For more than half a century, researchers have employed mathematical models to better understand the response of the Earth’s climate to internal fluctuations as well as external perturbations, whether anthropogenic or solar. These models range from highly complex and multi-dimensional general circulation models to simplified energy balance models. In recent years, significant effort has focused on multivariate stochastic models and statistical inference, and stochastic climate modelling using the Mori-Zwanzig formalism, which provides a general mathematical framework for the Langevin equation. Some of this work has benefitted from a formal equivalence between the Hasselmann model – an early energy-balance model – and the driven Langevin equation of statistical mechanics.

Much of this work has assumed spectrally white noise terms, which simplifies the mathematics, allowing use of the most mature tools of stochastic calculus. Even so, there have long been physically-motivated arguments for making the driving noise term spectrally red rather than white. As one researcher noted some 25 years ago, “unfortunately there is evidence that [a model of the white noise-driven Langevin type] would be unsatisfactory to capture some of the low frequency phenomena observed in the atmosphere. This is referred to as the infrared climate problem and appears to be caused by non-linear interactions of the chaotic internal weather frequencies that potentially induce a “piling-up” of extra variance at the low frequencies.”

In a recent paper, LML External Fellow Rainer Klages and colleagues examine a number of pathways to including more realistic non-white noise in stochastic climate models. As they note, modern evidence suggests long-range dependence in the climate system based on “1/f” power spectra and other related diagnostics, and a number of recent papers have used models based on fractional Brownian motion to introduce such features into models of climate time series. Yet, the authors argue, an ideal approach to stochastic modelling would permit a more general range of dependency structures than either just the shortest possible range (Markovian) or longest range (“1/f”) behaviours, while allowing both as limiting cases. Just such a formalism has long been available in statistical mechanics in the Mori-Kubo generalised Langevin equation, which extends the Langevin equation with an integral over a kernel replacing its constant damping term.

In the paper, Klages and colleagues propose extending the seminal Hasselmann model via this generalised Langevin equation route, and derive the analogous generalised Hasselmann equation. They then explore how taking a power-law form for the damping kernel leads to the fractional Langevin equation studied by researchers in topics such as anomalous diffusion, and also derive a corresponding fractional Hasselmann equation. Fractionally integrating both sides of this allows the researchers to identify how this perspective relates to a recently discussed fractional energy balance equation.

The paper is Watkins N.W., Chapman S.C., Chechkin A., Ford I., Klages R., Stainforth D.A. (2021) On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature. In: Braha D. et al. (eds) Unifying Themes in Complex Systems X. ICCS 2020. Springer Proceedings in Complexity. Springer, Cham.

It is available at https://arxiv.org/abs/2007.06464v2

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