Maximizing the statistical diversity of an ensemble of bred vectors by using the geometric norm

TL;DR

This paper demonstrates that using the geometric norm to construct ensembles of bred vectors enhances their statistical diversity, growth rate, and alignment with unstable directions, outperforming other norms in chaotic systems.

Contribution

It introduces the geometric norm as an optimal choice for ensemble construction, improving diversity and stability in bred vector methods for chaotic systems.

Findings

Geometric norm maximizes ensemble diversity.

It enhances the growth rate and alignment with Lyapunov vectors.

Optimal in minimizing ensemble dimension fluctuations.

Abstract

We show that the choice of the norm has a great impact on the construction of ensembles of bred vectors. The geometric norm maximizes (in comparison with other norms like the Euclidean one) the statistical diversity of the ensemble while, at the same time, enhances the growth rate of the bred vector and its projection on the linearly most unstable direction, i.e. the Lyapunov vector. The geometric norm is also optimal in providing the least fluctuating ensemble dimension among all the spectrum of q-norms studied. We exemplify our results with numerical integrations of a toy model of the atmosphere (the Lorenz-96 model), but our findings are expected to be generic for spatially extended chaotic systems.