# Non-trivial and trivial conservation laws in covariant formulations of geophysical fluid dynamics

## TL;DR

This paper investigates conservation laws in covariant geophysical fluid dynamics, revealing that some, like potential vorticity, are trivial and not linked to symmetries, and develops a Hamiltonian framework with constraints.

## Contribution

It introduces a covariant Lagrangian formulation for geophysical fluids, clarifies the nature of potential vorticity conservation, and constructs a Hamiltonian structure with explicit constraints.

## Key findings

- Potential vorticity conservation is a trivial law, not symmetry-related.
- All Dirac constraints are primary and second class, indicating no local gauge symmetry.
- Derived non-canonical Hamiltonian structure with time-dependent constraints and Casimir invariants.

## Abstract

Using a manifestly invariant Lagrangian density based on Clebsch fields and suitable for geophysical fluid dynamics, the conservation of mass, entropy, momentum and energy, and the associated symmetries are investigated. In contrast, it is shown that the conservation of Ertel's potential vorticity is not associated with any symmetry of the equations of motion, but is instead a trivial conservation law of the second kind. This is at odds with previous studies which claimed that potential vorticity conservation relates to a symmetry under particle-relabeling transformations. From the invariant Lagrangian density, a canonical Hamiltonian formulation is obtained in which Dirac constraints explicitly include the (possibly time-dependent) metric tensor. In this case, it is shown that all Dirac constraints are primary and of the second class, which implies that no local gauge symmetry transformations of Clebsch fields exist. Finally, the corresponding non-canonical Hamiltonian structure with time-dependent strong constraints is derived using tensor components. The existence of Casimir invariants is then investigated in arbitrary coordinates for two choices of dynamical variables in phase space.