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16
A statistical mechanical approach for the computation of the climatic response to general forcings
, 2010
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On stochastic parameterizing manifolds: pullback characterization and nonMarkovian reduced equations
 Mem Am Math Soc
"... Abstract. Part I of this article is devoted to the leading order approximations of stochastic critical manifolds associated with a broad class of stochastic partial differential equations (SPDEs) which are driven by linear multiplicative white noise. Stochastic critical manifolds are built naturally ..."
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Abstract. Part I of this article is devoted to the leading order approximations of stochastic critical manifolds associated with a broad class of stochastic partial differential equations (SPDEs) which are driven by linear multiplicative white noise. Stochastic critical manifolds are built naturally as random graphs over a fixed number of critical modes which lose their stability as a control parameter varies. Explicit formulas for the leadingorder Taylor approximation of such local manifolds about the basic state, are derived. It is shown that the corresponding approximating manifolds admit furthermore a pullback characterization, which provides a novel interpretation of such objects in terms of flows. The framework set up in this way allows us, furthermore, to unify the previous approximation approaches from the literature. The existence and attraction properties of oneparameter families of stochastic invariant manifolds are also revisited in this first part. In Part II, a point of view more global is adopted. In that respect, a general approach to provide approximate parameterizations of the “small ” scales by the “large ” ones, is developed for stochastic partial differential equations driven by linear multiplicative noise. This is accomplished via the concept of parameterizing manifolds (PMs) that are stochastic manifolds which improve in mean square error the partial knowledge of the full SPDE solution u when compared to the projection of u onto the
A Mathematical Theory of Climate Sensitivity or, How to Deal With Both Anthropogenic Forcing and Natural Variability?
"... Recent estimates of climate evolution over the coming century still differ by several degrees. This uncertainty motivates the work presented here. There are two basic approaches to apprehend the complexity of climate change: deterministically nonlinear and stochastically linear, i.e. the Lorenz and ..."
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Cited by 2 (1 self)
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Recent estimates of climate evolution over the coming century still differ by several degrees. This uncertainty motivates the work presented here. There are two basic approaches to apprehend the complexity of climate change: deterministically nonlinear and stochastically linear, i.e. the Lorenz and the Hasselmann approach. The grand unification of these two approaches relies on the theory of random dynamical systems. We apply this theory to study the random attractors of nonlinear, stochastically perturbed climate models. Doing so allows one to examine the interaction of internal climate variability with the forcing, whether natural or anthropogenic, and to take into account the climate system’s nonequilibrium behavior in determining climate sensitivity. This nonequilibrium behavior is due to a combination of nonlinear and random effects. We give here a unified treatment of such effects from the point of view of the theory of dynamical systems and of their bifurcations. Energy balance models are used to illustrate multiple equilibria, while multidecadal oscillations in the thermohaline circulation illustrate the transition from steady states to periodic behavior. Random effects are introduced in the setting of random dynamical systems, which permit a unified treatment of both nonlinearity and stochasticity. The combined treatment of nonlinear and random effects is applied to a stochastically perturbed version of the classical Lorenz
Climatic impacts of stochastic fluctuations in air–sea fluxes
"... [1] Air–sea fluxes vary partly on scales that are too small or fast to be resolved explicitly by global climate models. This paper proposes a nonlinear physical mechanism by which stochastic fluctuations in the air–sea buoyancy flux may modify the mean climate. The paper then demonstrates the mechan ..."
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[1] Air–sea fluxes vary partly on scales that are too small or fast to be resolved explicitly by global climate models. This paper proposes a nonlinear physical mechanism by which stochastic fluctuations in the air–sea buoyancy flux may modify the mean climate. The paper then demonstrates the mechanism in climate simulations with a comprehensive coupled general circulation model. Significant changes are detected in the timemean oceanic mixedlayer depth, seasurface temperature, atmospheric Hadley circulation, and net upward water flux at the sea surface. Also, El Niño Southern Oscillation (ENSO) variability is significantly increased. The findings demonstrate that noiseinduced drift and noiseenhanced variability, which are familiar concepts from simple climate models, continue to apply in comprehensive climate models with millions of degrees of freedom. The findings also suggest that the lack of representation of subgrid variability in air–sea fluxes may contribute to some of the biases exhibited by contemporary climate models. Citation: Williams, P. D. (2012), Climatic impacts of stochastic fluctuations in air–sea fluxes,Geophys. Res. Lett., 39, L10705, doi:10.1029/2012GL051813. 1.
NONMARKOVIAN REDUCED SYSTEMS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS: THE ADDITIVE NOISE CASE
"... Abstract. This article proposes for stochastic partial differential equations (SPDEs) driven by additive noise, a novel approach for the approximate parameterizations of the “small ” scales by the “large ” ones, along with the derivaton of the corresponding reduced systems. This is accomplished by s ..."
