We apply two independent data analysis methodologies to locate stable climate states in an intermediate complexity climate model and analyse their interplay. First, drawing from the theory of quasi-potentials, and viewing the state space as an energy landscape with valleys and mountain ridges, we infer the relative likelihood of the identified multistable climate states and investigate the most likely transition trajectories as well as the expected transition times between them. Second, harnessing techniques from data science, and specifically manifold learning, we characterize the data landscape of the simulation output to find climate states and basin boundaries within a fully agnostic and unsupervised framework. Both approaches show remarkable agreement, and reveal, apart from the well known warm and snowball earth states, a third intermediate stable state in one of the two versions of PLASIM, the climate model used in this study. The combination of our approaches allows to identify how the negative feedback of ocean heat transport and entropy production via the hydrological cycle drastically change the topography of the dynamical landscape of Earth’s climate.

We present a new method for sampling rare and large fluctuations in a non-equilibrium system governed by a stochastic partial differential equation (SPDE) with additive forcing. To this end, we deploy the so-called instanton formalism that corresponds to a saddle-point approximation of the action in the path integral formulation of the underlying SPDE. The crucial step in our approach is the formulation of an alternative SPDE that incorporates knowledge of the instanton solution such that we are able to constrain the dynamical evolutions around extreme flow configurations only. Finally, a reweighting procedure based on the Girsanov theorem is applied to recover the full distribution function of the original system. The entire procedure is demonstrated on the example of the one-dimensional Burgers equation. Furthermore, we compare our method to conventional direct numerical simulations as well as to Hybrid Monte Carlo methods. It will be shown that the instanton-based sampling method outperforms both approaches and allows for an accurate quantification of the whole probability density function of velocity gradients from the core to the very far tails.

We compare different approaches towards an effective description of multiscale velocity field correlations in turbulence. Predictions made by the operator-product expansion, the so-called fusion rules, are placed in juxtaposition to an approach that interprets the turbulent energy cascade in terms of a Markov process of velocity increments in scale. We explicitly show that the fusion rules are a direct consequence of the Markov property provided that the structure functions exhibit scaling in the inertial range. Furthermore, the limit case of joint velocity gradient and velocity increment statistics is discussed and put into the context of the notion of dissipative anomaly. We generalize a prediction made by the multifractal model derived by Benzi et al. [R. Benzi et al., Phys. Rev. Lett. 80, 3244 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.3244] to correlations among inertial range velocity increment and velocity gradients of any order. We show that for the case of squared velocity gradients such a relation can be derived from first principles in the case of Burgers equations. Our results are benchmarked by intensive direct numerical simulations of Burgers turbulence.

The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point and compute the properties of the tangent space (i.e. the Jacobian). The main goal of this paper is to propose a method that infers the Jacobian, thus, the stability properties, from observables (data). First, we propose the echo state network (ESN) with the Recycle validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the echo state network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution and its tangent space with negligible numerical errors. In detail, we compute from data only (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (iii) the Lyapunov spectrum; (iv) the finite-time Lyapunov exponents; (v) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations.

We introduce a variant of the Hybrid Monte Carlo (HMC) algorithm to address large deviation statistics in stochastic hydrodynamics. Based on the path integral approach to stochastic (partial) differential equations, our HMC algorithm samples space-time histories of the dynamical degrees of freedom under the influence of random noise. First, we validate and benchmark the HMC algorithm by reproducing multi-scale properties of the one-dimensional Burgers equation driven by Gaussian and white-in-time noise. Second, we show how to implement an importance sampling protocol to significantly enhance, by order-of-magnitudes, the probability to sample extreme and rare events, making it possible for the first time to estimate moments of field variables of extremely high order (up to 30 and more). By employing reweighting techniques, we map the biased configurations back to the original probability measure in order to probe their statistical importance. Finally, we show that by biasing the system towards very intense negative gradients, the HMC algorithm is able to explore the statistical fluctuations around instanton configurations. Our results will also be interesting and relevant in lattice gauge theory since they provide a new insight on reweighting techniques.

At the molecular level fluid motions are, by first principles, described by time reversible laws. On the other hand, the coarse grained macroscopic evolution is suitably described by the Navier-Stokes equations, which are inherently irreversible, due to the dissipation term. Here, a reversible version of three-dimensional Navier-Stokes is studied, by introducing a fluctuating viscosity constructed in such a way that enstrophy is conserved, along the lines of the paradigm of microcanonical versus canonical treatment in equilibrium statistical mechanics. Through systematic simulations we attack two important questions: (a) What are the conditions that must be satisfied in order to have a statistical equivalence between the two non-equilibrium ensembles? (b) What is the empirical distribution of the fluctuating viscosity observed by changing the Reynolds number and the number of modes used in the discretization of the evolution equation? The latter point is important also to establish regularity conditions for the reversible equations. We find that the probability to observe negative values of the fluctuating viscosity becomes very quickly extremely small when increasing the effective Reynolds number of the flow in the fully resolved hydro dynamical regime, at difference from what was observed previously. a

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.

<p>We apply two independent data analysis methodologies to locate stable climate states in an intermediate complexity climate model and analyze their interplay. First, drawing from the theory of quasipotentials, and viewing the state space as an energy landscape with valleys and mountain ridges, we infer the relative likelihood of the identified multistable climate states, and investigate the most likely transition trajectories as well as the expected transition times between them. Second, harnessing techniques from data science, specifically manifold learning, we characterize  the data landscape of the simulation output to find climate states and basin boundaries within a fully agnostic and unsupervised framework. Both approaches show remarkable agreement, and reveal, apart from the well known warm and snowball earth states, a third intermediate stable state in one of the two climate models we consider. The combination of our approaches allows to identify how the negative feedback of ocean heat transport and entropy production via the hydrological cycle drastically change the topography of the dynamical landscape of Earth's climate.</p>