Hidden symmetries, trivial conservation laws and Casimir invariants in geophysical fluid dynamics

Abstract

From a manifestly invariant Lagrangian density based on Clebsch fields and suitable for geophysical fluid dynamics, non-trivial conservation laws and their associated symmetries are described in arbitrary coordinates via Noether's first theorem. Potential vorticity conservation is however shown to be a trivial law of the second kind with no relevance to Noether's first theorem. A canonical Hamiltonian formulation is obtained in which Dirac constraints explicitly include the possibly timedependent metric tensor. It is shown that all Dirac constraints are primary and of the second class, which implies that no infinite-dimensional symmetry transformations of Clebsch fields exist and that Noether's second theorem does not apply to the governing equations. Therefore, the considered Lagrangian density does not admit a symmetry associated with potential vorticity conservation via Noether's two theorems. Finally, the corresponding non-canonical Hamiltonian structure with timedependent strong constraints is derived using tensor components for arbitrary coordinates. The existence of Casimir invariants is linked to trivial conservation laws of the second kind and to symmetries that become hidden after a transformation away from canonical dynamical fields.

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

“…In this example, because all non-trivial conservation laws are explicit and not hidden, the only Casimir invariants must be trivial. The volume integral of potential vorticity density provided by (5.70) in Charron and Zadra (2018a), being a boundary term, is such a trivial Casimir invariant. More generally, the integral over space of √ gρqA(s) is a trivial Casimir invariant (Appendix A, demonstration 3).…”

“…It was shown in Charron and Zadra (2018a) that the Lagrangian density L, based on Clebsch potentials and suitable for geophysical fluid dynamics, is invariant under certain internal transformations of dynamical fields also called global gauge transformations (see their subsection 3.4), giving rise via Noether's first theorem to mass, entropy and other Clebsch-related conservation. Those symmetry transformations were parameterized with an arbitrary and sufficiently smooth function F = F (s, γ (1) , λ (1) , ..., γ (N ) , λ (N ) ) * , which depends on specific entropy and other Clebsch pairs (but not on their space-time derivatives), all materially conserved on-shell.…”

“…where ǫ is an infinitesimal constant, are symmetry transformations of the dynamics † . This may be seen by establishing how the Lagrangian density L, given by (2.5) in Charron and Zadra (2018a), changes off-shell under these transformations. It turns out that δL = J µ :µ to order ǫ, where…”

“…An explicit association between, on the one hand, hidden global gauge symmetries and trivial conservation laws and, on the other hand, Casimir invariants in geophysical fluid dynamics was described in Charron and Zadra (2018a). However, the symmetry transformations proposed in Webb and Anco (2017); Charron and Zadra (2018a) are not as general as they may be, insofar as they do not involve a dependence on potential vorticity, which restricts the interpretation of a certain class of Casimir invariants as resulting from hidden symmetries. The internal symmetry presented in Charron and Zadra (2018a) is here generalized, allowing a complete characterization of known Casimir invariants in geophysical fluid dynamics.…”

“…However, the symmetry transformations proposed in Webb and Anco (2017); Charron and Zadra (2018a) are not as general as they may be, insofar as they do not involve a dependence on potential vorticity, which restricts the interpretation of a certain class of Casimir invariants as resulting from hidden symmetries. The internal symmetry presented in Charron and Zadra (2018a) is here generalized, allowing a complete characterization of known Casimir invariants in geophysical fluid dynamics. The generalization consists in symmetry transformations involving an arbitrary function of potential vorticity.…”