Conservation laws for potential vorticity in a salty ocean or cloudy atmosphere

TL;DR

This paper extends the fundamental conservation law of potential vorticity to more realistic oceanic and atmospheric conditions involving salinity and moisture, revealing new conserved quantities and symmetry relationships.

Contribution

It introduces generalized conservation laws for potential vorticity that include salinity and clouds, linking these laws to underlying symmetries in the fluid's Lagrangian.

Findings

Potential vorticity conserved over pancake-shaped volumes in salty and cloudy fluids.

Conservation laws relate to symmetries in the Lagrangian formulation.

Extended laws apply to more realistic oceanic and atmospheric scenarios.

Abstract

One of the most important conservation laws in atmospheric and oceanic science is conservation of potential vorticity. The original derivation is approximately a century old, in the work of Rossby and Ertel, and it is related to the celebrated circulation theorems of Kelvin and Bjerknes. However, the laws apply to idealized fluids, and extensions to more realistic scenarios have been problematic. Here, these laws are extended to hold with additional fundamental complexities, including salinity in the ocean, or moisture and clouds in the atmosphere. In the absence of these additional complexities, it is known that potential vorticity is conserved following each fluid parcel; here, for a salty ocean or cloudy atmosphere, the general conserved quantity is potential vorticity integrated over certain pancake-shaped volumes. Furthermore, the conservation laws are also related to a symmetry in the Lagrangian, which brings a connection to the symmetry-conservation relationships seen in other areas of physics.