Addendum: Hidden symmetries, trivial conservation laws and Casimir invariants in geophysical fluid dynamics (2018 J. Phys. Commun. 2 115018)

Abstract

An extension is proposed to the internal symmetry transformations associated with mass, entropy and other Clebsch-related conservation in geophysical fluid dynamics. Those symmetry transformations were previously parameterized with an arbitrary function $\mathcal{F}$ of materially conserved Clebsch potentials. The extension consists in adding potential vorticity $q$ to the list of fields on which a new arbitrary function $\mathcal{G}$ depends. If $\mathcal{G}=q\mathcal{A}(s)$, where $\mathcal{A}(s)$ is an arbitrary function of specific entropy $s$, then the symmetry is trivial and gives rise to a trivial conservation law. Otherwise, the symmetry is non-trivial and an associated non-trivial conservation law exists. Moreover, the notions of trivial and non-trivial Casimir invariants are defined. All non-trivial symmetries that become hidden following a reduction of phase space are associated with non-trivial Casimir invariants of a non-canonical Hamiltonian formulation for fluids, while all trivial conservation laws are associated with trivial Casimir invariants.