Spatiotemporal behavior of the TIGGE medium-range ensemble forecasts

Using the recently-developed mean–variance of logarithms (MVL) diagram, together with the TIGGE archive of medium-range ensemble forecasts from nine different centres, an analysis is presented of the spatiotemporal dynamics of their perturbations, showing how the differences between models and perturbation techniques can explain the shape of their characteristic MVL curves. In particular, a divide is seen between ensembles based on singular vectors or empirical orthogonal functions, and those based on bred vector, Ensemble Transform with Rescaling or Ensemble Kalman Filter techniques. Consideration is also given to the use of the MVL diagram to compare the growth of perturbations within the ensemble with the growth of the forecast error, showing that there is a much closer correspondence for some models than others. Finally, the use of the MVL technique to assist in selecting models for inclusion in a multi-model ensemble is discussed, and an experiment suggested to test its potential in this context.

Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.