Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances

Abstract

Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them-as filtered through an observable of the system-is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap-defined as the distance between the subdominant RP resonance and the unit circle-plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño-Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.climate dynamics | Markov operators | parametric dependence | sensitivity bounds | uncertainty quantification S ensitive behavior of long-term general circulation model (GCM) statistics is attracting increased attention (1-3), but its origin and fundamental mechanisms remain unclear. These sensitive-behavior issues are of practical, as well as theoretical, importance in climate dynamics and elsewhere (4). For some GCMs, involving millions of variables, circumstances have been found where certain climate observables vary smoothly through a plausible parameter range (5) or where linear response theory applies over some range (6). On the other hand, this may not hold for every observable or parameter, and concerns arise regarding the role of some type of "structural instability" in sensitive parameter dependence (1, 2, 4).The low-order Lorenz (L63) model (7) illustrates some of the relevant issues. Various statistics exhibit linear dependence over a broad range of parameters for which the dynamics is chaotic (e.g., figure 2 of ref. 8). The statistics' linear dependence coexists here with structural instability of this model's global attractor, as small variations in the parameters cause a plethora of topological changes (9). In particular, the unstable periodic orbits that appear or disappear as a parameter chan...

Predicting Critical Transitions in ENSO models. Part II: Spatially Dependent Models

“…In the case of the JNG model, the strongly mixing behavior also manifests itself by moderately energetic peaks in the power spectrum, as opposed to the nearly periodic behavior that is associated, in this model, with weak mixing [i.e., slow decay of correlations; see Fig. 2 in Chekroun et al (2014)]. The choice of d s 5 0.95 in this paper thus corresponds to putting the model in a less periodic and thus less predictable regime.…”

“…A brief description of this model is provided in appendix B; see Jin and Neelin (1993a,b), Sun et al (2002), and Kondrashov et al (2008) for further details on the original Jin-Neelin model with time-independent forcing and Jin et al (1994Jin et al ( , 1996 for the model formulation with seasonal forcing. Chekroun et al (2014) have called a further simplification of the JNG model the forced JN (fJN) model.…”

“…Strong mixing, also referred to as rapid mixing in dynamical systems theory (Chekroun et al 2014), corresponds to chaotic behavior that is, in addition, characterized by fast decay of lagged autocorrelations. In the case of the JNG model, the strongly mixing behavior also manifests itself by moderately energetic peaks in the power spectrum, as opposed to the nearly periodic behavior that is associated, in this model, with weak mixing [i.e., slow decay of correlations; see Fig.…”

Predicting Climate Change Using Response Theory: Global Averages and Spatial Patterns

“…[53], can be related to the presence of nearby critical transitions. Indeed, we have recently investigated such an issue in [54], thus providing the statistical mechanical embedding for the classical problem of multistability of the Earth's system, which had been previously studied using macroscopic thermodynamics in [55][56][57].…”

“…While the practical application of such an approach in a very high-dimensional problem like in the case of climate might in principle faces problems related to the curse of dimensionality, it has been advocated that it could provide an excellent framework for studying vicinity of the climate to critical transitions [53], i.e. anticipating where there is no smoothness of the invariant measure with respect to perturbations.…”