The climate system is a nonlinear chaotic system.  A steady-state linear response matrix (L) denotes the time-mean responses (x) of the system to weak time-invariant forcings (f), as x=Lf.  Such matrix L can be used to (1) tell the time-mean response given a time-invariant forcing, (2) force a specified mean state for hypothesis testing, (3) tell the most excitable mode (left singular vector of L with the largest singular value), and (4) tell the exponentially/spirally decaying eigenmodes of the system (eigenvectors of -L^-1), if the system can be approximated as linear Markov process.  Here, we discuss the steady-state linear response matrices of the Lorenz-63 model and a two-layer quasi-geostrophic (QG) model.
Counter-intuitively, direct computation (applying weak time-invariant forcings) shows that the steady-state linear response matrix of the Lorenz-63 model has a negative eigenvalue.  Specifically, a weak time-invariant forcing in z direction will give a steady-state response in the exact opposite direction.  This marks the failure of the linear Markov assumption.  While negative eigenvalue was also found in a convective system (Kuang 2024, doi:10.1175/JAS-D-23-0194.1), negative eigenvalue is more counter-intuitive for Lorenz-63 model as the model is not simplified (e.g., by horizonal average).
Methods to compute steady-state linear response matrix (for example, fluctuation-dissipation theorem, FDT) are sometimes inaccurate.  The reason behind the inaccuracy, especially the role of chaos, remains unclear.  Here, we propose to use two simple nonlinear chaotic models, the Lorenz-63 model and a two-layer quasi-geostrophic (QG) model, as unified testbeds to study the accuracy of those linearization methods.  We compute the steady-state linear response matrix by FDT, and test its accuracy against the directly computed matrix.  Finally, we will briefly introduce progress in computing the steady-state linear response matrix in the Lorenz-63 model and a two-layer QG model by sinusoidal forcings.  We hope that linearization methods, evaluated and/or improved in these testbeds, can be used for fast and accurate linearizations of more realistic atmospheric systems.

Lorenz-63 model,steady-state linear response matrix,chaos