The growth of linear disturbances to stable and unstable time-periodic basi
c states is analyzed in an asymptotic model of weakly nonlinear, baroclinic
wave-mean interaction. In this model, an ordinary differential equation fo
r the wave amplitude is coupled to a partial differential equation for the
zonal-flow correction. Floquet vectors, the eigenmodes for linear disturban
ces to the oscillatory basic states, split into wave-dynamical and decaying
zonal-flow modes. Singular vectors reflect the structure of the Floquet ve
ctors: the most rapid amplification and decay are associated with the wave-
dynamical Floquet vectors, while the intermediate singular vectors closely
follow the decaying zonal-flow Floquet vectors. Singular values depend stro
ngly on initial and optimization times. For initial times near wave amplitu
de maxima, the Floquet decomposition of the leading singular vector depends
relatively weakly on optimization time. For the unstable oscillatory basic
state in the chaotic regime, the leading Floquet vector is tangent to the
large-scale structure of the attractor, while the leading singular vector i
s not. However, corresponding inferences about the accessibility of disturb
ed states rely on the simple attractor geometry, and may not easily general
ize. The primary mechanism of disturbance growth on the wave timescale in t
his model involves a time-dependent phase shift along the basic wave cycle.
The Floquet vectors illustrate that modal disturbances to time-dependent b
asic states can have time-dependent spatial structure, and that the latter
need not indicate nonmodal dynamics. The dynamical splitting reduces the "b
utterfly effect,'' the ability of small-scale disturbances to influence the
evolution of an unstable large-scale flow.