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 NiñoSouthern Oscillation (ENSO), the dominant pattern of climate variability on interannual timescales. 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 stateoftheart climate models are able to resolve them and to give earlywarnings. The transition from a warm to a snowcovered 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 earlywarning indicators of crises to highdimensional 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 almostinvariant sets connected by preferred transition paths. These paths constitute a source of predictability allowing to design an earlywarning 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.
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 NiñoSouthern Oscillation (ENSO), the dominant pattern of climate variability on interannual timescales. 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 stateoftheart climate models are able to resolve them and to give earlywarnings. The transition from a warm to a snowcovered 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 earlywarning indicators of crises to highdimensional 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 almostinvariant sets connected by preferred transition paths. These paths constitute a source of predictability allowing to design an earlywarning 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.
Original language  English 

Awarding Institution 

Supervisors/Advisors 

Award date  25 Apr 2016 
Publisher  
Print ISBNs  9789039365052 
Publication status  Published  25 Apr 2016 
Keywords
 Climate
 Variability
 Transfer operator
 Ergodic Theory
 Dynamical System
 Stability
 Response