Transient and asymptotic perturbation growth in a simple model

The relationship between asymptotic and finite-interval instabilities in nonlinear, time-dependent flows is examined in the context of a quasi-geostrophic model. For a chosen interval and metric, perturbation growth is related to the singular vectors (SVs) of the linear propagator, while long-term or asymptotic growth is determined by the leading Lyapunov vector (LV), which is independent of metric and grows at the mean rate of the leading Lyapunov exponent. It is shown that the growth rates of the leading LVs vary significantly about their mean values over periods of a few days. For the leading LV, episodes of greater (or lesser) growth correspond to significant variations in vertical structure, and are highly correlated with its projection onto the leading SVs optimized for short (e.g. 24-hour) intervals. The LV exhibits maximum growth over east Asia and the Pacific, where its projection onto rapidly growing SVs in the vicinity of the Pacific storm track is greatest. For a given 24-hour interval, the 30 leading SVs at initial time account for only a few per cent of the total variance of the LV, but account for most of its growth. Filtering these leading SV components from the evolving LV structure over successive intervals reduces its mean (asymptotic) growth rate dramatically in a piecewise continuous sense. The results make clear the importance of transient growth in maintaining the positive mean growth rates of the leading LVs, and in determining the asymptotic stability properties of time-dependent flows. Copyright © 2002 Royal Meteorological Society.

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