PERIODIC-ORBITS, LYAPUNOV VECTORS, AND SINGULAR VECTORS IN THE LORENZSYSTEM

Some theoretical issues related to the problem of quantifying local pr edictability of atmospheric flow and the generation of perturbations f or ensemble forecasts are investigated in the Lorenz system. A periodi c orbit analysis and the study of the properties of the associated tan gent linear equations are performed. In this study a set of vectors ar e found that satisfy Oseledec theorem and reduce to Floquet eigenvecto rs in the particular case of a periodic orbit. These vectors, called L yapunov vectors, can be considered the generalization to aperiodic orb its of the normal modes of the instability problem and are not necessa rily mutually orthogonal. The relation between singular vectors and Ly apunov vectors is clarified, and transient or asymptotic error growth properties are investigated. The mechanism responsible for super-lyapu nov growth is shown to be related to the nonorthogonality of Lyapunov vectors. The leading Lyapunov vectors, as defined here, as well as the asymptotic final singular vectors, are tangent to the attractor, whil e the leading initial singular vectors, in general, point away from it . Perturbations that are on the attractor and maximize growth should b e considered in meteorological applications such as ensemble forecasti ng and adaptive observations. These perturbations can be found in the subspace of the leading Lyapunov vectors.