Balanced ellipsoidal vortex equilibria in a background shear flow at finite Rossby number

We consider a uniform ellipsoid of potential vorticity (PV), where we exploit analytical solutions derived for a balanced model at the second order in the Rossby number, the next order to quasi-geostrophic (QG) theory, the so-called QG+1 model. We consider this vortex in the presence of an external background shear flow, acting as a proxy for the effect of external vortices. For the QG model the system depends on four parameters, the height-to-width aspect ratio of the vortex, as well as three parameters characterising the background flow, the strain rate, the ratio of the background rotation rate to the strain, and the angle from which the flow is applied,. However, the QG+1 model also depends on the PV, as well as the Prandtl ratio, (and are the Coriolis and buoyancy frequencies, respectively). For QG and QG+1 we determine equilibria for different values of the background flow parameters for increasing values of the imposed strain rate up to the critical strain rate, beyond which equilibria do not exist. We also compute the linear stability of this vortex to second-order modes, determining the marginal strain at which ellipsoidal instability erupts. The results show that for QG+1 the most resilient cyclonic ellipsoids are slightly prolate, while anticyclonic ellipsoids tend to be more oblate. The highest values of occur as. For large values of, changes in the marginal strain rates occur, stabilising anticyclonic ellipsoids while destabilising cyclonic ellipsoids.