Lyapunov, Floquet, and singular vectors for baroclinic waves

Abstract

Abstract. 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 φ 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 λ ≈ 0.016. Relatively large deviations of the Lyapunov vector from parallel to φ 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 φ 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 wavedynamical 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.

Covariant Lyapunov vectors of a quasi‐geostrophic baroclinic model: analysis of instabilities and feedbacks

“…For stationary states, the CLVs reduce to the normal modes. In the case of periodic orbits, the CLVs coincide with the Floquet vectors, which for example have been obtained for the weakly unstable Pedlosky model (Samelson, ). Samelson also extended this analysis to unstable periodic orbits (Samelson, ).…”

An efficient method for recovering Lyapunov vectors from singular vectors

“…While there are, in principle, an infinite number of mean flow correction terms, in practice, only a finite number J are retained. We use J = 6, the same value used by Samelson (2001a,b); the system considered here is thus 8‐dimensional. The behaviour of the model is controlled by three parameters: the zonal and meridional wavenumbers ( k , m ) of the fundamental wave and the dissipation parameter γ.…”

“…The behaviour of the model is controlled by three parameters: the zonal and meridional wavenumbers ( k , m ) of the fundamental wave and the dissipation parameter γ. For ( k , m ) = (π, 1) and γ= 0.1315, the model undergoes a baroclinic wave‐mean oscillation of chaotically vacillating amplitude with mean period T p ≈ 24.4 (for details, see Samelson, 2001a,b).…”

Singular Vectors and Time-Dependent Normal Modes of a Baroclinic Wave-Mean Oscillation

“…It should be noted that the majority of the studies that examine the relationship between SVs and LVs either use an orthogonalized LV (Vastano and Moser 1991;Vannitsem and Nicolis 1997;Reynolds and Errico 1999) or focus only on the leading LV (Szunyogh et al 1997;Samelson 2001a). The former case presents difficulties with interpretation since orthogonalized LVs are essentially identical to SVs with asymptotically long optimization intervals (Trevisan and Pancotti 1998).…”