Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems

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 behaviour, 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 towards the potential use of the blended response algorithms in operational long-term climate change projection.