Maximizing the Statistical Diversity of an Ensemble of Bred Vectors by Using the Geometric Norm

It is shown 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 such as the Euclidean one) the statistical diversity of the ensemble while at the same time it 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 norms studied. The results are exemplified with numerical integrations of a toy model of the atmosphere (the Lorenz-96 model), but these findings are expected to be generic for spatially extended chaotic systems.