Lyapunov analysis of multiscale dynamics: The slow bundle of the two-scale Lorenz 96 model

TL;DR

This paper uses Lyapunov analysis to identify a slow bundle in the tangent space of the two-scale Lorenz 96 model, revealing how slow and fast dynamics interact and influence stability.

Contribution

It uncovers the existence and properties of a slow bundle of covariant Lyapunov vectors, linking stability features across multiple time scales in a multiscale model.

Findings

The slow bundle is extensive in size, involving both slow and fast degrees of freedom.

The slow bundle vectors overlap with regions of the Lyapunov spectrum where slow and fast instability rates intersect.

Results suggest the slow-variable behavior is governed by a non-trivial subset of degrees of freedom.

Abstract

We investigate the geometrical structure of instabilities in the two-scales Lorenz 96 model through the prism of Lyapunov analysis. Our detailed study of the full spectrum of covariant Lyapunov vectors reveals the presence of a slow bundle in tangent space, composed by a set of vectors with a significant projection on the slow degrees of freedom; they correspond to the smallest (in absolute value) Lyapunov exponents and thereby to the longer time scales. We show that the dimension of the slow bundle is extensive in the number of both slow and fast degrees of freedom, and discuss its relationship with the results of a finite-size analysis of instabilities, supporting the conjecture that the slow-variable behavior is effectively determined by a non-trivial subset of degrees of freedom. More precisely, we show that the slow bundle corresponds to the Lyapunov spectrum region where fast and slow instability rates overlap, "mixing" their evolution into a set of vectors which simultaneously carry information on both scales. We suggest these results may pave the way for future applications to ensemble forecasting and data assimilations in weather and climate models.