Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2 / b / Solution

The first-order perturbation of the vortensity isTaking the curl of the linearized momentum equation and using the linearized continuity equation yieldsThis is linearized vortensity conservation. It is the perturbative form of material conservation of potential vorticity in an inviscid barotropic fluid; the forcing $-\nabla \Psi$ is a gradient and therefore creates no vorticity.