Physical basis of long range persistence in the climate system

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 qt represent the average water inflow discharge over the period (t − ∆t, t), into an ideal reser-voir, which is neither empty nor full. Let the record length be n, ∆t = 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 q̄. Therefore, the continuity equation gives the vol-ume of water stored in the reservoir at time t as, St = St−1 + (qt − q̄). Let us further assume S0 = 0. For these conditions, define the range of Sn as, Rn = max 0≤t≤nSt − min0≤t≤nSt. (1) Harold E. Hurst was interested in estimating the design capacity of a reservoir for the Nile River in Egypt. He considered Rn as a measure of the reservoir capacity. He re-scaled the range by the sample standard deviation and called R∗n = Rn/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