Spatial Instability of the Barotropic Jet with Slow Streamwise Variation

Abstract

A two-scale expansion technique is used to study the barotropic instability of basic flows with slow streamwise variation. Disturbances in nonparallel flow possess properties that differ from those calculated from parallel flow theory. The difference, which is obtained at higher order in the parameter that measures the nonparallelism, depends on the first derivative of the parallel flow properties with respect to the streamwise direction. This higher order correction shifts the spatial growth rate profile for the nonparallel flow downstream relative to the spatial growth rate profile for parallel flow. These results are compared with a previous numerical study by Tupaz, Williams and Chang and some of their conclusions are modified. Physically, the difference in the spatial instability for parallel and nonparallel flow is subject to two combined effects. The first is the lag effect discussed by Tupaz et al., which causes the disturbance structure to lag the parallel-flow solution structure in regions where the mean flow changes rapidly downstream. This causes the downstream shifting of the nonparallel growth rate profile. The second is related to the phase speed difference between the parallel and nonparallel flows. If the disturbance propagates faster than predicted by the parallel flow theory, the local spatial growth rate will be smaller than that calculated by the parallel flow and vice versa.