We investigate the effects of a time-correlated noise on an extended chaotic system. The chosen model is the Lorenz'96, a kind of toy model used for climate studies. The system is subjected to both temporal and spatiotemporal perturbations. Through the analysis of the system's time evolution and its time correlations, we have obtained numerical evidence for two stochastic resonance-like behaviors. Such behavior is seen when a generalized signal-to-noise ratio function are depicted as a function of the external noise intensity or as function of the system size. The underlying mechanism seems to be associated to a noise-induced chaos reduction. The possible relevance of those findings for an optimal climate prediction are discussed, using an analysis of the noise effects on the evolution of finite perturbations and errors.
It is well known that noise and chaos represent, respectively, two kinds of essentially different phenomena. The former is induced by genuine stochastic sources, while the randomness of the later is pseudo and is deterministic in its origin. The spatiotemporal chaos is intrinsically irregular in both space and time and represents a prototype of deterministic randomness. It is interesting to see what would come about as a result of the interaction between these two irregularities that are essentially distinct.
As the influence of noises on low-dimensional dynamics systems has been studied extensively [1,11], much research interest has nowadays shifted to spatially extended system, a situation that is apparently much more complicated [12].
In the spatially extended situations, the way in which the noise takes effect is not obvious and the deterministic description usually cannot give the right results. It is known that noiseinduced phenomena have come about as a consequence of nonlinear interaction between the noise and the deterministic dynamics. The spatiotemporal stochastic resonance are believed to have potential importance, for instance, in the area of signal and image processing, pattern formation, social and economical as well as climate dynamics [1,3].
Here we consider a fully study on the Lorenz'96 model, driven by two kinds of perturbations, a deterministic perturbation given by the own chaotic behavior of the model and a stochastic one which we have assumed as an effective way of including a more realistic evolution. The relevance of this model rest on the fact that it represents a simple but still realistic description of some physical properties of global atmospheric models.
Manifestations of noise on other characteristics of spatiotemporal chaos such as Lyapunov exponents and dimensions have not been considered. The results presented here provided a first step in order to explore the possibilities of complex dynamics coming out from the interaction between chaos and noise clearly. Further investigation along this line is desirable. This work is organized as follows: in section II we describe the Lorenz'96 model assuming that its evolution is governed by both a deterministic and a stochastic processes. In section III we present and discuss numerical simulations of the Lorenz'96 equation, describing qualitatively the interaction of the real noise and the deterministic noise on the time evolution of the system. In section IV we discuss the important problem of the perturbations and errors in the Lorenz'96 evolution. Finally in section V we present the conclusions of our work as well as possible implications and the relevance of this study on the actual climate evolution.
The equations corresponding to the Lorenz'96 model are
where ẋj indicates the time derivative of x j
with Ψ j (t) a dichotomic process. That is, Ψ j (t) adopts the values ±∆ with a transition rate γ: each state changes according to the waiting time distribution ϕ i (t) ∼ e -γt . The noise intensity for this process is defined through ξ = ∆ 2 2γ . In this work we have supposed that the system is subjected to a spatiotemporal perturbation as well as a temporal one.
The first perturbation is achieved when F depends on both j and t variables, meanwhile for temporal perturbation the F function only depends on t. In order to simulate a scalar meteorological quantity extended around a latitude circle, we consider periodic boundary
As indicated before, the Lorenz'96 model has been heuristically formulated as the simplest way to take into account certain properties of global atmospheric models. The terms included in the equation intend to simulate advection, dissipation, and forcing respectively.
In contrast with other toy models used in the analysis of extended chaotic systems and based on coupled map lattices, the Lorenz'96 model exhibits extended chaos when the F parameter exceeds a determinate threshold value (F > 9/8) with a spatial structure in the form of moving waves. The length of these waves is close to 5 spatial units. It is worth noting that the system has scaled variables with unit coefficients, hence the time unit is the dissipative decay time. In addition we adjust the value of the parameter F to give a reasonable signal to noise ratio (Lorenz considered F = 8), so the model could be most adequate to perform basic studies of predictability.
As a measure of the SR system’s response we have used the signal-to-noise ratio (SNR) [1]. To obtain the SNR we need to previously evaluate S(ω), the power spectral density (psd), defined as the Fourier transform of the correlation function [21,22]
where indicates the average over realizations. As we have periodic boundary conditions simulating a closed system, x j (0)x j (τ ) has a homogeneous spatial behavior. Hence, it is enough to analyze the response in a single site.
We consider two forms of SNR. In one hand the usual SNR measure at the resonant frequency ω o (that is, i
