On the triviality of potential vorticity conservation in geophysical fluid dynamics

Abstract

Using a four-dimensional manifestly covariant formalism suitable for classical fluid dynamics, it is shown that potential vorticity conservation is an algebraic identity that takes the form of a trivial law of the second kind. Noether's first theorem is therefore irrelevant to associate the conservation of potential vorticity with a symmetry. The demonstration is provided in arbitrary coordinates and applies to comoving (or label) coordinates. Previous studies claimed that potential vorticity conservation is associated with particle-relabeling via Noether's first theorem. Since the present paper contradicts these studies, a discussion on relabeling transformations is also presented. 0 0 00 a symmetric contravariant tensor for purely spatial distances with h 0 μ =h μ0 =0, p the pressure, and s the fluid specific entropy. The absolute nature OPEN ACCESS RECEIVED

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

“…The conservation law (4.9) is an identity and is demonstrated without assuming that the equations of motion are satisfied. The triviality of potential vorticity conservation was also demonstrated using the fields u , i r, and s-instead of Clebsch potentials-by Charron and Zadra (2018). When the equations of motion are under-determined, trivial conservation laws of the second kind may be associated with infinite-dimensional symmetries via Noether's second theorem.…”

“…The conservation laws associated with these active and passive transformations are called non-trivial: they exist onshell only, and they arise from internal or space-time symmetries of the equations of motion. These non-trivial conservation laws may be contrasted with trivial conservation laws of the second kind, which exist off-shell and cannot be obtained via Noether's first theorem (Olver 1993, Charron andZadra 2018).…”

“…Another fundamental conservation law in geophysical fluid dynamics is that of Ertel's potential vorticity. As will be shown below, the equation describing potential vorticity conservation is an off-shell identity and a trivial law (see also Charron and Zadra 2018). Therefore, Noether's first theorem is inadequate to associate potential vorticity conservation with a symmetry.…”

“…The Lagrangian density IIa  ( ) in Zadra and Charron (2015) leads to 16 Dirac constraints (13 primary, 3 secondary) but none are of the first class. In addition, the Lagrangian density I  ( ) in Zadra and Charron (2015)-the particle-like formulation used, for instance, by Padhye and Morrison (1996a, 1996b), Charron and Zadra (2018)-leads to three equations of motion x i L ( ) that are linear combinations of the Euler-Lagrange equations (2.4) (and the first-order derivative of one of them) obtained from the Lagrangian density (2.5):…”

“…Symmetries of the equations of motion leading to mass, entropy, momentum, and energy conservation are studied in arbitrary coordinates. In section 4, potential vorticity conservation-a trivial law of the second kind-is obtained independently of Noether's first theorem (see also Charron and Zadra 2018). In section 5, canonical and non-canonical Hamiltonian formulations are presented in arbitrary coordinates.…”