No abstract

The existence of persistent midlatitude atmospheric flow regimes with time-scales larger than 5-10 days and indications of preferred transitions between them motivates to develop early warning indicators for such regime transitions. In this paper, we use a hemispheric barotropic model together with estimates of transfer operators on a reduced phase space to develop an early warning indicator of the zonal to blocked flow transition in this model. It is shown that the spectrum of the transfer operators can be used to study the slow dynamics of the flow as well as the non-Markovian character of the reduction. The slowest motions are thereby found to have time scales of three to six weeks and to be associated with meta-stable regimes (and their transitions) which can be detected as almost-invariant sets of the transfer operator. From the energy budget of the model, we are able to explain the meta-stability of the regimes and the existence of preferred transition paths. Even though the model is highly simplified, the skill of the early warning indicator is promising, suggesting that the transfer operator approach can be used in parallel to an operational deterministic model for stochastic prediction or to assess forecast uncertainty.

The destruction of a chaotic attractor leading to rough changes in the dynamics of a dynamical system is studied. Local bifurcations are known to be characterised by a single or a pair of characteristic exponents crossing the imaginary axis. As a result, the approach of such bifurcations in the presence of noise can be inferred from the slowing down of the decay of correlations [1]. On the other hand, little is known about global bifurcations involving high-dimensional attractors with several positive Lyapunov exponents. It is known that the global stability of chaotic attractors may be characterised by the spectral properties of the Koopman [2] or the transfer operators governing the evolution of statistical ensembles. Accordingly, it has recently been shown [3] that a boundary crisis in the Lorenz flow coincides with the approach to the unit circle of the eigenvalues of these operators associated with motions about the attractor, the stable resonances. A second type of resonances, the unstable resonances, is responsible for the decay of correlations and mixing on the attractor. In the deterministic case, those cannot be expected to be affected by general boundary crises. Here, however, we give an example of chaotic system in which slowing down of the decay of correlations of some observables does occur at the approach of a boundary crisis. The system considered is a high-dimensional, chaotic climate model of physical relevance. Moreover, coarsegrained approximations of the transfer operators on a reduced space, constructed from a long time series of the system, give evidence that this behaviour is due to the approach of unstable resonances to the unit circle. That the unstable resonances are affected by the crisis can be physically understood from the fact that the process responsible for the instability, the ice-albedo feedback, is also active on the attractor. Implications regarding response theory and the design of early-warning signals are finally discussed.

A theory of Ruelle-Pollicott (RP) resonances for stochastic systems is introduced. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal's oscillatory components manifested by peaks in the PSD.It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space V . These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space V , and can be estimated from series. When the sampling rate (in time) at which the observations are collected is either sufficiently small or large, it is shown that the reduced RP resonances approximate the RP resonances of the generator of the conditional expectation in V , i.e. the optimal reduced system in V obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak.The companions articles, Part II[TCND19a] and Part III [TCND19b], deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances offer. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear signature in terms of the geometric organization of the RP resonances in the left half plane. This analysis provides thus an unambiguous signature of nonlinear oscillations contained in a noisy signal and that can be extracted from time series. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses then the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane-Zebiak model of El Niño-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.