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

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.