The stability of a two-dimensional steady-state flow: Local growth of perturbations

Abstract

The linear instability of a weakly non-zonal barotropic flow is investigated numerically using the spectral method and analytically via WKB theory. The basic steady-state is the consequence of the interaction of a constant mean zonal wind and a sinusoidal topography. The (numerically determined) slowest growing eigenmode is stationary, on the scale of the topography and depends crucially on a form drag mechanism. The fastest growing eigenmode, on the other hand, does not involve a form drag mechanism but depends rather on the presence of local shears inherent in the basic flow itself. A large part of the discussion involves an analysis of the structure and energetics of the fastest growing eigenmode.

The fastest growing eigenmode has a simple meridional structure and is strongly modulated in the zonal direction. The peak in the modulation pattern is associated with an easterly jet in the basic flow which is particularly unstable as a result of the beta effect. A secondary maximum in the modulation pattern is associated with regions of basic flow confluence and dimuence.

Because of a zonal spatial scale separation between the disturbance and the basic flow, a simple locally-zonal approach is also used. The theory, which relies on some techniques developed by Pierrehumbert (1984) in connection with baroclinic instability, correctly reproduces the qualitative nature of the modulation pattern of the fastest-growing eigenmode.

Key Words:

• Barotropic flows
• stability
• topographic effects