Particle relabelling symmetries and Noether's theorem for vertical slice models

TL;DR

This paper explores the variational structure of vertical slice models, revealing that their Kelvin circulation and potential vorticity conservation stem from a relabelling symmetry related to vertical slice diffeomorphisms.

Contribution

It demonstrates that vertical slice models possess a broader relabelling symmetry than previously recognized, linking this symmetry to conservation laws via Noether's theorem.

Findings

Kelvin circulation theorem holds on all materially-transported loops

Potential vorticity conservation derived from circulation theorem

Relabelling symmetry explained for vertical slice models

Abstract

We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc Roy Soc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closed loops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can be derived directly from this circulation theorem. In this paper, we show that this property is due to these models having a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, not just those diffeomorphisms that preserve the potential temperature. This is developed using the methodology of Cotter and Holm (Foundations of Computational Mathematics, 2012).