Logarithmic bred vectors in spatiotemporal chaos: structure and growth

TL;DR

This paper investigates the structure and growth behavior of logarithmic bred vectors in spatiotemporal chaos, revealing their relation to Lyapunov vectors and deriving a scaling law for their growth rates.

Contribution

It introduces a novel analysis of logarithmic bred vectors, showing they resemble scaled Lyapunov vectors and establishing a growth rate scaling law based on their amplitude.

Findings

Logarithmic bred vectors are piecewise copies of the leading Lyapunov vector.

A scaling law for bred vector growth rates as a function of amplitude is derived.

Growth rates are related to the spectrum of Lyapunov exponents.

Abstract

Bred vectors are a type of finite perturbation used in prediction studies of atmospheric models that exhibit spatially extended chaos. We study the structure, spatial correlations, and the growth- rates of logarithmic bred vectors (which are constructed by using a given norm). We find that, after a suitable transformation, logarithmic bred vectors are roughly piecewise copies of the leading Lyapunov vector. This fact allows us to deduce a scaling law for the bred vector growth rate as a function of their amplitude. In addition, we relate growth rates with the spectrum of Lyapunov exponents corresponding to the most expanding directions. We illustrate our results with simulations of the Lorenz '96 model.