Fractional relaxation noises, motions and the fractional energy balance equation

Abstract. We consider the statistical properties of solutions of the stochastic fractional relaxation equation that has been proposed as a model for the earth's energy balance. In this equation, the (scaling) fractional derivative term models energy storage processes that occur over a wide range of space and time scales. Up until now, stochastic fractional relaxation processes have only been considered with Riemann-Liouville fractional derivatives in the context of random walk processes where it yields highly nonstationary behaviour. For our purposes we require the stationary processes that are the solutions of the Weyl fractional relaxation equations whose domain is −∞ to t rather than 0 to t. We develop a framework for handling fractional equations driven by white noise forcings. To avoid divergences, we follow the approach used in fractional Brownian motion (fBm). The resulting fractional relaxation motions (fRm) and fractional relaxation noises (fRn) generalize the more familiar fBm and fGn (fractional Gaussian noise). We analytically determine both the small and large scale limits and show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations. Finally, we discuss the prediction of fRn, fRm which – due to long memories is a past value problem, not an initial value problem. We develop an analytic formula for the fRn forecast skill and compare it to fGn. Although the large scale limit is an (unpredictable) white noise that is attained in a slow power law manner, when the temporal resolution of the series is small compared to the relaxation time, fRn can mimick a long memory process with a wide range of exponents ranging from fGn to fBm and beyond. We discuss the implications for monthly, seasonal, annual forecasts of the earth's temperature.