On the dissociation between potential vorticity conservation and symmetries

TL;DR

This paper demonstrates that potential vorticity conservation in fluid dynamics is a trivial conservation law unrelated to symmetries, challenging previous claims linking it to particle-relabeling symmetries.

Contribution

It clarifies the nature of potential vorticity conservation using a covariant formalism, showing it is not connected to any symmetry of the equations of motion.

Findings

Potential vorticity conservation is a trivial conservation law.

It is not associated with any symmetry of the equations.

The formalism applies in arbitrary coordinates, including comoving coordinates.

Abstract

Using a four-dimensional manifestly covariant formalism suitable for classical fluid dynamics, it is shown that the conservation of potential vorticity is not associated with any symmetry of the equations of motion but is instead a trivial conservation law of the second kind. The demonstration is provided in arbitrary coordinates and therefore applies to comoving (or label) coordinates. Since this is at odds with previous studies, which claimed that potential vorticity conservation is associated with a symmetry under particle-relabeling, a detailed discussion on relabeling transformations is also presented.