ON THE FLUCTUATION-RESPONSE RELATION IN GEOPHYSICAL SYSTEMS

International Journal of Modern Physics B Vol. 23, Nos. 28 & 29 (2009) 5515 5529 c World Scientific Publishing Company ON THE FLUCTUATION-RESPONSE RELATION IN GEOPHYSICAL SYSTEMS GUGLIELMO LACORATA Institute of Atmospheric and Climate Sciences, National Research Council, Str. Monteroni, I-73100 Lecce, Italy g.lacorata@isac.cnr.it ANDREA PUGLISI CNR-INFM-SMC and Dipartimento di Fisica, Universitá La Sapienza, p.le A. Moro 2, I-00185, Roma, Italy andrea.puglisi@roma1.infn.it ANGELO VULPIANI Dipartimento di Fisica, Universitá La Sapienza, INFN and CNISM, p.le A. Moro 2, I-00185, Roma, Italy angelo.vulpiani@roma1.infn.it Received 21 October 2009 One of the major issues concerning the study of a dynamical system is the response to perturbations. In climate dynamics, for example, it is of major interest to understand how a given variable, e.g., the temperature, is sensitive to alterations of some other component of the system, e.g., the greenhouse gas concentration. We review the connection between equilibrium and non-equilibrium properties, also known as Fluctuation-Relaxation Relation, and its main aspects in chaotic and turbulent systems. We consider, in particular, the effects of the fast variables on the slow variables in a multiscale system, as far as the sensitivity properties are concerned. Two examples about (widely speaking) climate modelling are discussed: the Lorenz-96 model and the double-potential well model. Both of them, despite their apparent simplicity, hide the same kind of interesting features of much more complex systems. Keywords: Climate dynamics; Lorenz-96 model; Fluctuation. 1. Introduction One important aspect of climate dynamics is understanding the features of the response to perturbations of the external forcings, or of the control parameters. An interesting aspect is the so-called Fluctuation-Response relation (FRR), i.e., the possibility, at least in principle, to understand the behavior of a system under perturbations (e.g., a volcanic eruption, or a change of the CO 2 concentration) in terms of the knowledge obtained from its past time history 17,18,9. The basic point is how to express the response in terms of correlation functions of the unperturbed system. 5515

5516 G. Lacorata, A. Puglisi & A. Vulpiani This field has been initially developed in the context of equilibrium statistical mechanics of Hamiltonian systems; this generated some confusion and misleading ideas on its validity. For instance, some authors claimed that in fully developed turbulence, which is a non-hamiltonian and non-equilibrium system, there is no relaxation between spontaneous fluctuations and relaxation to the statistical steady state 30. As a matter of fact, it is possible to show that a generalized FRR holds under rather general hypotheses 8,10,24 : the FRR is also valid in non Hamiltonian systems. Starting from the seminal works of Leith 17,18, who proposed the use of FRR for the response of the climatic system to changes in the external forcing, many authors tried to apply this relation to different geophysical problems, ranging from simplified models 2, to general circulation models 27,7 and to the covariance of satellite radiance spectra 12. Often, the FRR has been invoked in its Gaussian version, see below, which has been used as a kind of approximation, often without a precise idea of its limits of applicability. For recent works on the application of the FRR to the sensitivity problem and the predictability see 1. The paper is organised as follows. In section 2 we present a general FRR which holds under very general hypothesis. Section 3 is devoted to a discussion of gaussian and non-gaussian hydrodynamic models. Section 4 deals with a climatic model and the problem of the fast variable modeling. In section 5 one can find some conclusions and considerations. 2. A General Fluctuation-Relaxation Relation Let us consider a dynamical system X(0) X(t) = U t X(0) whose time evolution can even be not completely deterministic (e.g., stochastic differential equations), with states X belonging to a N-dimensional vector space. We assume: a) the existence of an invariant probability distribution ρ(x), for which some absolute continuity type conditions are required; b) the mixing character of the system (from which its ergodicity follows). At time t = 0 we introduce a perturbation δx(0) on the variable X(0). For the quantity δx i (t), in the case of an infinitesimal perturbation δx(0) = (δx 1 (0) δx N (0)) one obtains: δx i (t) = j R ij (t)δx j (0). (1) where the linear response functions (according to FRR) are ln ρ(x) R ij (t) = X i (t). (2) X j In the following () indicates the average on the unperturbed system, while () indicates the mean value of perturbed quantities. Let us derive Eq. (2). At the first step we study the behavior of one component of x, say x i, when the system, described by ρ(x), is subjected to an initial (non- t=0

