Theory and computation of covariant Lyapunov vectors

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.Comment: 21 pages, 5 figure

On open boundary conditions for a limited-area coastal model off Oregon. Part 2: Response to wind forcing from a regional mesoscale atmospheric model

This is the second part of a study of open boundary conditions (OBCs) in a limited-area high resolution coastal model off Oregon. In this paper, the OBCs developed in Part 1 [Ocean Modeling, 2005] are further evaluated by an application in which the coastal ocean model is forced with time- and space-dependent wind fields from a regional mesoscale atmospheric model [Journal of Geophysical Research--Oceans 107 (2002)]. The response during summer 1999 of the wind-driven upwelling flow field over the variable shelf bottom topography off Oregon coast is described. Satisfactory performance of the model and of the OBCs in this experiment with complex spatially and temporally varying atmospheric forcing is indicated by the production of physically reasonable fields in the ocean variables and by favorable model/data comparisons. Additional experiments forced by realistic, time-variable, but spatially uniform winds are included to allow a direct comparison of solutions obtained with OBCs and with cyclic boundary condition (CBCs). The general similarity of the results in these two cases provides additional support for the effectiveness of the OBCs in integrating the outer fluxes into, and radiating coastal trapped waves and advective disturbances out of, the computational domain. © 2004 Elsevier Ltd. All rights reserved