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Dynamical vectors characterizing instability and applicable as ensembleperturbations for prediction with geophysical fluid dynamical models areanalysed. The relationships between covariant Lyapunov vectors (CLVs),orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectorsand finite-time normal modes (FTNMs) are examined for periodic and aperiodicsystems. In the phase-space of FTNM coefficients, SVs are found to equate withunit norm FTNMs at certain times. In the long-time limit, when SVs approachOLVs, the Oseledec theorem and the relationships between OLVs and CLVs are usedto connect CLVs to FTNMs in this phase-space. The covariant properties of boththe CLVs, and the FTNMs, together with their phase-space independence, and thenorm independence of global Lyapunov exponents and FTNM growth rates,establishes their asymptotic convergence. Conditions on the dynamical systemsfor the validity of these results, particularly ergodicity, boundedness andnon-singular FTNM characteristic matrix and propagator, are documented. Systemswith nondegenerate OLVs, and with degenerate Lyapunov spectrum as is the rulein the presence of waves such as Rossby waves, are examined, and efficientnumerical methods for the calculation of leading CLVs proposed. Normindependent finite-time versions of the Kolmogorov-Sinai entropy production andKaplan-Yorke dimension are presented.

Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems