On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature

Abstract

Climate science employs a hierarchy of models, trading the tractability of simplified energy balance models (EBMs) against the detail of Global Circulation Models. Since the pioneering work of Hasselmann, stochastic EBMs have allowed treatment of climate fluctuations and noise. However, it has recently been claimed that observations motivate heavy-tailed temporal response functions in global mean temperature to perturbations. Our complementary approach exploits the correspondence between Hasselmann's EBM and the original mean-reverting stochastic model in physics, Langevin's equation of 1908. We propose mapping a model well known in statistical mechanics, the Mori-Kubo Generalised Langevin Equation (GLE) to generalise the Hasselmann EBM. If present, long range memory then simplifies the GLE to a fractional Langevin equation (FLE). We describe the corresponding EBMs that map to the GLE and FLE, briefly discuss their solutions, and relate them to Lovejoy et al's new Fractional Energy Balance Model (FEBE).

Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus

“…This computation will additionally show that assuming the constancy of σ t as in [7] is not strictly needed, at least from the mathematical viewpoint, although this result can be recovered as a particular case in the present setting. Let us also note that many generalizations of the classical Langevin equation have been studied along the years, since classical developments [16,17] to more recent extensions [18,19], but however none of them, to the best of our knowledge, includes the backward stochastic differential equation approach considered herein. On the other hand, the use of a stochastic amplitude of fluctuations seems not to be frequently used in this field, while stochastic volatility models are popular in finance [20], a field in which backward stochastic differential equations have been commonly employed too [12].…”