Sensitivity analysis of the climate of a chaotic system

This paper addresses some fundamental methodological issues concerning the sensitivity analysisof chaotic geophysical systems. We show, using the Lorenz system as an example, that anaïve approach to variational (“adjoint”) sensitivity analysis is of limited utility. Applied totrajectories which are long relative to the predictability time scales of the system, cumulativeerror growth means that adjoint results diverge exponentially from the “macroscopic climatesensitivity” (that is, the sensitivity of time-averaged properties of the system to finite-amplitudeperturbations). This problem occurs even for time-averaged quantities and given infinite computingresources. Alternatively, applied to very short trajectories, the adjoint provides an incorrectestimate of the sensitivity, even if averaged over large numbers of initial conditions, becausea finite time scale is required for the model climate to respond fully to certain perturbations.In the Lorenz (1963) system, an intermediate time scale is found on which an ensemble ofadjoint gradients can give a reasonably accurate (O(10%)) estimate of the macroscopic climatesensitivity. While this ensemble-adjoint approach is unlikely to be reliable for more complexsystems, it may provide useful guidance in identifying important parameter-combinations to beexplored further through direct finite-amplitude perturbations.