Lorenz-63 Model as a Metaphor for Transient Complexity in Climate

Sergey Kravtsov,Anastasios A. Tsonis

doi:10.3390/e23080951

Abstract

Dynamical systems like the one described by the three-variable Lorenz-63 model may serve as metaphors for complex natural systems such as climate systems. When these systems are perturbed by external forcing factors, they tend to relax back to their equilibrium conditions after the forcing has shut off. Here we investigate the behavior of such transients in the Lorenz-63 model by studying its trajectories initialized far away from the asymptotic attractor. Counterintuitively, these transient trajectories exhibit complex routes and, in particular, the sensitivity to initial conditions is akin to that of the asymptotic behavior on the attractor. Thus, similar extreme events may lead to widely different variations before the perturbed system returns back to its statistical equilibrium.

Highlights

IntroductionThe famous Lorenz-63 system [1] (hereafter, the Lorenz system or the Lorenz model), which arises via a truncation of Saltzman’s equations [2] for convective motion—a paramount feature in climate—is described by the following system of ordinary differential equations:
The famous Lorenz-63 system [1], which arises via a truncation of Saltzman’s equations [2] for convective motion—a paramount feature in climate—is described by the following system of ordinary differential equations: x = −σx + σy, (1)y = − xz + rx − hy, z = xy − bz.Here the dot denotes the time derivative, while the parameters σ and r correspond to the Rayleigh and Prandtl numbers, respectively
We studied transient behavior in numerical simulations of the three-variable Lorenz model (1) initialized far away from the region of its asymptotic chaotic attractor

Summary

Introduction

The famous Lorenz-63 system [1] (hereafter, the Lorenz system or the Lorenz model), which arises via a truncation of Saltzman’s equations [2] for convective motion—a paramount feature in climate—is described by the following system of ordinary differential equations:. Because of the great interest in the structural details of this—and other—chaotic attractors, their numerical simulations are usually initialized near the attractor itself In this case the transients, defined as phase-space trajectories connecting the initial condition and the attractor, are short and uninteresting [4]. Less attention far was, paid to the transient behavior in situations when the Lorenz system is numerically integrated from the states located far from its asymptotic attractor. Investigating such transients is important because extreme far-from-equilibrium events do occur in nature due to either external forcing factors or due to self-amplifying interactions between various subcomponents of complex natural systems. The purpose of this note is to point out some interesting properties of post-extreme-event transients in the Lorenz model

Duration of Transients and Its Relationship to Trajectory-Averaged Local

Types of Transient

Geometric Complexity of Transients

Summary and Discussion