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In a recent paper the authors developed and tested two novel computational algorithms for predicting the mean linear response of a chaotic dynamical system to small changes in external forcing via the fluctuation-dissipation theorem (FDT): the short-time FDT (ST-FDT), and the hybrid Axiom A FDT (hA-FDT). Unlike the earlier work in developing fluctuation-dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, these two new methods are based on the theory of Sinai-Ruelle-Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. These two algorithms take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian (qG-FDT) approximation of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. It has been discovered that the ST-FDT algorithm is an extremely precise linear response approximation for short response times, but numerically unstable for longer response times. On the other hand, the hA-FDT method is numerically stable for all times, but is less accurate for short times. Here we develop blended linear response algorithms, by combining accurate prediction of the ST-FDT method at short response times with numerical stability of qG-FDT and hA-FDT methods at longer response times. The new blended linear response algorithms are tested on the nonlinear Lorenz 96 model with 40 degrees of freedom, chaotic behavior, forcing, dissipation, and mimicking large-scale features of real-world geophysical models in a wide range of dynamical regimes varying from weakly to strongly chaotic, and to fully turbulent. The results below for the blended response algorithms have a high level of accuracy for the linear response of both mean state and variance throughout all the different chaotic regimes of the 40-mode model. These results point the way toward the potential use of the blended response algorithms in operational long-term climate change projection.

We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation-dissipation theorem. Unlike the earlier work in developing fluctuationdissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai-Ruelle-Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forceddissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation-dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.

The low-frequency response to changes in external forcing for the climate system is a fundamental issue. In two recent papers the authors developed a new blended response algorithm for predicting the response of a nonlinear chaotic forced-dissipative system to small changes in external forcing. This new algorithm is based on the fluctuation–dissipation theorem and combines the geometrically exact general response formula using integration of a linear tangent model at short response times and the classical quasi-Gaussian response algorithm at longer response times. This algorithm overcomes the inherent numerical instability in the geometric formula arising because of positive Lyapunov exponents at longer times while removing potentially large systematic errors from the classical quasi-Gaussian approximation at moderate times. Here the new blended method is tested on the low-frequency response for a T21 barotropic truncation on the sphere with realistic topography in two dynamical regimes corresponding to the mean climate behavior at 300- and 500-hPa geopotential height. The 300-hPa regime has robust strongly mixing behavior with a nearly Gaussian equilibrium state distribution, whereas the 500-hPa regime is weakly chaotic with slowly decaying time autocorrelation functions and strongly non-Gaussian climatology. It is found that the blended response algorithm has significant skill beyond the classical quasi-Gaussian algorithm for the response of the climate mean state and its variance. Additionally, the predicted response of the T21 barotropic model in the low-frequency regime for these functionals does not seem to be affected by the model’s structural instability. Thus, the results here suggest the use of the fluctuation–dissipation theorem–based blended response algorithm for more complex nonlinear geophysical models.