On the Fluctuation-Response Relation in Geophysical Systems 5517 random) perturbation such that x(0) x(0) + x 0. This instantaneous kick a modifies the density of the system into ρ (x), related to the invariant distribution by ρ (x) = ρ(x x 0 ). We introduce the probability of transition from x 0 at time 0 to x at time t, W (x 0, 0 x, t). For a deterministic system, with evolution law x(t) = U t x(0), the probability of transition reduces to W (x 0, 0 x, t) = δ(x U t x 0 ), where δ( ) is the Dirac s delta. Then we can write an expression for the mean value of the variable x i, computed with the density of the perturbed system: x i (t) = x i ρ (x 0 )W (x 0, 0 x, t) dx dx 0. (3) The mean value of x i during the unperturbed evolution can be written in a similar way: x i (t) = x i ρ(x 0 )W (x 0, 0 x, t) dx dx 0. (4) Therefore, defining δx i = x i x i, we have: δx i (t) = x i F (x 0, x 0 ) ρ(x 0 )W (x 0, 0 x, t) dx dx 0 = x i (t) F (x 0, x 0 ) (5) where [ ] ρ(x0 x 0 ) ρ(x 0 ) F (x 0, x 0 ) =. (6) ρ(x 0 ) Let us note here that the mixing property of the system is required so that the decay to zero of the time-correlation functions assures the switching off of the deviations from equilibrium. For an infinitesimal perturbation δx(0) = (δx 1 (0) δx N (0)), if ρ(x) is nonvanishing and differentiable, the function in (6) can be expanded to first order and one obtains Eq. (1) with a response given by Eq. (2). One can easily repeat the computation for a generic observable A(x): δa (t) = j A(x(t)) ln ρ(x) x j t=0 δx j (0). (7) For Langevin equations, the differentiability of ρ(x) is well established. On the contrary, one could argue that in a chaotic deterministic dissipative system the above machinery cannot be applied, because the invariant measure is not smooth at all. Typically in chaotic dissipative systems the invariant measure is singular, however the previous derivation of the FRR is still valid if one considers perturbations along the expanding directions. For a mathematically oriented presentation see Ruelle 31 and Cessac and Sepulchre 6. a The study of an impulsive perturbation is not a severe limitation, e.g., in the linear regime from the (differential) linear response one understands the effect of a generic perturbation.

