On the stability of a barotropic shear flow to nongeostrophic disturbances

The stability of a barotropic shear flow ū* = U tanh (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_gt)² > 0, where c_gi denotes the phase speed of a gravityinertia wave which is characterized by zonal and vertical wave numbers α^* and nπ/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 - π^-1 = 0.682, well-outside the quasi-geostrophic range. Characteristics of the instability are presented and discussed.