Ruelle's linear response formula, ensemble adjoint schemes and Lévy flights

A traditional subject in statistical physics is the linear response of a molecular dynamical system to changes in an external forcing agency, e.g. the Ohmic response of an electrical conductor to an applied electric field. For molecular systems the linear response matrices, such as the electrical conductivity, can be represented by Green–Kubo formulae as improper time-integrals of 2-time correlation functions in the system. Recently, Ruelle has extended the Green–Kubo formalism to describe the statistical, steady-state response of a 'sufficiently chaotic' nonlinear dynamical system to changes in its parameters. This formalism potentially has a number of important applications. For instance, in studies of global warming one wants to calculate the response of climate-mean temperature to a change in the atmospheric concentration of greenhouse gases. In general, a climate sensitivity is defined as the linear response of a long-time average to changes in external forces. We show that Ruelle's linear response formula can be computed by an ensemble adjoint technique and that this algorithm is equivalent to a more standard ensemble adjoint method proposed by Lea, Allen and Haine to calculate climate sensitivities.In a numerical implementation for the 3-variable, chaotic Lorenz model it is shown that the two methods perform very similarly. However, because of a power-law tail in the histogram of adjoint gradients their sum over ensemble members becomes a Levy flight, and the central limit theorem breaks down. The law of large numbers still holds and the ensemble-average converges to the desired sensitivity, but only very slowly, as the number of samples is increased. We discuss the implications of this example more generally for ensemble adjoint techniques and for the important practical issue of calculating climate sensitivities.