Lyapunov, singular and bred vectors in a multi-scale system: An empirical exploration of vectors related to instabilities

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Title:	Lyapunov, singular and bred vectors in a multi-scale system: An empirical exploration of vectors related to instabilities
Authors:	楊舒芝;Norwood, Adrienne;Kalnay, Eugenia;Ide, Kayo;Yang, Shu-Chih;Wolfe, Christopher
Contributors:	地球科學學院大氣科學學系
Keywords:	bred vectors;Coupled modes;Dynamical systems;Dynamics;Exploration;General Physics and Astronomy;Instability;Lyapunov vectors;Mathematical analysis;Mathematical models;Mathematical Physics;Modeling and Simulation;singular vectors;Stability;Statistical and Nonlinear Physics;Statistics and Probability;Vectors (mathematics)
Date:	2013-06-28
Issue Date:	2026-04-21 13:40:11 (UTC+8)
Publisher:	IOP Publishing Ltd.;IOP Publishing
Abstract:	摘要： We compute and compare the three types of vectors frequently used to explore the instability properties of dynamical models, namely Lyapunov vectors (LVs), singular vectors (SVs) and bred vectors (BVs) in two systems, using the Wolfe-Samelson (2007 Tellus A 59 355-66) algorithm to compute all of the Lyapunov vectors. The first system is the Lorenz (1963 J. Atmos. Sci. 20 130-41) three-variable model. Although the leading Lyapunov vector, LV1, grows fastest globally, the second Lyapunov vector, LV2, which has zero growth globally, often grows faster than LV1 locally. Whenever this happens, BVs grow closer to LV2, suggesting that in larger atmospheric or oceanic models where several instabilities can grow in different areas of the world, BVs will grow toward the fastest growing local unstable mode. A comparison of their growth rates at different times shows that all three types of dynamical vectors have the ability to predict regime changes and the duration of the new regime based on their growth rates in the last orbit of the old regime, as shown for BVs by Evans et al (2004 Bull. Am. Meteorol. Soc. 520-4). LV1 and BVs have similar predictive skill, LV2 has a tendency to produce false alarms, and even LV3 shows that maximum decay is also associated with regime change. Initial and final SVs grow much faster and are the most accurate predictors of regime change, although the characteristics of the initial SVs are strongly dependent on the length of the optimization window. The second system is the toy 'ocean-atmosphere' model developed by Peña and Kalnay (2004 Nonlinear Process. Geophys. 11 319-27) coupling three Lorenz (1963 J. Atmos. Sci. 20 130-41) systems with different time scales, in order to test the effects of fast and slow modes of growth on the dynamical vectors. A fast 'extratropical atmosphere' is weakly coupled to a fast 'tropical atmosphere' which is, in turn, strongly coupled to a slow 'ocean' system, the latter coupling imitating the tropical El Niño-Southern Oscillation. The bred vectors are able to separate the fast and slow modes of growth through appropriate selection of the breeding perturbation size and rescaling interval. The Lyapunov vectors are able to successfully separate the fast 'extratropical atmosphere', but are unable to completely decouple the 'tropical atmosphere' from the 'ocean'. This leads to 'coupled' Lyapunov vectors that are mainly useful in the (slow) 'ocean' system, but are still affected by changes in the (fast) 'tropical' system. The singular vectors are excellent in capturing the fast modes, but are unable to capture the slow modes of growth. The dissimilar behavior of the three types of vectors leads to a degradation in the similarities of the subspaces they inhabit and affects their relative ability of representing the coupled modes. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Lyapunov analysis: from dynamical systems theory to applications'. 其他題名： JPhysA 其他題名： J. Phys. A: Math. Theor 出版者： IOP Publishing 出版日期： 2013-06-28 出處： Journal of Physics A: Mathematical and Theoretical, 2013-06, Vol.46 (25), p.254021-20 資源來源： Institute of Physics Journals 版權： 2013 IOP Publishing Ltd 識別號： ISSN: 1751-8113 識別號： EISSN: 1751-8121 識別號： DOI: 10.1088/1751-8113/46/25/254021 識別號： CODEN: JPHAC5
Appears in Collections:	[Department of Atmospheric Sciences] journal & Dissertation

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