On the triviality of potential vorticity conservation in geophysical fluid dynamics

Using a four-dimensional manifestly covariant formalism suitable for classical fluid dynamics, it is shown that potential vorticity conservation is an algebraic identity that takes the form of a trivial law of the second kind. Noether’s first theorem is therefore irrelevant to associate the conservation of potential vorticity with a symmetry. The demonstration is provided in arbitrary coordinates and applies to comoving (or label) coordinates. Previous studies claimed that potential vorticity conservation is associated with particle-relabeling via Noether’s first theorem. Since the present paper contradicts these studies, a discussion on relabeling transformations is also presented.