Physical basis of long range persistence in the climate system Oscar J. Mesa and Vijay K. Gupta Universidad Nacional de Colombia, Medell´ ın, Colombia, Area Metropolitana del Valle de Aburra, Medell´ ın, Colombia, CIRES, University of Colorado, Boulder, Colorado ojmesa@unal.edu.co Abstract The Hurst effect remains unsolved, despite its practical and theoretical importance. Non uniformly mixing dynamical systems produce long range correlations when orbits expend long time near neutral fixed points in overall mixing sys- tems. The simple auto oscillatory Daisyworld model that has been used to explore the importance of the atmosphere—biosphere feedbacks is one exam- ple. Depending on the parameters, time series resulting from the numerical integration of these examples exhibit the Hurst effect. We interpret this behavior as a consequence of the existence of fixed neutral points. The Hurst Effect Let q t represent the average water inflow discharge over the period (t - Δt, t), into an ideal reser- voir, which is neither empty nor full. Let the record length be n, Δt =1 and suppose that in each time period the same amount of water is extracted from the reservoir that is equal to the sample mean ¯ q . Therefore, the continuity equation gives the vol- ume of water stored in the reservoir at time t as, S t = S t-1 +(q t - ¯ q ). Let us further assume S 0 =0. For these conditions, define the range of S n as, R n = max 0≤t≤n S t - min 0≤t≤n S t . (1) Harold E. Hurst was interested in estimating the design capacity of a reservoir for the Nile River in Egypt. He considered R n as a measure of the reservoir capacity. He re-scaled the range by the sample standard deviation and called R * n = R n /σ the rescaled adjusted range. He analyzed records of different geophysical variables, and found that R * n grows with the record length n to a power h greater that 0.5, typically close to 0.7. Whereas according to the Functional Central Limit Theo- rem of Probability Theory for a very general class of stochastic processes, the following holds [Bhat- tacharya et al., 1983] E [R * n ] ∼ ( 1 2 πθn) 0.5 , with θ = ∞ X k =-∞ ρ k . (2) ρ k denotes the sequence of lag–k correlation coef- ficients. θ is also known as the “scale of fluctuation of the process” (Taylor [1922]). The “Hurst Effect” is the deviation of the empirically observed values of power h from 0.5. Because of the FCLT either non stationarity or long memory are necessary for the Hurst effect. Finite θ , also called finite memory, depends on the rate of convergence of ρ k to zero. Examples Fractional Brownian noise A stationary Gaussian process with correlation function (Mandelbrot and Wallis [1969]) ρ(s)= 1 2  |s +1| 2H - 2|s| 2H + |s - 1| 2H  , with H ∈ (0, 1). This process is the derivative of a frac- tional Brownian motion (FBM) B H (x), a non- stationary Gaussian process with normally dis- tributed independent increments, B H (x) - B H (y ), with zero mean and variance |x - y | 2H . The FBM is self–similar in the sense that (B H (x) - B H (0)) . = s -H (B H (sx) - B H (0)). Now, the local properties of realizations depend on ρ(s) near s =0. If 1 - ρ(s) ∼|s| -α as s → 0, for some α ∈ (0, 2], then the fractal dimension of the trajectories in an m dimensional space is D = m +1 - α/2 [Gneiting and Schlather, 2004]. The asymptotic behavior of ρ(s) at s →∞ deter- mines presence or absence of long range depen- dence. If ρ(s) ∼|s| -β as s →∞, (3) for some β ∈ (0, 1], then H =1 - β/2. Non–stationary processes Bhattacharya et al. [1983] proved that under cer- tain class of non–stationarity the Hurst effect also arises in models with finite memory, or short- range dependence. Intermittency maps One dimensional interval maps, expanding at every point but at a neutral fixed point p, this means that f (p)= p and that |f 0 (x)| > 1 for every x 6= p and f 0 (p)=1. The fixed point p is repelling, but nearby points remain close to p much longer than in the uniformly expanding case. This is characteristic of intermittency ([Alves and Viana, 2000; Luzzatto, 2004, 2003]). Intermittency maps exhibit slow rates of decay of correlations. Therefore this is a very simple deterministic dy- namical system that produces the Hurst effect. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,2 0,4 0,6 0,8 1 Long memory in Daisyworld The Daisyworld is a simple but well-known physi- cal model of climate [Watson and Lovelock, 1983; Lovelock, 1995; Wood et al., 2008]. We force the system with a time dependant deterministic quasi- periodic solar forcing that includes the annual cy- cle, and the effect of long-time changes in the or- bital parameters of earth [Hartmann, 1994; Laskar et al., 2004]. Our experiments indicate that the forcing is necessary for the mixing to occur in the phase space, and for the trajectories to get near the neutral fixed points that produces long range correlations. In addition, we explore a case in which we add random noise to the solar forcing. It represents the effect of all those physical pro- cesses that are not explicitly represented in the dy- namical equations. H51B-0795, American Geophysical Union Fall Meting, December 2008, San Francisco, California