Nonlinear dynamics of forced baroclinic critical layers II

Abstract

Baroclinic critical levels arise as singularities in the inviscid linear theory of waves propagating through a stratified, horizontally directed and sheared flow. For a steady wave forcing, disturbances grow secularly over the criticial layers surrounding these levels, generating a jet-like defect in the mean flow. We use a matched asymptotic expansion to furnish a reduced model of the nonlinear dynamics of such defects. By solving the linear initial-value problem for small perturbations to the defect, we establish that secondary instabilities appear at later times. Because the defect is time-dependent, conventional normal-mode analysis is quantitatively inaccurate, but does successfully predict the occurrence of the secondary instability. The instability has a singular character in that disturbances with the shortest horizontal wavelength grow most vigorously at late times, unless dissipation is included. The instability can be suppressed by weak viscosity; by itself, thermal dissipation delays, but does not arrest instability. Numerical computations with the dissipative reduced model demonstrate that the secondary instability saturates as the defect rolls up into a coherent vortical structure. This structure excites a new wave propagating at a different phase speed, thereby forcing a new set of baroclinic critical levels. The implications for self-replication are discussed.