Convergence of singular vectors toward Lyapunov vectors

The rate at which the leading singular vectors converge toward a single pat tern for increasing optimization times is examined within the context of a T21 L3 quasigeostrophic model. As expected, the final-time backward singula r vectors converge toward the backward Lyapunov vector, while the initial-t ime forward singular vectors converge toward the forward Lyapunov vector. A lthough there is significant case-to-case variability, in general this conv ergence does not occur over timescales for which the tangent approximation is valid (i.e., less than 5 days). However, a significant portion of the le ading Lyapunov vector is contained within the subspace spanned by an ensemb le composed of the first 30 singular vectors optimized over 2 or 3 days. Al so as expected, the final-time leading singular vectors become independent of metric as optimization time is increased. Given an initial perturbation that has a white spectrum with respect to the initial-time singular vectors , the percent of the final-time perturbation explained by the leading singu lar vector is significant and increases as optimization time increases. How ever, even for 10-day optimization times, the leading singular vector accou nts for, on average, only 23% to 28% of the total evolved global perturbati on variance depending on the metric and trajectory.