Periodic Orbits and Disturbance Growth for Baroclinic Waves

The growth of linear disturbances to stable and unstable time-periodic basic states is analyzed in an asymptotic\ud model of weakly nonlinear, baroclinic wave–mean interaction. In this model, an ordinary differential equation\ud for the wave amplitude is coupled to a partial differential equation for the zonal-flow correction. Floquet vectors,\ud the eigenmodes for linear disturbances to the oscillatory basic states, split into wave-dynamical and decaying\ud zonal-flow modes. Singular vectors reflect the structure of the Floquet vectors: the most rapid amplification and\ud decay are associated with the wave-dynamical Floquet vectors, while the intermediate singular vectors closely\ud follow the decaying zonal-flow Floquet vectors. Singular values depend strongly on initial and optimization\ud times. For initial times near wave amplitude maxima, the Floquet decomposition of the leading singular vector\ud depends relatively weakly on optimization time. For the unstable oscillatory basic state in the chaotic regime,\ud the leading Floquet vector is tangent to the large-scale structure of the attractor, while the leading singular vector\ud is not. However, corresponding inferences about the accessibility of disturbed states rely on the simple attractor\ud geometry, and may not easily generalize. The primary mechanism of disturbance growth on the wave timescale\ud in this model involves a time-dependent phase shift along the basic wave cycle. The Floquet vectors illustrate\ud that modal disturbances to time-dependent basic states can have time-dependent spatial structure, and that the\ud latter need not indicate nonmodal dynamics. The dynamical splitting reduces the ‘‘butterfly effect,’’ the ability\ud of small-scale disturbances to influence the evolution of an unstable large-scale flow