5518 G. Lacorata, A. Puglisi & A. Vulpiani Let us notice that a small amount of noise, that is always present in a physical system, smoothen the ρ(x) and the FRR can be derived. We recall that this beneficial noise has the important role of selecting the natural measure, and, in the numerical experiments, it is provided by the round-off errors of the computer. Usually, in non Hamiltonian systems, the shape of ρ(x) is not known, therefore relation (2) does not give a very detailed information. On the other hand (2) shows that, anyway, there exists a connection between the mean response function R ij and some suitable correlation function, computed in the unperturbed systems. 3. Fluctuation-Relaxation Relation in Chaotic Fluid Dynamics Systems In this section we present some numerical results on chaotic systems which are able to catch the basic statistical features of fluids. Such systems are not Hamiltonian and can be even non-gaussian, therefore they are natural candidates for a discussion of the FRR in a non-standard context. One important nontrivial class of systems with a Gaussian invariant measure, i.e., ln ρ(x) = i,j α ijx i X j + const. where {α ij } is a positive symmetric matrix, is the inviscid hydrodynamics, where the Liouville theorem holds, and a quadratic invariant exists 13,15,5. In these cases the elements of the linear response matrix can be written in terms of the usual correlation functions, C ik (t) = X i (t)x k (0), as: R ij (t) = α jk X i (t)x k (0). (8) k Let us first discuss some numerical tests of the FDR in simplified models of fluid dynamics. Such systems are far from being realistic, but are nontrivial and share the main features of the Navier- Stokes, or the Euler equations. Consider the model: dy n dt = k n (Y n+1 Y n+2 + Y n 1 Y n 2 2Y n+1 Y n 1 ), (9) with n = (1, 2,..., N), the periodic condition Y n+n = Y n and k n = α β n, with β > 1, for n = 1, 2,..., N/2, with the mirror property k n+n/2 = k N/2+1 n. This system, originally introduced as a toy model for chaos in fluid mechanics 28,4, contains some of the main features of inviscid hydrodynamics: (a) quadratic interactions; (b) a quadratic invariant, I = N n=1 (Y n 2 /k n ); (c) the Liouville theorem holds. For sufficiently large N the distribution of each variable Y n is Gaussian, and < Y n Y m >=< Yn 2 > δ nm k n. As a consequence the simplest FDR, i.e., Eq. (8), holds for each of the variables. On the other hand, each Y n has here its own characteristic time. Figure 1 shows the (expected) validity of (8), however the shape of C nn (t) changes with n. The correlation time of a variable Y n behaves as τ C (k n ) kn 3/2. The exponent of this scaling law can be easily explained with a dimensional argument, by noticing that < Yn 2 > k n, and from (9) the characteristic time results to be just τ C (k n ) < Yn 2 >/(k n < Yn 2 >) k 3/2 n.

On the Fluctuation-Response Relation in Geophysical Systems 5519 1 0.8 C n,n (t), R n n (t) 0.6 0.4 0.2 0-0.2 0.01 0.1 1 10 100 t Fig. 1. FDRs for the six fastest variables of the model (9), with N = 20, α = 0.005 and β = 1.7, n = 5, 6, 7, 8, 9, 10. Thin lines represent the normalized correlation functions C n,n(t)/c n,n(0). Self-response functions R n,n are plotted with statistically computed error bars. Let us now discuss more interesting cases, i.e. the chaotic dissipative systems with non Gaussian statistics, called shell models which have been introduced to study the turbulent energy cascade and share many statistical properties with turbulent three dimensional velocity fields 5,21. Let us consider a set of wave-numbers k n = 2 n k 0 with n = 0,..., N, and the shell-velocity (complex) variables u n (t) which must be understood as the velocity fluctuation over a distance l n = kn 1. We present numerical results for a particular choice, the so-called Sabra model 21,3 : ( ) d dt + νk2 n u n = i[k n u n+1u n+2 + bk n 1 u n+1 u n 1 + (1 + b)k n 2 u n 2 u n 1 ] + f n (10) where b is a free parameter, ν is the molecular viscosity and f n is an external forcing acting only at large scales, necessary to maintain a statistically stationary state. We note that the shell models have the same type of quadratic nonlinearities as the Navier-Stokes equations in the Fourier space. In addition in the inviscid and n u n 2 ) is conserved and the volumes unforced limit (ν = f n = 0) the energy ( 1 2 in phase space are preserved. Because of the above properties, the set of coupled ordinary differential equations (11) possesses the features necessary to mimic the Navier-Stokes non-linear evolution. The main, strong, difference with the previous inviscid model (9) is the existence of a mean energy flux from large to small scales, which drives the system towards a strongly non-gaussian stationary state 5. The shell models present the same qualitative difficulties as the original Navier-Stokes eqs.: strong non-linearity and far from equilibrium statistical fluctuations. The most

