On the stability of a barotropic shear flow to nongeostrophic disturbances

Abstract

The stability of a barotropic shear flow u^* = Utanh (y^*/L) of a stably stratified fluid of depth H to three-dimensional disturbances, is investigated by a linear analysis. Both the basic-state and perturbed motion are hydrostatic and the Brunt-Vàisàlàsfrequency and Coriolis parameter are assumed constant. A neutral solution is found, under the condition that 1 - (U/c_gi)2 > 0, where c_gi denotes the phase speed of a gravityinertia wave which is characterized by zonal and vertical wave numbers a^* and n/H respectively. With the aid of this solution, some characteristics of the unstable waves are found by numerical methods. It is then shown that, for values of the parameters characteristic of large-scale mid-latitude flow, the basic current is stable to the internal baroclinic modes (n = 0) but unstable to the barotropic mode (n = 0) in the quasi-geostrophic range, R₀ = U/fL < 1. Instability to the internal modes does not set in until R₀ > 1 - ^-1 = 0.682, well-outside the quasi-geostrophic range. Characteristics of the instability are presented and discussed.