Instability and wave propagation in compressible quasi-geostrophic dynamics

The vertical dependence of background density in the quasi-geostrophic system has a strong influence on the dynamics, though the inversion relation between potential vorticity (PV) and streamfunction: for constant background density, the Boussinesq case, the inversion relation is up-down symmetric; when the background density decreases exponentially with height, the compressible case relevant to large scale atmospheric dynamics, that symmetry is broken. Recently, the authors derived a closed form Green's function solution for the compressible quasi-geostrophic system, which shows explicitly the spatial dependence of the streamfunction response to a general distribution of PV. The Green's function decays algebraically like 1/r above the source point, compared with exponentially in all other directions. The stronger upward influence of PV means that upper levels of a finite volume vortex patch rotate faster than lower levels, stabilizing the upper vortex and destabilizing the lower vortex. As a consequence, wave propagation on a finite vortex is preferentially downward. For ellipsoidal vortices the effect is particularly dramatic with all wave breaking occurring in the lower vortex levels.

The effect is present over a wide range of parameters and is relevant to the situation of wave propagation on the stratospheric polar vortex. On vortices resembling the stratospheric polar vortex in size and aspect ratio, a disturbance placed centrally almost always exhibits significantly stronger downward propagation than upward. The preference for downward propagation is remarkably robust, occurring for a range of planetary wavenumbers, vortex shapes, and disturbance amplitudes.

The presence of a horizontal lower boundary introduces a strong barotropic component that is absent in the unbounded Green's function (the presence of an upper boundary has almost no effect). The lower boundary also alters the differential rotation in the lower vortex with important consequences for the nonlinear evolution: for very small separation between the lower boundary and the vortex the differential rotation is reversed leading to strong deformations of the middle vortex; for a critical separation the vortex is stabilized by the reduction of the differential rotation, and remains coherent over remarkably long times.