Linear and Finite-Amplitude Localized Baroclinic Instability

Abstract

The linear and finite-amplitude dissipative dynamics of unstable, zonally localized baroclinic disturbances is investigated in cases where the supercriticality varies in the zonal direction. The zonal confinement occurs due to O(1) variations of the frictional influence on the current's instability. A two-layer f-plane model is used. No meridional shear is present in the basic shear flow. When the basic current is equal and opposite in the two layers, two zonally localized modes with the same growth rate and opposite symmetries exist for all unstable parameter values. Thus, an infinite family of unstable modes formed from an arbitrary linear combination of these two modes exists. This degeneracy persists in finite amplitude. Hence, the phase of individual crests in the disturbance is a function of initial conditions even for dissipative localized instabilities. The presence of a mean barotropic flow reduces the growth rates of the localized disturbances and expunges the symmetry properties of the mode and the resulting degeneracy. The disturbance becomes time dependent due to phase translation of crests. Localized modes exist even when the flow in both layers is in the same direction. In finite amplitude there is a weak vacillation in energy level. A discussion of the appropriate boundary condition for the localized modes suggests that the total geostrophic perturbation streamfunction should vanish on the flow boundaries.