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Stochastic Climate Dynamics: Random Attractors and Timedependent Invariant Measures
, 2010
"... 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 detai ..."
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Cited by 17 (8 self)
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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 highresolution 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 timedependent invariant measures supported by these attractors; the latter are shown to be random SinaiRuelleBowen (SRB) measures. The first of the two models is a stochastically forced version of the classical Lorenz model. The second one is a lowdimensional, nonlinear stochastic model of the El Niño–Southern Oscillation (ENSO).
Punishment i n o r g a n i z a t i o n s : A rev iew, p r o p o s i t i o n s and r e s e a r c h s u g g e s t i o n s
 Academy of Management Review
, 1980
"... Dissociation, as the editor of this important volume reminds us, "challenges many comfortable assumptions." From a theoretical vantage, it demands great conceptual clarity and a knowledge ofmany areas of import in psychology; from a clinical vantage, it has brought about one ofthe most co ..."
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Cited by 6 (0 self)
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Dissociation, as the editor of this important volume reminds us, "challenges many comfortable assumptions." From a theoretical vantage, it demands great conceptual clarity and a knowledge ofmany areas of import in psychology; from a clinical vantage, it has brought about one ofthe most contentious debates in recent history, whether or notimportant memories can be "forgotten " only to appear as habits, behaviors, and dreams, or even later as fullfledged remembrances. But above all, dissociative phenomena challenge the cherished notion that our conscious self is an allknowing, integrated entity. This volume is the result of a 1991 conference at the Center for Advanced Study in the Behavioral Sciences, sponsored by the MacArthur Foundation. I was fortunate to be present at this truly multidisciplinary meeting.
Reduced models of atmospheric lowfrequency variability: Parameter estimation and . . .
, 2008
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Lowfrequency variability, coherence resonance and phase selection in a loworder model of the winddriven ocean circulation
 J. Phys. Oceanogr
"... In this paper, a loworder spectral quasigeostrophic (QG) model of the winddriven ocean circulation is derived and used to analyze the loworder character of the intrinsic lowfrequency variability of the midlatitude doublegyre ocean circulation and of the related coherence resonance and phase se ..."
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Cited by 4 (4 self)
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In this paper, a loworder spectral quasigeostrophic (QG) model of the winddriven ocean circulation is derived and used to analyze the loworder character of the intrinsic lowfrequency variability of the midlatitude doublegyre ocean circulation and of the related coherence resonance and phase selection phenomena. The model includes an exponential in the basis functions that allows for westward intensification, retains only four modes in the Galerkin projection, is defined in a rectangular domain, and is forced by deterministic and stochastic winds, thus extending previous loworder QG ocean models. The solution under steady forcing is first obtained, and the results are also analyzed in terms of dynamical systems theory. A homoclinic bifurcation (with the wind amplitude chosen as the control parameter) leads to intrinsic decadal relaxation oscillations (ROs) similar in several respects to those obtainedwith primitive equationmodels. The system is then forced with an additional red noise wind, and, in a parameter range preceding the global bifurcation, a coherence resonance scenario very similar to the one found with a primitive equation model of the Kuroshio Extension is obtained: this suggests that such a phenomenon is of loworder character. To study the RO excitation mechanism, a method denoted as phase selection is proposed. The system is forced with additional fictitious periodic winds that produce an emergence of ROs yielding strong phase dependence with the periodic forcing. The subsequent analysis reveals the character of the wind forcing that is most likely to excite a RO. All the results are discussed within the general framework of climate dynamics. 1.
Review The LiangKleeman Information Flow: Theory and Applications
, 2013
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A CONFORMING FINITE ELEMENT DISCRETIZATION OF THE STREAMFUNCTION FORM OF THE UNSTEADY QUASIGEOSTROPHIC EQUATIONS
"... Abstract. This paper presents a conforming finite element semidiscretization of the streamfunction form of the onelayer unsteady quasigeostrophic equations, which are a commonly used model for largescale winddriven ocean circulation. We derive optimal error estimates and present numerical resul ..."
