Linear and nonlinear stability of a quasi-geostrophic mixing layer subject to a uniform background shear

The aim of this work is to shed light by revisiting - through the kernel-wave (KW) perspective - the breakdown of a quasi-geostrophic (QG) mixing layer (or vortex strip/filament) in atmosphere under the influence of a background shear. The QG mixing layer is modelled with a family of quasi-Rayleigh velocity profiles in which the potential vorticity (PV) is constant in patches. In the KW perspective a counter-propagating Rossby wave (CRW) is created at each PV edge, i.e. the edge where a PV jump is located. The important parameters of our study are (i) the vorticity of the uniform shear m and (ii) the Rossby deformation radius Ld, which indicates how far the pressure perturbations can vertically propagate. While an adverse shear (m < 0) stabilizes the system, a favorable shear (m > 0) strengthens the instability. This is due to how the background shear affects the two uncoupled CRWs by shifting the optimal phase difference towards large (small) wavenumber when m < 0 (m > 0). As the QG environment is introduced a general weakening of the instability is noticed, particularly for m > 0...