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

Abstract

Potential vorticity (PV) is usually defined as α??·?grad?, where α is the specific volume, ? is vorticity, and ? 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 ? surfaces. Since PV can also be expressed as α div(u ? grad?) and α div(??), 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 |grad?|) of the equation of motion onto the direction of grad?, or, alternately, as the divergence of a ?-weighted vorticity equation. The first of these interpretations is closely related to the procedure of considering every ? 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 as the material description of the fields. The kinematical foundations of the PV theorem in the material description are especially simple because they only involve derivative commutations with respect to the material variables. It is also provided a precise mathematical expression for the so-called impermeability theorem, clarifying the sense in which such a theorem can be understood. In order to do so it is necessary to introduce a suitable transformation of the fluid velocity. An immediate consequence of such a transformation is that the quantity ? and the quantity ??·?grad? (also called potential vorticity substance per unit volume) behave as a label of the particles and as the ?density,? respectively, of the transformed fluid. The impermeability theorem is then an expression of the conservation of the ?mass? and of the conservation of the identity of the particles in the transformed fluid.