Linking response to forcing to natural variability using a Koopman operator formalism

Response theory has been shown to be a powerful tool in determining the impact of external forcing on the earth’s climate. High sensitivity to perturbations and the slow decay of response functions is associated with critical behaviour and tipping points. Despite the nonlinear nature of the climate dynamics, a generalisation of the fluctuation-dissipation theorem provides a direct connection between these response functions and the natural variability of the system. We show how response functions for a complex dynamical system may be written as a sum of terms that depend on the eigenvalues and eigenfunctions of the Koopman operator of the system, each term corresponding to a mode of variability. We demonstrate in a number of low-dimensional examples how extended dynamic mode decomposition may be used to accurately compute response and correlation functions of various observables, given only a set of snapshot data.