Ergodic theory of climate: variability, stability and response

Abstract

Climate studies deal with changes in the statistics rather than in the exact state of the Earth system. Ergodic theory is the art of relating the dynamics of a system to its statistical properties and is thus particularly suitable to a unifying theory of climate. This work introduces concepts of ergodic theory to the study of climate variability, stability and response to forcing. The focus is on the spectrum of transfer operators, governing the evolution of statistics by the dynamics. It is shown that crucial inform- ation about a system’s stability can be extracted from approximations of the transfer operators and their ergodicity spectrum. These results are first applied to stochastic systems relevant to climate and in particular to El Niño-Southern Oscillation (ENSO), the dominant pattern of climate variability on interannual time-scales. Novel analyt- ical formulas are found for the ergodicity spectrum of these systems. A key result is the direct relation between the shrinkage of this spectrum and the slowing down of a system undergoing a crisis, i.e. a dramatic change in its statistics. In addition, it is shown that the ergodicity eigenvectors are very sensitive to nonlinear effects and could allow to unravel the nature of dominant patterns of variability such as ENSO. It is a matter of great debate whether the Earth system can undergo such dramatic events and if state-of-the-art climate models are able to resolve them and to give early-warnings. The transition from a warm to a snow-covered Earth due to the ice- albedo feedback found in the Planet Simulator, a General Circulation Model (GCM) of intermediate complexity, is an example of such crisis. We explain for the first time how slowing down occurring at the crisis is due to the shrinkage of the ergodicity spectrum. The study of the ergodicity spectrum thus allows to discuss the validity of concepts such as climate sensitivity or linear response as well as the applicability of early-warning indicators of crises to high-dimensional systems such as climate. Lastly, we show that transfer operators can be applied to the study of midlatitude blocking events. These atmospheric regimes of increased persistence have a large impact on the climate of western Europe and North America. In a hemispheric baro- tropic model of the troposphere with realistic forcing, it is found that these regimes are associated with the ergodicity spectrum responsible for a slow decay of correlations. This allows to extract them as almost-invariant sets connected by preferred transition paths. These paths constitute a source of predictability allowing to design an early-warning indicator of transition from the zonal to the blocked regime. As a general conclusion, this work supports ergodic and dynamical systems theory as a unifying mathematical framework to climate dynamics and the study of its natural variability, stability and response to forcing.