.” SIAM J. Appl. Math. 65, no. 5 (2005): 1576–1596.
Kao, Jim, Dawn Flicker, Rudy Henninger, Sarah Frey, Michael Ghil, and Kayo Ide. “Data assimilation with an extended Kalman filter for impact-produced shock-wave dynamics.” Journal of Computational Physics 196, no. 2 (2004): 705–723.
PDF (text only) PDF (figures)
Loeuille, Nicolas, and Michael Ghil. “Intrinsic and climatic factors in North-American animal population dynamics.” BMC Ecology 4, no. 1 (2004): 1. Publisher's Version