Steady-State Linear Response Matrix of the Lorenz-63 System

A steady-state linear response matrix ( L ) characterizes the time-mean responses (〈Δ x 〉) of a dynamical system to weak, time-invariant forcings ( f ), as 〈Δ x 〉 = Lf . For the Lorenz-63 system, direct simulations give We find l zz = −0.00111±0.00014 to be a negative eigenvalue of L , which is incompatible with the linear Markov assumption or the linear inverse modeling approach for calculating L . Such negative eigenvalue arises not from numerical errors nor reduction of prognostic variables, as previously suggested. We also review a few direct applications of L and discuss the prospect of using the Lorenz-63 L as a benchmark in a unified testbed. In this framework, other linearization methods can be examined and refined to enable faster and more accurate computation of the steady-state linear response matrix L for more realistic atmospheric systems.