Abstract/Summary

Understanding how a physical system responds to external stimuli is fundamental in every area of science. To this end, the diverse theories of response in statistical physics come into play in order to predict the change in the average behaviour of a system undergoing perturbations. The general goal of this thesis is to apply the theory of response into the modelling and conceptual understanding of Earth’s climate system. Perturbation theory of operator semigroups shall be employed to derive response formulas in a variety of contexts. Under a stochastic framework, we shall study the effects of adding external forcing in terms of the unperturbed regime, following the spirit of linear response theory. Furthermore, the yielding response formulas are shown to decompose according to the spectral features of the generator of the transfer operator semigroup, allowing to simplify the expressions. Finite dimensional representations of transfer operators lead to stochastic matrices whose properties give useful information about the system up to finite precision. Thus, it is possible to define a coarse-grained linear response, whose conditions for well-posedness and computability are investigated. This methodology is applied to an Ornstein-Uhlenbeck process and the Lorenz 63 atmospheric convection model, whose linear responses are calculated in agreement with observed simulations of the systems. Reduced-order equations are derived using operator expansions. The latter provide non-Markovian closures that preserve the statistical properties of the model in question and are proved to posses the structure of multilevel stochastic models. Such structure is also present in the Empirical Model Reduction (EMR), which constructs non-Markovian models out of partially observed data. This analogy is illustrated in a conceptual climate model, suggesting a formal link between the response theoretic methodology and the EMR data-driven protocol.