Lévy noise versus Gaussian-noise-induced transitions in the Ghil–Sellers energy balance model

Abstract

Abstract. We study the impact of applying stochastic forcing to the Ghil–Sellers energy balance climate model in the form of a fluctuating solar irradiance. Through numerical simulations, we explore the noise-induced transitions between the competing warm and snowball climate states. We consider multiplicative stochastic forcing driven by Gaussian and α-stable Lévy – α∈(0,2) – noise laws, examine the statistics of transition times, and estimate the most probable transition paths. While the Gaussian noise case – used here as a reference – has been carefully studied in a plethora of investigations on metastable systems, much less is known about the Lévy case, both in terms of mathematical theory and heuristics, especially in the case of high- and infinite-dimensional systems. In the weak noise limit, the expected residence time in each metastable state scales in a fundamentally different way in the Gaussian vs. Lévy noise case with respect to the intensity of the noise. In the former case, the classical Kramers-like exponential law is recovered. In the latter case, power laws are found, with the exponent equal to −α, in apparent agreement with rigorous results obtained for additive noise in a related – yet different – reaction–diffusion equation and in simpler models. This can be better understood by treating the Lévy noise as a compound Poisson process. The transition paths are studied in a projection of the state space, and remarkable differences are observed between the two different types of noise. The snowball-to-warm and the warm-to-snowball most probable transition paths cross at the single unstable edge state on the basin boundary. In the case of Lévy noise, the most probable transition paths in the two directions are wholly separated, as transitions apparently take place via the closest basin boundary region to the outgoing attractor. This property can be better elucidated by considering singular perturbations to the solar irradiance.

Sensitivity of Neoproterozoic snowball-Earth inceptions to continental configuration, orbital geometry, and volcanism

“…It is worth noting that a recent study investigated noiseinduced tipping towards snowball states in an energy-balance model under different noise types (Lucarini et al, 2022). For noise characterised by probability distributions with long tails, allowing for extreme events such as volcanic eruptions, the shape of the noise was found to be much less important for the tipping behaviour than single extreme events that trigger the tipping.…”

Variational Techniques for a One-Dimensional Energy Balance Model

“…The first consists of changes in factors influencing the climate system, such as variations in greenhouse gas (GHG) concentrations like carbon dioxide (CO 2 ), altering the balance of incoming and outgoing radiation and amplifying the greenhouse effect. Mathematically, this mechanism can be described by assuming that the model depends on one additional parameter, and changes in the parameter lead the model to undergo a bifurcation (Ashwin et al (2012)); the second consists in noise-induced transitions resulting from unresolved processes in climate models or the representation of short-timescale weather as stochastic forcing acting on slow variables, as observed in stochastic reduced models (Imkeller (2001); Lucarini et al (2022)). These two types of transitions correspond to mechanisms recognized to induce climate tipping, that is rapid non-linear changes in the climate system with potentially irreversible and catastrophic consequences (Lenton et al (2008); Scheffer et al (2009); Lenton et al (2012); Lucarini and Bódai (2019); Ghil and Lucarini (2020)).…”

Review article: Large fluctuations in non-equilibrium physics

“…In climate science, the models are comparatively more complicated. However the application of large deviation theory to rather complex models as in Galfi et al (2019), , and Lucarini et al (2022) is encouraging.…”

“…Non-equilibrium is dominant in geophysical and climate phenomena, and we refer to the following papers for the relevance of large deviation estimates in this domain (Galfi et al, 2019;; see also Lucarini et al (2022). In Galfi et al (2019), a simplified model of the general circulation of the atmosphere is adopted and assumed to be a chaotic dynamical system so that probabilistic concepts and methods can be applied.…”