  Dynamical vectors characterizing instability and applicable as ensemble
perturbations for prediction with geophysical fluid dynamical models are
analysed. The relationships between covariant Lyapunov vectors (CLVs),
orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors
and finite-time normal modes (FTNMs) are examined for periodic and aperiodic
systems. In the phase-space of FTNM coefficients, SVs are found to equate with
unit norm FTNMs at certain times. In the long-time limit, when SVs approach
OLVs, the Oseledec theorem and the relationships between OLVs and CLVs are used
to connect CLVs to FTNMs in this phase-space. The covariant properties of both
the CLVs, and the FTNMs, together with their phase-space independence, and the
norm independence of global Lyapunov exponents and FTNM growth rates,
establishes their asymptotic convergence. Conditions on the dynamical systems
for the validity of these results, particularly ergodicity, boundedness and
non-singular FTNM characteristic matrix and propagator, are documented. Systems
with nondegenerate OLVs, and with degenerate Lyapunov spectrum as is the rule
in the presence of waves such as Rossby waves, are examined, and efficient
numerical methods for the calculation of leading CLVs proposed. Norm
independent finite-time versions of the Kolmogorov-Sinai entropy production and
Kaplan-Yorke dimension are presented.
