A simple Lagrangian for shallow water equations with constant potential vorticity

Inspired by a recent gauge field description of shallow water equations [SciPost Phys. 14 (2023) 102], in this work we build a simple Lagrangian formulation for the case of constant potential vorticity. An attractive feature of the construction is that it does not appeal to a Clebsch-like parametrization or the use of auxiliary fields. It is also emphasized that on the equator, where the Coriolis parameter vanishes, the shallow water equations exhibit extra SO(2,1) conformal invariance. Using the group-theoretic method, a new exact solution is found, which is associated with the scaling symmetry transformation entering the conformal group.