5520 G. Lacorata, A. Puglisi & A. Vulpiani striking quantitative feature of the fully developed turbulence, i.e., the highly non- Gaussian statistics and the existence of anomalous scaling laws for the velocity moments, is reproduced in shell models, in good quantitative agreement with the experimental results 5. One has: < u n p > k ζ(p) n, (11) where ζ(p) (p/2)ζ(2), which implies that the velocity PDF s at different scales cannot be rescaled by any change of variables. Let us now examine the numerical results concerning the response functions in the shell model 4. A study of the normalized diagonal responses, R nn (t), for k n in the inertial range, n [7, 14], shows that the characteristic response time τ R (k n ) decreases as k n increases. In Figure 2, one sees a clear difference between the response and the correlation function: this is an additional indication that the inertial-range statistics are very far from Gaussian. Although the R nn (t) do not coincide with the (normalized) C nn (t)/c nn (0), the times τ C (k n ) and τ R (k n ) are not very different τ C (k n ) τ R (k n ), in agreement with the numerical result obtained by Kraichnan with the DIA approximation at moderate Reynolds numbers 14. In addition, one has a scaling behavior τ R (k n ) τ C (k n ) kn γ, where γ is close to the value 2/3 of the Kolmogorov scaling. 4. An Application to Climate Climate dynamics is described by systems with a large number of components and with many very different time scales. Even using modern supercomputers, it is not possible to simulate all the relevant scales of the climate dynamics, which involves processes with characteristic times ranging from days (atmosphere) to 10 2-10 3 years (deep ocean and ice shields) 23. Fig. 2. Comparison between the averaged Response Function, R nn(t), (top) and the (normalized) self-correlation, C n,n(t)/c nn(0) (bottom) for the shell n = 10.

On the Fluctuation-Response Relation in Geophysical Systems 5521 For the sake of simplicity, we consider the FRR issue in the case in which the state variables evolve over two very different time scales: dx s dt = f(x s, X f ) (12) dx f = 1 dt ɛ g(x s, X f ), (13) where X s and X f indicate the slow and fast state vectors, respectively, ɛ 1 is the ratio between fast and slow characteristic times, and both f and g are O(1). From the practical point of view, one basic question is to derive effective equations for the slow variables, e.g., the climatic observable, in which the effects of the fast variables, e.g., high frequency forcings, are taken into account by means of stochastic parameterization. Under rather general conditions 11 one has the result that, in the limit of small ɛ, the slow dynamics is ruled by a Langevin equation with multiplicative noise: dx s = f eff (X s ) + σ(x s )η, (14) dt where η is a white-noise vector, i.e., its components are Gaussian processes such that η i (t) = 0 and η i (t)η j (t ) = δ ij δ(t t ). Although there exist general mathematical results 11 on the possibility to derive eq. (14) from (12) and (13), in practice one has to invoke (rather crude) approximations based on physical intuition to determine the shape of f eff and σ 25. For a more rigorous approach in some climate problems see 22. In the following, we analyse and discuss two models which, in spite of their apparent simplicity, contain the basic features, and the same difficulties, of the general multiscale approach: the Lorenz-96 model 19 and a double-well potential with deterministic chaotic forcing. 4.1. The Lorenz-96 model First, let us consider the Lorenz-96 system 19, introduced as a simplified model for the atmospheric circulation. Define the set {x k (t)}, for k = 1,..., N k, and {y k,j (t)}, for j = 1,..., N j, as the slow large-scale variables and the fast small-scale variables, respectively (being N k = 36 and N j = 10). Roughly speaking, the {x k } s represent the synoptic scales while the {y k,j } s represent the convective scales. The forced dissipative equations of motion are: dx k dt N j = x k 1 (x k 2 x k+1 ) νx k + F + c 1 y k,j (15) dy k,j = cby k,j+1 (y k,j+2 y k,j 1 ) cνy k,j + c 1 x k, (16) dt where: F = 10 is the forcing term, ν = 1 is the linear damping coefficient, c = 10 is the ratio between slow and fast characteristic times, b = 10 is the relative amplitude j=1

