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

On the dissociation between potential vorticity conservation and symmetries