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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Discussion papers
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.

Submitted as: research article 20 Aug 2019

Submitted as: research article | 20 Aug 2019

Review status
This discussion paper is a preprint. It is a manuscript under review for the journal Nonlinear Processes in Geophysics (NPG).

Fractional relaxation noises, motions and the fractional energy balance equation

Shaun Lovejoy Shaun Lovejoy
  • Physics, McGill University, 3600 University st. Montreal, Que. H3A 2T8 Canada

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.

Shaun Lovejoy
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Shaun Lovejoy
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Publications Copernicus
Short summary
The difference between the energy that the earth receives from the sun and the energy it emits as black body radiation is stored in a scaling hierarchies of structures in the ocean, soil and hydrosphere. The simplest scaling storage model leads to the Fractional Energy Balance Equation (FEBE). This paper examines the statistical properties of FEBE temperature responses when it is driven by random fluctuations.
The difference between the energy that the earth receives from the sun and the energy it emits...