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

The climate system comprises the following five interacting subdomains: the atmosphere, the hydrosphere (water in liquid form), the upper layer of the lithosphere, the cryosphere (water in solid form), and the biosphere (ecosystems and living organisms). The climate is driven by the inhomogeneous absorption of incoming solar radiation, which sets up nonequilibrium conditions. The system reaches an approximate steady state, where macroscopic fluxes of energy, momentum, and mass are present throughout its domain, and entropy is continuously generated and expelled into the outer space. The climate features variability on a vast range of spatial and temporal scales as a result of the interplay of forcing, dissipation, feedbacks, mixing, transport,
chemical reactions, phase changes, and exchange processes between the subdomains (see

In the late 1960s

Only later were these predictions confirmed by actual data. Indeed, geological and paleomagnetic evidence suggests that, during the Neoproterozoic era, between

Additionally, several results indicate that the phase space of the climate system might well be more complex than the scenario of bistability described above. Various studies

Clearly, in the case of autonomous systems where the phase space is partitioned in more than one basin of attraction of the corresponding attractors and the basin boundaries, the asymptotic state of the system is determined by its initial conditions. Things change dramatically when one includes time-dependent forcing which allows for transitions between competing metastable states

Metastability is ubiquitous in nature, and advancing its understanding is a key challenge in complex system science at large

A major limitation of this mathematical framework is the need to rigidly consider Gaussian noise laws, even if considerable freedom is left as to the choice of the spatial correlation properties of the noise. It seems natural to attempt a generalisation by considering the whole class of

Note that

Following

The contribution by

We remark that Gaussian and Lévy noise can be associated with stochastic forcings of a fundamentally different nature. One might think of Gaussian noise as being associated to the impact of very rapid unresolved scales of motion on the resolved ones

We consider here the Ghil–Sellers Earth's EBM

The main challenges of the problem are (a) the fact that we are considering dynamical processes occurring in infinite dimensions

The results obtained confirm that, in the weak noise limit

Furthermore, we find clear confirmation that, in the case of Gaussian noise in the weak noise limit, the escape from either attractor's basin takes place through the edge state. Indeed, the most probable paths for both thawing and freezing processes meet at the edge state and have distinct instantonic and relaxation sections. In turn, for Lévy noise in the weak noise limit, the escapes from a given basin of attraction occur through the
boundary region closest to the outgoing attractor. Hence, the paths are very different from the Gaussian case (especially so for the freezing transition) and, somewhat surprisingly, are identical regardless of the value of

The rest of the paper is organised as follows. In Sect.

The Ghil–Sellers EBM

The equation does not depend explicitly on the time

The empirical functions

The second term –

The term

The last term –

In this study, we consider

Following previous studies

Such a representation allows for a minimal yet still physically relevant description of the system. Indeed, changes in the energy budget of the system (warming versus cooling) are, to a first approximation, related to variations in

In order to analyse the influence of random perturbations on the deterministic dynamics of the climate model described in Sect.

We define

First, let us define the concept of a mild solution in this context. Let

As mentioned above, things are radically different for the special case

By incorporating stochastic forcing into the system, its long-time dynamics change significantly, allowing transitions between the competing basins. This dynamical behaviour is called metastability and is graphically captured by Fig.

In a complete probability space

The mean escape time is then expressed by

We now consider the case

We assume that steady-state conditions and ergodicity are met, and we also assume that the analysing system is bistable and a unique edge state is present at the basin boundary, as in the case studied here. In the case of Gaussian noise, transitions between the competing basins of attraction are not determined by a single event as in the

In the weak noise limit, the most probable path to escape an attractor is defined by a class of trajectories named instantons

In our case, the theory indicates that, if the stochastic forcing is Gaussian, under a rather general hypothesis, the instanton will connect the attractor W/SB with the edge state M, which then acts as gateway for noise-induced transitions. Once the quasi-potential barrier is overcome, a free-fall relaxation trajectory links M with the competing attractor SB/W. For equilibrium systems, (e.g. for gradient flows), where a detailed balance is achieved, the relaxation and instantonic trajectories within the same basin of attraction are identical. On the contrary, for non-equilibrium systems, the relaxation and instantonic trajectories will differ and will only meet at the attractor. (See a detailed discussion of this aspect and of the
dynamical interpretation of the quasi-potential

In general, the maximum likelihood transition trajectory

In the following section, for practical purposes, we construct such optimal transition path in the coarse-grained 2D phase space
(

The metastable behaviour of the solution path of a stochastic energy balance model (Eq.

