Crisis of the chaotic attractor of a climate model: a transfer operator approach

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 (Held and Kleinen 2004 Geophys. Res. Lett. 31 1–4). 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 (Mauroy and Mezić 2016 IEEE Trans. Autom. Control 61 3356–69) or the transfer operators governing the evolution of statistical ensembles. Accordingly, it has recently been shown (Tantet 2017 J. Stat. Phys. 1–33) 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 class of resonances, the unstable resonances, are responsible for the decay of correlations and mixing on the attractor. In the deterministic case, these cannot be expected to be affected by general boundary crises.

Here, however, we give an example of a 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, coarse-grained 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. Finally, we discuss implications regarding response theory and the design of early-warning signals.