The Relation between Beltrami's Material Vorticity and Rossby–Ertel's Potential Vorticity

It is shown that there is an exact correspondence between the scalar Rossby–Ertel's potential vorticity (PV) for a field ε, and the component of Beltrami's material vorticity along the ε-coordinate line (first equivalence theorem). Thus, Rossby–Ertel's PV can be interpreted as a particular case (scalar) of the vectorial Beltrami's material vorticity. The rate of change of Beltrami's vorticity only depends on the curl of the acceleration (or baroclinic–diffusive terms in the rate of change of vorticity) and not on the convective rate of change (involving advection, stretching, and divergence terms). When the motion is circulation preserving (the acceleration is irrotational) Cauchy's vorticity formula states that Beltrami's material vorticity is conserved. However, when the curl of the acceleration is zero only along the direction normal to certain ε surfaces, only the ε component of Beltrami's material vorticity is conserved. Thus, a second equivalence theorem states that Ertel's PV conservation theorem is equivalent to Cauchy's vorticity formula along the ε-coordinate line. Beltrami's material vorticity is first introduced via Piola's transformations from the spatial vorticity field, but it is shown that a direct definition of Beltrami's material vorticity using the referential velocity and in terms of material variables is also possible. In this approach the current definition of specific PV in terms of the spatial vorticity and the gradient of ε becomes instead a relation between both vorticities that can be derived from their respective definitions.