We solve Eq. (

The time span of integration

For the numerical simulations discussed below, we consider

We remark that, when we consider Lévy noise, it does happen that, for some years, the solar irradiance has negative values. Of course these conditions bear no physical relevance, and are a necessary result of considering unbounded noise. Nonetheless, we have allowed for this to occur in our simulations in order to be able to stick to the desired mathematical framework. We remind the reader that this study does not aim at capturing, with any high degree of realism, the description of the actual evolution of climate. At any rate, as can be understood from the discussion below in Sect.

In what follows, we aim at addressing three main questions: (1) what are the temporal statistics of the

Our analysis confirms that there is a fundamental dichotomy in the statistics of mean escape times between Lévy noise and Gaussian noise-induced transitions.

Figure

The Gaussian case – where no jumps are present – is portrayed in Fig.

Estimates of the mean escape time

Estimates of the exponent

We now explore the geometry of the transition paths associated with the metastable behaviour of the system. We first discuss the case of Gaussian noise because it is indeed more familiar and more extensively studied.

We estimate the transition paths by averaging among the escape plus relaxation trajectories using the run performed with the weakest noise (see Table

In the background of Fig.

Let us provide some physical interpretation of how the transitions occur. Looking at the

The global freezing of the planet associated with the

Figure

There is scarcity of rigorous mathematical results regarding the weak noise limit of the transition paths between competing states in metastable stochastic systems forced by multiplicative Lévy noise. Indeed, the derivation of analytical results for this type of system largely remains an open problem. Recently, for stochastic partial differential equations with additive Lévy and Gaussian noise, the Onsager–Machlup action functional
has been derived in

A two-dimensional projection of the invariant measure on

A striking feature in Fig.

The

Comparison between the supercritical and subcritical paths that are constricted, using singular perturbations and the ensemble of trajectories corresponding to Lévy noise-induced transitions.

Same as Fig.

Based on what is discussed in Sect.

The projections on the 2D phase space spanned by

In Fig.

Further support to the viewpoint proposed here is given by Fig.

The transitions shown in Figs.

It is a well-known that, as a result of the competition between the Boltzmann stabilising feedback and the ice–albedo destabilising feedback, under current astronomical and astrophysical conditions, the climate system is multistable, as at least two competing and distinct climates are present, i.e. the W and the SB. More recent investigations indicate that the partition of the phase space of the climate system might be more complex, as more than two asymptotic states might be present, some of them, possibly, associated with small basins of attraction.

For deterministic multistable systems, the asymptotic state of an orbit depends uniquely on the initial condition, and, specifically, on which basin of attraction it belongs to. The presence of stochastic forcing allows for transitions to occur between competing basins, thus giving rise to the phenomenon of metastability. Gaussian noise as a source of stochastic perturbations has been widely studied by the scientific community in recent years and provided very fruitful insight into the multiscale nature of the climatic time series. However, it has become apparent that more general classes of

As a starting point in this direction, we have studied the influence of different noise laws on the metastability properties of the randomly forced Ghil–Sellers EBM, which is governed by a nonlinear, parabolic, reaction–diffusion PDE. In the deterministic version of the model, we have three steady-state solutions, i.e. two stable, attractive climate states and one unstable saddle, corresponding to the edge state. The stable states correspond to the well-known W and SB climates. There is a fundamental dichotomy in the properties of the noise-induced transitions determined by whether we consider a stochastic forcing of intensity

