On Ertel’s Potential Vorticity Theorem. On the Impermeability Theorem for Potential Vorticity

Abstract

Abstract Potential vorticity (PV) is usually defined as αω · gradϕ, where α is the specific volume, ω is vorticity, and ϕ is any quantity, usually a conserved one. The most common derivation of the PV theorem therefore uses the component of the vorticity equation normal to the ϕ surfaces. Since PV can also be expressed as α div(u × gradϕ) and α div(ωϕ), alternative derivations of the PV conservation law are introduced. In these derivations the PV conservation theorem is considered as the divergence of the projection (weighted by |gradϕ|) of the equation of motion onto the direction of gradϕ, or, alternately, as the divergence of a ϕ-weighted vorticity equation. The first of these interpretations is closely related to the procedure of considering every ϕ surface as a surface of constraint for the infinitesimal virtual displacements used in variational methods, and therefore it is closely related to a Hamiltonian derivation of the PV theorem. The different expressions are presented using the spatial as well...