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

TL;DR

This paper extends the understanding of symmetries and conservation laws in geophysical fluid dynamics by introducing a new potential vorticity-dependent transformation, distinguishing trivial and non-trivial invariants.

Contribution

It proposes an extension to internal symmetry transformations involving potential vorticity, clarifies the distinction between trivial and non-trivial Casimir invariants, and links hidden symmetries to non-canonical Hamiltonian structures.

Findings

Extended symmetry transformations with potential vorticity dependence.

Defined trivial and non-trivial Casimir invariants.

Linked hidden symmetries to non-trivial Casimir invariants.

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 $\cal F$ of materially conserved Clebsch potentials. The extension consists in adding potential vorticity $q$ to the list of fields on which a new arbitrary function $\cal G$ depends. If ${\cal G}=q{\cal A}(s)$, where ${\cal A}(s)$ 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.