Convergence of Singular Vectors toward Lyapunov Vectors

The rate at which the leading singular vectors converge toward a single pattern for increasing optimization times is examined within the context of a T21 L3 quasigeostrophic model. As expected, the final-time backward singular vectors converge toward the backward Lyapunov vector, while the initial-time forward singular vectors converge toward the forward Lyapunov vector. Although there is significant case-to-case variability, in general this convergence does not occur over timescales for which the tangent approximation is valid (i.e., less than 5 days). However, a significant portion of the leading Lyapunov vector is contained within the subspace spanned by an ensemble composed of the first 30 singular vectors optimized over 2 or 3 days. Also 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 singular vector is significant and increases as optimization time increases. However, even for 10-day optimization times, the leading singular vector accounts for, on average, only 23% to 28% of the total evolved global perturbation variance depending on the metric and trajectory.