Lyapunov, Floquet, and singular vectors for baroclinic waves

• Main Author: R. M. Samelson
• Format: Article
• Language: English
• Published: Copernicus Publications 2001-01-01
• Series: Nonlinear Processes in Geophysics
• Online Access: http://www.nonlin-processes-geophys.net/8/439/2001/npg-8-439-2001.pdf

The dynamics of the growth of linear disturbances to a chaotic basic state is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave-mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal flow correction. The leading Lyapunov vector is nearly parallel to the leading Floquet vector <font face='Symbol'><i>f</i></font>₁ of the lowest-order unstable periodic orbit over most of the attractor. Departures of the Lyapunov vector from this orientation are primarily rotations of the vector in an approximate tangent plane to the large-scale attractor structure. Exponential growth and decay rates of the Lyapunov vector during individual Poincaré section returns are an order of magnitude larger than the Lyapunov exponent <font face='Symbol'>l</font> ≈ 0.016. Relatively large deviations of the Lyapunov vector from parallel to <font face='Symbol'><i>f</i></font>₁ are generally associated with relatively large transient decays. The transient growth and decay of the Lyapunov vector is well described by the transient growth and decay of the leading Floquet vectors of the set of unstable periodic orbits associated with the attractor. Each of these vectors is also nearly parallel to <font face='Symbol'><i>f</i></font>₁. The dynamical splitting of the complete sets of Floquet vectors for the higher-order cycles follows the previous results on the lowest-order cycle, with the vectors divided into wave-dynamical and decaying zonal flow modes. Singular vectors and singular values also generally follow this split. The primary difference between the leading Lyapunov and singular vectors is the contribution of decaying, inviscidly-damped wave-dynamical structures to the singular vectors.