First, in the weak noise limit

Second, the results obtained for the most probable transition paths confirm that, in the weak noise limit, escapes from basins of attraction driven by Gaussian noise take place through the edge state. Additionally, instantonic and relaxation portions within each basin of attraction are clearly distinct, indicating nonequilibrium conditions that are, yet, qualitatively similar. In turn, Lévy diffusions leave the basin through the boundary region closest to the outgoing attractor, which seems to be the vicinity of the edge state when the thawing transition is considered. The freezing transition, instead, proceeds along a path that is fundamentally different. Finally, the most probable transition paths for the Lévy case appear to depend very weakly on the value of the stability parameter

Our findings provide strong evidence that choosing noise laws other than Gaussian leads to fundamental changes in the metastability properties of a system, both in terms of statistics of the transitions between competing basins of attraction and most probable paths for such transitions. Leaving the door open for general noise laws might be relevant, both for interpreting observational data and for performing modelling exercises for the climate system and complex systems in general.

Let us give an example of the impact of making a wrong assumption on the nature of the acting stochastic forcing. Were we to naively interpret one of the panels in Fig.

Recent developments in data-driven methods based on the formalism of the Kramers–Moyal equation allow one to test accurately whether data are compatible with the hypothesis that stochasticity in the dynamics enters as a result of Gaussian noise or more general form of random forcing

In this section, we provide a summary of the basic properties of a symmetric

Independent increments – for any

Stationary increments – for

This law in

Sample paths are continuous in probability, i.e. for any

For

Although neither the incremental nor the marginal distributions of a Lévy process in general are representable by the elementary functions, the Lévy motion is completely determined by the Lévy–Khintchine formula, which specifies the characteristic function of the Lévy process.

If

The characteristic function of the Lévy–Khintchine formula is as follows:

In the Lévy–Itô decomposition, for any sequence of positive radii

If

Its Lévy jump measure

One can come to a more intuitive interpretation of the stability parameter

We briefly recapitulate here the main ideas behind the proof given in

One proceeds by considering the decomposition in the driving Lévy process by regularly varying the jump measure

The waiting times between successive

Small jump processes

When a first large jump occurs, the solution process moves to the neighbouring domain of attraction with the following probability:

This is the probability that, at time

In

In Sect.

Values of supercritical

Comparison between the supercritical paths constructed using singular perturbations of different duration

We performed additional simulations to locate the supercritical and subcritical values of

Finally, as stated earlier, from the third column of Table

We report in Table

Estimates and 95 % confidence intervals for the mean escape time

All the data used to produce the figures contained in this paper are publicly available in the supplement

Illustrative animations portraying noise-induced transitions can be found in

VL conceptualized the paper, developed the methodology, validated the results, conducted the data analysis. and took the lead role in writing and revising the paper. LS and GM conceptualized the paper, developed the methodology and software, and helped with the writing and revising of the paper.

At least one of the (co-)authors is a member of the editorial board of

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This article is part of the special issue “Centennial issue on nonlinear geophysics: accomplishments of the past, challenges of the future”. It is not associated with a conference.

The authors wish to thank Peter Ashwin, Reyk Börner, Jesus I. Diaz, Jinqiao Duan, Michael, Ghil, Tobias Grafke, Serafim Kalliadasis, Alessandro Laio, Xue-Mei Li, and Greg Pavliotis, for the useful exchanges on various topics covered in this paper. Valerio Lucarini wishes to thank Myles Allen, for suggesting that we look into 3D projections of the phase space when studying transitions paths. The authors acknowledge the support provided by the EU Horizon 2020 project TiPES (grant no. 820970). Valerio Lucarini acknowledges the support provided by the EPSRC project (grant no. EP/T018178/1). This paper is dedicated to K. Hasselmann.

This research has been supported by the Horizon 2020 (TiPES; grant no. 820970) and the Engineering and Physical Sciences Research Council (grant no. EP/T018178/1).

This paper was edited by Daniel Schertzer and reviewed by two anonymous referees.