A conservation theorem for the f plane

TL;DR

This paper introduces a new conserved scalar, Lq, that improves the understanding of vorticity and fluid parcel behavior at small scales in oceanic and atmospheric flows, addressing limitations of Ertel's potential vorticity theorem.

Contribution

The paper derives a new conservation theorem involving Lq that better captures small-scale vorticity dynamics by incorporating angular momentum considerations.

Findings

Lq is conserved at small scales where Ertel's theorem fails.

The new scalar provides a more accurate tracer for submesoscale flows.

Application insights for oceanic and polar mesoscale dynamics.

Abstract

Ertel's potential vorticity theorem is essentially a clever combination of two conservation principles. The result is a conserved scalar $q$ that accurately reflects possible vorticity values that fluid parcels can possess and acts as a tracer for fluid flow. While true at large scales in the ocean and atmosphere, at increasingly smaller scales and in sharply curved fronts, its accuracy breaks down. This is because Earth's rotation imparts angular momentum to fluid parcels and the conservation of absolute angular momentum $L$ restricts the range of centripetal accelerations possible in balanced flow; this correspondingly restricts vorticity. To address this discrepancy, we revisit Ertel's derivation and obtain a new conserved scalar $Lq$ that more properly reflects the behavior of fluid parcels at these small horizontal scales. Application of the theorem is briefly discussed, with an emphasis on better understanding oceanic submesoscale and polar mesoscale flows.