On the fluctuation-response relation in Geophysical systems

One of the major issues concerning the study of a dynamical system is the response to perturbations. In climate dynamics, for example, it is of major interest to understand how a given variable, e.g., the temperature, is sensitive to alterations of some other component of the system, e.g., the greenhouse gas concentration. We review the connection between equilibrium and non-equilibrium properties, also known as Fluctuation-Relaxation Relation, and its main aspects in chaotic and turbulent systems. We consider, in particular, the effects of the fast variables on the slow variables in a multiscale system, as far as the sensitivity properties are concerned. Two examples about (widely speaking) climate modelling are discussed: the Lorenz-96 model and the double-potential well model. Both of them, despite their apparent simplicity, hide the same kind of interesting features of much more complex systems.