Fractional relaxation noises, motions and the fractional energy balance equation

[摘要] We consider the statistical properties of solutions of the stochasticfractional relaxation equation and its fractionally integrated extensionsthat are models for the Earth's energy balance. In these equations, thehighest-order derivative term is fractional, and it models the energy storage processes that are scaling over a wide range. When driven stochastically, the system is a fractional Langevin equation (FLE) that has been consideredin the context of random walks where it yields highly nonstationarybehaviour. An important difference with the usual applications is that we instead consider the stationary solutions of the Weyl fractional relaxationequations whose domain is −∞ to t rather than 0 to t . An additional key difference is that, unlike the (usual) FLEs – where the highest-order term is of integer order and the fractional term represents a scaling damping – in the fractional relaxation equation, the fractional termis of the highest order. When its order is less than 1 / 2 (this is the main empirically relevant range), the solutions are noises (generalizedfunctions) whose high-frequency limits are fractional Gaussian noises (fGn). In order to yield physical processes, they must be smoothed, and this is conveniently done by considering their integrals. Whereas the basicprocesses are (stationary) fractional relaxation noises (fRn), theirintegrals are (nonstationary) fractional relaxation motions (fRm) that generalize both fractional Brownian motion (fBm) as well as Ornstein–Uhlenbeck processes. Since these processes are Gaussian, their properties are determined by theirsecond-order statistics; using Fourier and Laplace techniques, we analytically develop corresponding power series expansions for fRn and fRm and their fractionally integrated extensions needed to model energy storageprocesses. We show extensive analytic and numerical results on theautocorrelation functions, Haar fluctuations and spectra. We display samplerealizations. Finally, we discuss the predictability of these processes which – due tolong memories – is a past value problem, not an initial value problem (that is used forexample in highly skillful monthly and seasonal temperature forecasts). Wedevelop an analytic formula for the fRn forecast skills and compare it tofGn skill. The large-scale white noise and fGn limits are attained in a slow power law manner so that when the temporal resolution of the series is smallcompared to the relaxation time (of the order of a few years on the Earth), fRn and its extensions can mimic a long memory process with a range ofexponents wider than possible with fGn or fBm. We discuss the implicationsfor monthly, seasonal, and annual forecasts of the Earth's temperature as well as for projecting the temperature to 2050 and 2100.

[学科分类] 自动化工程