Periodic Orbits, Lyapunov Vectors, and Singular Vectors in the Lorenz System

Some theoretical issues related to the problem of quantifying local predictability of atmospheric flow and the generation of perturbations for ensemble forecasts are investigated in the Lorenz system. A periodic orbit analysis and the study of the properties of the associated tangent linear equations are performed. In this study a set of vectors are found that satisfy Oseledec theorem and reduce to Floquet eigenvectors in the particular case of a periodic orbit. These vectors, called Lyapunov vectors, can be considered the generalization to aperiodic orbits of the normal modes of the instability problem and are not necessarily mutually orthogonal. The relation between singular vectors and Lyapunov vectors is clarified, and transient or asymptotic error growth properties are investigated. The mechanism responsible for super-Lyapunov 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, while the leading initial singular vectors, in general, point away from it. Perturbations that are on the attractor and maximize growth should be considered in meteorological applications such as ensemble forecasting and adaptive observations. These perturbations can be found in the subspace of the leading Lyapunov vectors.