5522 G. Lacorata, A. Puglisi & A. Vulpiani 1 0.8 0.6 C jj (t), R jj (t) 0.4 0.2 0-0.2-0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 3. Lorenz-96 model: autocorrelation C jj (t) (full line) and self-response R jj (t) (+) of the fast variable y k,j (t) (k = 3, j = 3). The statistical error bars on R jj (t) are of the same size as the graphic symbols used in the plot. t between large scale and small scale variables, and c 1 = c/b = 1 is the coupling constant that determines the amount of reciprocal feedback. Let us consider, first, the response properties of fast and slow variables, see Figs. 3 and 4. In Fig. 3, the autocorrelation C jj (t) and self-response R jj (t) refer to the fast variable y k,j (t), with fixed k and j. It is well evident how, even in absence of a precise agreement between autocorrelations and self-response functions (due to the non Gaussian character of the system), the correlation of the slow (fast) variables have at least a qualitative resemblance with the response of the slow (fast) variables themselves. The structure of the Lorenz-96 model includes a rather natural set of quantities that suggests how to parameterize the effects of the fast variables on the slow variables, for each k. Let us indicate with z k = N j j=1 y k,j the term containing all the N j fast terms in the equations for the N k slow modes. Replacing the deterministic terms {z k } s in the equations for the {x k } s with suitable stochastic processes, one obtains an effective model able to reproduce the main statistical features of the slow components of the original system 16. Basically, in the effective model for the slow variables, one parameterizes the effects of the fast variables with a suitable renormalization of the forcing, F F + F, of the viscosity, ν ν + ν, and the

On the Fluctuation-Response Relation in Geophysical Systems 5523 1 0.8 0.6 C kk (t), R kk (t) 0.4 0.2 0-0.2-0.4 0 0.5 1 1.5 2 2.5 t Fig. 4. Lorenz-96 model: autocorrelation C kk (t) (full line) and self-response R kk (t), with statistical error bars, of the slow variable x k (t) (k = 3). addition of a random term. In other words, we replace the z k = N j j=1 y k,j terms in (15) with stochastic processes z k depending on the slow variables x k : dx k dt = x k 1 (x k 2 x k+1 ) νx k + F + c 1 z k (17) where z k = 1 c 1 ( ν x k + F + c 2 η k ) (18) with c 1 is a new coupling constant, and η k is a normalized white noise. We notice that eq. (17) has the same form of eq. (15). With a proper choice of ν, F and c 2 one can reproduce the statistics of x k and z k to a very good extent. Of course the above described parameterization of the fast variables is inspired to the general philosophy of the Large-Eddy Simulation of turbulent geophysical flows at high Reynolds numbers 26,32. Let us come back to the response problem. Of course the mean response of a slow variable to a perturbation on a fast variable is zero. However, this does not mean that the effect of the fast variables on the slow dynamics is not statistically relevant. Let us introduce the quadratic response of x k (t) with respect to an infinitesimal

5524 G. Lacorata, A. Puglisi & A. Vulpiani 6 5 4 R sf (q) (t) 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 t Fig. 5. Lorenz-96 model: quadratic cross-response function R (q) sf (t) for the deterministic model (full line), for the stochastic model when the slow variables evolve with the same noise realization for all components except one (dashed line), and when the slow variables evolve with independent noise realization for each components (dotted line). perturbation on y k,j (0), for fixed k and j: ] 1/2 [δx k (t) 2 R (q) kj (t) = δy k,j (0) Considered that in all simulations the initial impulsive perturbations on the y k,j is kept constant, δy k,j (0) =, with y 2 k,j 1/2, it is convenient to take the average of (19) over all j s, at a fixed k, and introduce the quantity: R (q) sf (t) = N j N j j=1 (19) R (q) kj (t) (20) where with s and f we label the slow and fast variables, respectively. In the case of the Lorenz-96 system, all the y k,j variables, at fixed k, are statistically equivalent, and have identical coupling with x k, so that R (q) sf (t)/ coincides with R(q) (t). We report in Fig. 5 the behavior of R (q) sf (t), for both (15) and (17). This dependence of the slow dynamics on the fast variables is the analogous of the strong dependence on the noise realization in the context of noisy systems 29. As regards to the stochastic model, the analogous of (20) is defined as follows. One studies the evolution of kj