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Abstract. This paper presents a conforming finite element semidiscretization of the streamfunction form of the onelayer unsteady quasigeostrophic equations, which are a commonly used model for largescale winddriven ocean circulation. We derive optimal error estimates and present numerical results. Key words. Quasigeostrophic equations, finite element method, Argyris element. 1.
Probing chaos and biodiversity in a simple competition model
"... Recent theoretical work has reported that chaos facilitates biodiversity. In this paper, we study the lowestdimensional LotkaVolterra competition model that exhibits chaotic trajectories, a model with four species. We observe that interaction and growth parameters leading respectively to extinctio ..."
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Recent theoretical work has reported that chaos facilitates biodiversity. In this paper, we study the lowestdimensional LotkaVolterra competition model that exhibits chaotic trajectories, a model with four species. We observe that interaction and growth parameters leading respectively to extinction of three species, or coexistence of two, three or four species, are for the most part arranged in large regions with clear boundaries. Small islands of parameters that lead to chaos are also found. These regions where chaos occurs are, in the three cases presented here, situated at the interface between a nonchaotic fourspecies region and a region where extinction occurs. This implies a high sensitivity of biodiversity with respect to parameter variations in the chaotic regions. Additionally, in regions where extinction occurs which are adjacent to chaotic regions, the computation of local Lyapunov exponents reveals that a possible cause of extinction is the overly strong fluctuations in species abundances induced by local chaos at the beginning of the interval of study. For this model, we conclude that biodiversity is a necessary condition for chaos rather than a consequence of chaos, which can be seen as a signal of a high extinction risk.
the Creative Commons Attribution 3.0 License. Nonlinear Processes in Geophysics Another look at climate sensitivity
"... Abstract. We revisit a recent claim that the Earth’s climate system is characterized by sensitive dependence to parameters; in particular, that the system exhibits an asymmetric, largeamplitude response to normally distributed feedback forcing. Such a response would imply irreducible uncertainty in ..."
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Abstract. We revisit a recent claim that the Earth’s climate system is characterized by sensitive dependence to parameters; in particular, that the system exhibits an asymmetric, largeamplitude response to normally distributed feedback forcing. Such a response would imply irreducible uncertainty in climate change predictions and thus have notable implications for climate science and climaterelated policy making. We show that equilibrium climate sensitivity in all generality does not support such an intrinsic indeterminacy; the latter appears only in essentially linear systems. The main flaw in the analysis that led to this claim is inappropriate linearization of an intrinsically nonlinear model; there is no room for physical interpretations or policy conclusions based on this mathematical error. Sensitive dependence nonetheless does exist in the climate system, as well
Manuscript prepared for Nonlin. Processes Geophys. with version 1.3 of the L ATEX class copernicus.cls.
, 2009
"... Abstract. We analyze here a classical energy balance model (EBM) with nonlinear feedbacks and revisit in this context a recent claim that — in a modelindependent sense — the climate system may exhibit an asymmetric, largeamplitude response to normally distributed forcing; such a response would imp ..."
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Abstract. We analyze here a classical energy balance model (EBM) with nonlinear feedbacks and revisit in this context a recent claim that — in a modelindependent sense — the climate system may exhibit an asymmetric, largeamplitude response to normally distributed forcing; such a response would imply irreducible uncertainty in climate change predictions. We show that equilibrium climate sensitivity in all generality does not support such an intrinsic indeterminacy. EBMs, though, do exhibit a saddlenode bifurcation that gives rise to several steadystate climates; this behavior is, furthermore, supported by results from more realistic, nonequilibrium climate models. In a truly nonlinear setting, indeterminacy in the size of the response is associated with the vicinity to such a bifurcation. We discuss this bifurcation in the EBM context and recall some wellknown results about the stability and equilibrium sensitivity of such models. It is shown here, moreover, that small disturbances cannot result in a largeamplitude response, unless the system is at or near the bifurcation point. Our EBM suggests that the current Earth climate is relatively close to this bifurcation, and a moderate decrease of the global radiative input may bring the Earth to a “snowball” state; results of general circulation models point in the same direction. We discuss briefly how the distance to the bifurcation may be related to the Earth’s icealbedo feedback.