Quasi-linear instability theory for a weakly supercritical zonal shear flow on the β-plane

Abstract

A quasi-linear approach to the problem of the evolution of a shear flow on the β-plane is considered for the case when a large number of waves with close phase velocities are excited. Their interaction with the flow occurs mainly in a narrow resonance region due to the presence of a positive derivative on the absolute vorticity profile Ω _a (y).

It is shown that a stabilization of the oscillation level is possible only for infinitely large Reynolds numbers. In the case of large, but finite Reynolds numbers, the quasilinear relaxation takes the evolution into a regime in which the noise level and the mean velocity perturbation in the resonance region increase with time [∼(Re)⁻¹•t]. In both cases, a “plateau” (dΩ _a /dy=0) is formed on the absolute vorticity profile in the resonance region.

The range of applicability for the adopted approximation is discussed.

Key Words:

• Shear flow
• β-plane
• stability