On the Fluctuation-Response Relation in Geophysical Systems 5525 δx k (t) as difference of two trajectories obtained with two different realizations of the {η k } s. 4.2. A simplified model In order to grasp the essence of systems with fast and slow variables, we discuss now a very simple model in which the climatic variable fluctuates between two states. Consider a four dimensional state vector q = (q 0, q 1, q 2, q 3 ) whose evolution is given by: dq 0 dt = V q 0 + cq 1 V = Hq 2 0 + q3 0 4/4 + H 2 (21) dq 1 dt = 1 ɛ [ σ L(q 1 q 2 )] (22) dq 2 dt = 1 ɛ [ q 1q 3 + r L q 1 q 2 ] (23) dq 3 dt = 1 ɛ [q 1q 2 b L q 3 ] (24) The above equation system will be named the deterministic DW model. The subsystem formed by (22), (23) and (24) is nothing but the well-known Lorenz-63 model 20, in which the constant ɛ has the function of rescaling the characteristic time. The presence of the coupling (c 0) between slow and fast variables can induce transitions between the two valleys. The parameters in (21-24) are fixed to the following values: σ L = 10, r L = 28, b L = 8/3, i.e., the classical set-up corresponding to the chaotic regime for the Lorenz-63 system; H = 4, the height of the barrier; c = 0.5, the coupling constant that rules the transition time scale of q 0 (t) between the two valleys; by setting ɛ = 1, the ratio ɛ between fast and slow characteristic times, see (12) and (13), is O(10 1 ). Since the time scale of the q 0 (t) well-to-well transitions may be considerably longer, depending on 1/c, than the characteristic time of q 1 (t), of order O(1), we refer to q 0 as the slow variable, or the low-frequency observable, and to q 1 as the fast variable, or the high-frequency forcing, of the deterministic DW model. In the limit of time scales separation we can consider a stochastic model for the slow variable q 0 (t), obtained by replacing the fast variable q 1, in the equation for q 0, with a white noise: dq 0 dv (t) = + σ ξ(t), (25) dt dq 0 where ξ(t) is a Gaussian process with ξ(t) = 0 and ξ(t)ξ(t ) = δ(t t ). We call eq. (25) the WNDW model. The value of σ is determined by requiring that the PDFs of the well-to-well transition times have the same asymptotic behavior (i.e., exponential tail with the same exponent). Because of the skew structure of the system (21-24), i.e., the fast dynamics drives the slow dynamics but without counter-feedback, one expects that, at the least in

5526 G. Lacorata, A. Puglisi & A. Vulpiani 0.7 0.6 0.5 (q 0 ) 0.4 0.3 0.2 0.1 0-3 -2-1 0 1 2 3 q 0 Fig. 6. PDFs of the slow variable q 0 for the DW model with ɛ = 1, i.e., ɛ 0.1 (full line), the WNDW model (dashed line) and the DW model with ɛ = 10 2, i.e., ɛ 10 3 (dotted line). In the limit ɛ 0, the PDFs of the deterministic model and of the stochastic model coincide. the limit of large time scale separation, the joint PDF can be factorized, with an asymptotic PDF for q 0 of the form ρ 0 = K e V eff (q 0), where K is a normalization constant. Since the statistics is far from being Gaussian, the correct correlation function which satisfies the FR theorem, for the slow variable, has the form: C(t) = q 0 (t) ρ ɛ(q 0, q 1, q 2, q 3 ) q 0 t=0 where ρ ɛ (q 0, q 1, q 2, q 3 ) is the (unknown) joint PDF of the state variable of the system at a fixed ɛ. In the limit of large time separation, i.e., for ɛ 0, one expects that the asymptotic PDF ρ 0 (q 0, q 1, q 2, q 3 ) is factorized: (26) ρ 0 (q 0, q 1, q 2, q 3 ) = Ke V eff (q 0) ρ L (q 1, q 2, q 3 ) (27) where K is a normalization constant, and ρ L is the PDF of the Lorenz-63 state variable. Under this condition, the right correlation function predicted by the FRR has a relatively simple form: C(t) = q 0 (t) V eff (q 0 ) q 0 t=0 (28)