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Abstract. This article proposes for stochastic partial differential equations (SPDEs) driven by additive noise, a novel approach for the approximate parameterizations of the “small ” scales by the “large ” ones, along with the derivaton of the corresponding reduced systems. This is accomplished by seeking for stochastic parameterizing manifolds (PMs) introduced in [CLW13] which are random manifolds aiming to provide — in a mean square sense — such approximate parameterizations. Backwardforward systems are designed to give access to such PMs as pullback limits depending through the nonlinear terms on the timehistory of the dynamics of the low modes when the latter is simply approximated by its stochastic linear component. It is shown that the corresponding pullback limits can be efficiently determined, leading in turn to an operational procedure for the derivation of nonMarkovian reduced systems able to achieve good modeling performances in practice. This is illustrated on a stochastic Burgerstype equation, where it is shown that the corresponding nonMarkovian features of these reduced systems play a key role to reach such performances. 1.
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications∗
, 2012
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Mathematical Theory of Climate Sensitivity
, 2012
"... Abstract. Recent estimates of climate evolution over the coming century still differ by several degrees. This uncertainty motivates the work presented here. There are two basic approaches to apprehend the complexity of climate change: deterministically nonlinear and stochastically linear, i.e. the L ..."
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Abstract. Recent estimates of climate evolution over the coming century still differ by several degrees. This uncertainty motivates the work presented here. There are two basic approaches to apprehend the complexity of climate change: deterministically nonlinear and stochastically linear, i.e. the Lorenz and the Hasselmann approach. The “grand unification ” of these two approaches relies on the theory of random dynamical systems. We apply this theory to study the random attractors of nonlinear, stochastically perturbed climate models. Doing so allows one to examine the interaction of internal climate variability with the forcing, whether natural or anthropogenic, and to take into account the climate system’s nonequilibrium behavior in determining climate sensitivity. This nonequilibrium behavior is due to a combination of nonlinear and random effects. We give here a unified treatment of such effects from the point of view of the theory of dynamical systems and of their bifurcations. Energy balance models are used to illustrate multiple equilibria, while multidecadal oscillations in the thermohaline circulation illustrate the transition from steady states to periodic behavior. Random effects are introduced in the setting of random dynamical systems, which permit a unified treatment of both nonlinearity and stochasticity. The combined treatment of nonlinear and random effects is applied to a stochastically perturbed version of the classical Lorenz convection model. Climate sensitivity is then defined mathematically as the derivative of an appropriate functional or other function of the system’s state with respect to the bifurcation parameter. 1.
Whitepaper Fostering Interactions Between the Geosciences and
, 2012
"... Over the past two decades the geosciences have acquired a wealth of new and high quality data from new and greatly improved observing technologies. These datasets have been paramount in enabling improved understanding and modeling but have also strikingly demonstrated important knowledge gaps and th ..."
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Over the past two decades the geosciences have acquired a wealth of new and high quality data from new and greatly improved observing technologies. These datasets have been paramount in enabling improved understanding and modeling but have also strikingly demonstrated important knowledge gaps and the limitations of our current conceptual models to explain key aspects of these observations. This situation limits progress on questions that have both fundamental scientific and societal significance, including climate and weather, natural disaster mitigation, earthquake and volcano dynamics, earth structure and geodynamics, resource exploration, and planetary evolution. The challenge in modeling accurately these processes is not only one of computational power. Powerful computations based on existing models are incapable of reproducing the observations faithfully. Instead, we need to develop new conceptual approaches to describe the complexities of these natural systems. Essentially, we need ways to characterize complicated systems that vary strongly in space and time in ways that are not adequately accounted for in our current paradigms. A fourday workshop in October 2011 explored this issue and reached a consensus that significant advances can result from accelerating the traditional interaction between geoscientists, mathematicians, statisticians, and computer scientists. We considered key challenges that confront the geosciences and major areas of rapid development in mathematics, statistics and computer science that offer the potential for significant advances in meeting these challenges. This type of research will require combining stochastic and deterministic models, improving methods of model validation and verification, developing inverse methods and techniques for the identification of extreme events and critical transitions, and formulating novel numerical algorithms and implementations, along with the greatly enhanced use of data from the rapidly evolving observing systems. To advance these goals, we propose establishing a geographically distributed but wellfocused institute with a novel operational, educational and training structure that can foster and promote these valuable interactions.
Toward a Mathematical Theory of Climate Sensitivity
"... The first attempt at a consensus estimate of the equilibrium sensitivity of climate to changes in atmospheric carbon dioxide concentrations appeared in 1979, in the U.S. National Research Council report of J.G. Charney and associates. The result was the now famous range for an increase of 1.5–4.5 K ..."
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The first attempt at a consensus estimate of the equilibrium sensitivity of climate to changes in atmospheric carbon dioxide concentrations appeared in 1979, in the U.S. National Research Council report of J.G. Charney and associates. The result was the now famous range for an increase of 1.5–4.5 K in global temperatures, given a doubling of CO2 concentrations. Earth’s climate, however, never was and is unlikely ever to be in equilibrium. The Intergovernmental Panel on Climate Change, therefore, in addition to estimates of equilibrium sensitivity, focused on estimates of climate change in the 21st century. The latter estimates of temperature increase in the coming 100 years still range over several degrees Celsius. This difficulty in narrowing the range of estimates is clearly connected to the complexity of the climate system, the nonlinearity of the processes involved, and the obstacles to a faithful representation of these processes and feedbacks in global climate models, as described in [4].