A conservation theorem for the f plane

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.