On the Fluctuation-Response Relation in Geophysical Systems 5527 1 0.8 R 00 (t), C 00 (t), C(t) 0.6 0.4 0.2 0-0.2 0 2 4 6 8 10 12 14 16 18 20 t Fig. 7. WNDW model: autocorrelation C 00 (t) (dashed line), self-response R 00 (t), with statistical error bars, and the correlation function C(t) predicted by the FRR (full line). where V eff indicates the effective potential. For ɛ 10 1 (corresponding to ɛ = 1) we have checked numerically that the joint PDF is not yet factorized, while for a very small ratio between the characteristic times, ɛ 10 3 (corresponding to ɛ = 10 2 ), the form (27) holds and, taking V eff V, we obtain a very good agreement between R 00 (t) and C(t), we do not show the result which is rather similar to that in Fig. 7. Let us notice that, in this case, because of the skew structure of the original system, the stochastic modeling is (relatively) simple and, differently from the generic case, the noise is additive. The shape of the PDF of the slow variable changes drastically when passing from small to large time scale separation, as shown in Fig. 6. The FR properties of the WNDW model are reported in Fig. 7. The slow variable is distributed according to e V (q0)/k, with K = σ 2 /2, and the FR theorem prediction is verified, i.e., one has a good agreement between R 00 (t) and the correlation function C(t). As far as the sensitivity to noise is concerned, even this simple model displays properties qualitatively similar to those seen for the Lorenz-96 model, that is a mean quadratic separation in time between two low-frequency climatic states, starting from the same initial conditions but evolving with different realizations of the random forcings. This determines an intrinsic time scale after which even the optimal model can only give statistical predictions.

5528 G. Lacorata, A. Puglisi & A. Vulpiani 5. Conclusive Remarks We discussed the general vaildity of the FRR in non-hamiltonian systems with particular emphasis on geophysics. The existence of a generalized FRR allows for a link between the average relaxation of perturbations and statistical features of the unperturbed systems (namely correlation functions). Although one has non Gaussian statistics, the correlation functions of the slow (fast) variables have at least a qualitative resemblance with the response functions to perturbations on the slow (fast) degrees of freedom. The average response function of a slow variable to perturbations of the fast degrees of freedom is zero, nevertheless the impact of the fast dynamics on the slowly varying components cannot be neglected. This fact is clearly highlighted by the behavior of a suitable quadratic response function. Such a phenomenon, which can be regarded as a sort of sensitivity of the slow variables to variations of the fast components, has an important consequence for the modeling of the slow dynamics in terms of a Langevin equation. Even an optimal model (i.e., able to mimic autocorrelation and self-response of the slow variable), beyond a certain intrinsic time interval, can give just statistical predictions, in the sense that, at most, one can hope to have an agreement among the statistical features of system and model. In stochastic dynamical systems, one has to deal with a similar behavior: the relevant complexity of the systems is obtained by considering the divergence of nearby trajectories evolving under two different noise realizations 29. Therefore a good model for the slow dynamics (e.g., a Langevin equation) must show a sensitivity to the noise. References 1. R. Abramov and A. J. Majda, New approximations and tests of linear fluctuationresponse for chaotic nonlinear forced-dissipative dynamical systems. J. Nonlinear Sci., 18:303, 2008. 2. R. E. Bell, Climate sensitivity from fluctuation dissipation: Some simple model tests. J. Atmos. Sci., 37:1700, 1980. 3. L. Biferale, Shell models of energy cascade in turbulence. Ann. Rev. of Fluid Mech., 35:441, 2003. 4. L. Biferale, I. Daumont, G. Lacorata, and A. Vulpiani, Fluctuation-response relation in turbulent systems. Phys. Rev. E, 65:016302, 2002. 5. T. Bohr, M. H. Jensen, G. Paladin, and A. Vulpiani, Dynamical Systems Approach to Turbulence. (Cambridge University Press, Cambridge U.K., 1998. 6. B. Cessac and J.-A. Sepulchre, Linear response, susceptibility and resonance in chaotic toy models. Physica D, 225:13, 2007. 7. I. Cionni, G. Visconti, and F. Sassi, Fluctuation dissipation theorem in a general circulation model. Geophys. Res. Lett., 31:L09206, 2004. 8. U. Deker and F. Haake, Fluctuation-dissipation theorems for classical processes. Phys. Rev. A, 11:2043, 1975. 9. V. P. Dymnikov and A. S. Gritsoun, Climate model attractors: chaos, quasi-regularity and sensitivity to small perturbations of external forcing. Nonlin. Proc. in Geoph., 8:201, 2001. 10. M. Falcioni, S. Isola, and A. Vulpiani, Correlation functions and relaxation properties

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