Piecewise uniform potential vorticity pancake shielded vortices

Shielded vortices consist of a core of potential vorticity (PV) of a given sign surrounded (or shielded) by a layer of opposite-signed PV. Such vortices have specific properties and have been the focus of numerous studies, first in two dimensional geometries (where PV is just the vertical component of the vorticity vector) and in geophysical applications (mostly in layered models). The present paper focuses on three-dimensional, spheroidal shielded vortices. In particular, we focus on vortical structures whose overall volume-integrated PV is zero. We restrict attention to vortices of piecewise uniform PV in the present research. We first revisit the problem within the quasi-geostrophic model, then we extend the results to the non-hydrostatic regime. We show that the stability of the structure depends on the ratio of PV between the inner core and the outer shield. In particular it depends on the polarity of the core and of the wavenumber of the azimuthal mode perturbed.