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C. Nicolis and G. Nicolis

Abstract

Traditionally, climate is defined by the properties of the averages of the meteorological fields over an appropriate time interval. In this paper the properties of the time-averaged observables of a red noise atmosphere and of a simplified model of thermal convection are investigated both analytically and numerically and are compared to those of the original finescale variables. It is shown that averaging tends to reduce the domain of variability and the attractor dimension favors persistence of initial correlations and enhances predictability. The implications of these findings in the real-world atmosphere are briefly assessed.

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C. Nicolis and G. Nicolis

Abstract

A linear response theory of systems of interest in atmospheric and climate dynamics taking fully into account the nonlinearities of the underlying processes is developed. Under the assumption that the source of intrinsic variability can be modeled as a white-noise process, a Fokker–Planck equation approach leads to fluctuation–dissipation-type expressions in the form of time cross-correlation functions, linking the perturbation-induced shift of an observable to the statistical and dynamical properties of the reference system. These expressions feature a generalized potential function and enable one to go systematically beyond the Gaussian approximation usually adopted in the literature. Full solutions and explicit expressions are derived for different subclasses of systems, including a global climate model giving rise to oscillatory behavior.

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C. Nicolis

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C. Nicolis

Abstract

The dynamical and probabilistic aspects of model error arising from the neglect of unresolved scales and the concomitant reduction of phase space dimensionality of a reference system are analyzed. A general expression of the mean quadratic error involving the time correlation function of the excess phase space velocity is derived. In the short time regime, this expression reduces to the sum of a t 2 contribution reflecting the structure of the invariant distribution of the reference system and of a t 3 part bearing the signature of the Lyapunov exponents of both the reduced and the reference models. The approach is illustrated on two classes of systems involving, successively, multiple time and space scales. A comparison between the purely deterministic analysis and the one in which the model system is augmented by error sources assimilated to Gaussian random noises is carried out. It is shown that such a representation leads to a deterioration of the predictive skill of the model as far as mean values are concerned, but may enhance its variability properties, bringing them closer to the variability of the reference system.

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C. Nicolis

Abstract

A systematic formulation of model error is developed accounting for both its dynamical and probabilistic aspects. Some generic features, largely independent of the particular model under consideration, are brought out in connection with power-law behavior in the short time regime. The lifetime of this regime is shown to be related to the spectrum of the Lyapunov exponents and the decay characteristics of the time correlation functions involved. The results are illustrated and tested on representative low-order models of atmospheric dynamics exhibiting oscillations, bistability, and chaotic behavior.

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C. Nicolis

Abstract

The formalism of irreversible thermodynamics is applied to the shallow water model. Entropy production and entropy flow terms are identified, describing the ways dissipation and exchange processes unfold in space and time. Explicit evaluations are carried out in the case of Lorenz’s nine-mode truncation and in the quasigeostrophic limit of the model. A number of systematic trends are identified by studying the way dissipation and kinetic energy vary as the forcing is increased and the system undergoes qualitative changes of behavior between different regimes, from simple symmetric flow to intermittent chaos. The constraints imposed by thermodynamics on the structure of the model equations and, especially, on the parameterization schemes are brought out.

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C. Nicolis

Abstract

A dynamical approach to atmospheric analogs extending the statistical formulation by Toth and Van den Dool is developed. Explicit analytical formulas for the probability and the mean recurrence time of analogs displaying the system’s intrinsic time scales are provided for both discrete and continuous time dynamical systems and are evaluated numerically on a representative model. The analysis reveals strong dependence of recurrence times of analogs on the local properties of the attractor and a pronounced variability around their mean. Finally, the formulation is extended to stochastically forced systems such as a red noise atmosphere.

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C. Nicolis

Abstract

The global observational network is known to be highly inhomogeneous and fractally distributed in space. In this paper the spatial distribution of stations in a network is studied in connection with the type of dynamics exhibited by the atmospheric fields. The existence of an optimal distribution capable of capturing the complexity of this dynamics with a small prescribed error, and the role of deviations of the network distribution from a homogeneous one in this observational error are explored. For sparsely distributed stations like those of the present-day network, it is found that inhomogeneous distributions may generate a record that is closer to reality than homogeneous ones.

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C. Nicolis

Abstract

The different modes of the early stages of the response of a forecasting model to a small error in the boundary conditions are analyzed. A general formulation of the problem based on the use of Green’s functions is developed and implemented on systems in which the operators acting on the spatial coordinates of the fields involved are diffusion-like and advection-like. It is shown that the generic behavior displays a nonanalytic structure not reducible to a power of time for short times.

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S. Vannitsem and C. Nicolis

Abstract

The dynamical properties of forecasts corrected using model output statistics (MOS) schemes are explored, with emphasis on the respective role of model and initial condition uncertainties. Analytical and numerical investigations of low-order systems displaying chaos indicate that MOS schemes are able to partly correct the impact of both initial and model errors on model forecasting. Nevertheless the amplitude of the correction is much more sensitive to the presence of (state dependent) model errors, and if initial condition errors are much larger than model uncertainties then MOS schemes become less effective. Furthermore, the amplitude of the MOS correction depends strongly on the statistical properties of the phase space velocity difference between the model and reference systems, such as its mean and its covariance with the model predictors in the MOS scheme. Large corrections are expected when the predictors are closely related to the sources of model errors. The practical implications of these results are briefly discussed.

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