A Linear State-Space Model of the Lorenz-63 System and Its Applications

The climate is a nonlinear system, but it is sometimes useful to approximate it as a linear system.  Considering the climate response under steady forcing (e.g., heating tendency), a linear Markov model (a model without memory effect) should never give a response opposite to the forcing, because it implies an unstable mode.  Here, using the Lorenz-63 system, a 3-variable nonlinear system simplified from 2D convection, as testbed, we show that the climate response of a nonlinear system can be exactly opposite to the forcing, demonstrating a shortcoming of linear Markov model which cannot tolerate an opposite response.  Such opposite response arises not from numerical errors nor reduction of prognostic variables, as previously suggested.  We build a linear state-space model (SSM, a model with memory effect) and quantitatively explain how memory effect gives rise to an opposite response.  Our linear SSM can serve as a benchmark in a unified testbed, where other indirect methods to compute climate response, e.g., fluctuation-dissipation theorem (FDT), can be examined and refined.  Our linear SSM can also be applied to accurately predict response under periodic forcing.  With this, the resonant frequencies of the system can be identified.  The Lorenz-63 system may be far from real world.  Yet, the same approach can be applied to quantitatively analyze the dynamics of natural variability of the climate system, such as annular mode.

Published/submitted:

Hou, Y., Chen, J., Ma, D., & Chan, P. W. (2025). Steady-state linear response matrix of the Lorenz-63 system. J. Atmos. Sci., 82(12), 2667-2675. https://doi.org/10.1175/JAS-D-25-0016.1

Hou, Y., & Chan, P. W. (submitted). A linear state-space model of the Lorenz-63 system and its applications. https://doi.org/10.6084/m9.figshare.30271819.v1