Quantifying linear disturbance growth in periodic and aperiodic systems

Graduation date: 2007The mathematical and physical connections between three different ways of quantifying linear predictability in geophysical fluid systems are studied in a series of analytical and numerical models. Normal modes, as they are traditionally formulated in the instabilities theories of geophysical fluid dynamics, characterize the asymptotic development of disturbances to stationary flows. Singular vectors, currently used to generate initial conditions for ensemble forecasting systems at some operational centers, characterize the transient evolution of disturbances to flows with arbitrary time dependence. Lyapunov vectors are an attempt to associate a physical structure with the Lyapunov exponents, which give the rate at which the trajectories of dynamical systems diverge. It is shown that these seemingly divergent ways of quantifying linear disturbance growth are closely related. It is argued that Lyapunov vectors are a natural generalization of normal modes to flows with arbitrary time dependence. Singular vectors are shown to asymptotically converge to orthogonalizations of the Lyapunov vectors. A direct, efficient, and norm-independent method for constructing the n most rapidly growing Lyapunov vectors from the n most rapidly growing forward and the n most rapidly decaying backward asymptotic singular vectors is proposed and demonstrated using several models of geophysical flows.\ud \ud These connections are further studied using a (time-periodic) wave-mean oscillation in an intermediate complexity baroclinic channel model. For time-periodic systems, normal modes may be defined in terms of Floquet vectors. It is argued that Floquet vectors are equivalent to Lyapunov vectors for time-periodic flows. The Floquet vectors of the wave-mean oscillation are found to split into two dynamically distinct classes that have analogs in the classical theories of the baroclinic instability and parallel shear flow. The singular vectors of the oscillation are found to preserve this dynamical splitting. The representations of the singular vectors in terms of the forward and adjoint Floquet vectors display much simpler temporal behavior than the singular vectors or the Floquet vectors individually. \ud It is further demonstrated that while the Floquet vectors point 'onto' the local system attractor, the singular vectors point 'off' the attractor