Stationary vortices in three-dimensional quasi-geostrophic shear flow

An existence theorem for localized stationary vortex solutions in an external shear flow is proved. The flow is three-dimensional and quasi-geostrophic in an unbounded domain. The external flow is unidirectional, with linear horizontal and vertical shear. The flow conserves an infinite family of Casimir integrals. Flows that have the same value of all Casimir integrals are called isovortical flows, and the potential vorticity (PV) fields of isovortical flows are stratified rearrangements of one another. The theorem guarantees the existence of a maximum-energy flow in any family of isovortical flows that satisfies the following conditions: the PV-anomaly must have compact support, it must have the same sign everywhere, and this sign must be the same as the sign of the external horizontal shear over the vertical interval to which the support of the PV-anomaly is confined. This flow represents a stationary and localized vortex, and the maximum-energy property implies that the vortex is stable. The PV-anomaly decreases monotonically outward from the vortex centre in each horizontal plane, but apart from this the profile is arbitrary.