# A general theorem on angular-momentum changes due to potential vorticity mixing and on potential-energy changes due to buoyancy mixing

## TL;DR

This paper proves a general theorem showing that potential vorticity mixing always decreases angular momentum in quasigeostrophic dynamics, with implications for jet sharpening, PV staircases, and stability of shear flows.

## Contribution

It establishes a universal theorem relating PV mixing to angular momentum and energy changes, extending to multiple layers and stratification, and introduces a new nonlinear stability criterion.

## Key findings

- PV mixing always reduces angular momentum when dq_i/dy > 0.
- The theorem applies to both horizontal PV mixing and vertical buoyancy mixing.
- PV reconfigurations involving unmixing are necessary for certain PV staircase formations.

## Abstract

An initial zonally symmetric quasigeostrophic potential-vorticity (PV) distribution q_i(y) is subjected to complete or partial mixing within some finite zone |y| < L, where y is latitude. The change in M, the total absolute angular momentum, between the initial and any later time is considered. For standard quasigeostrophic shallow-water beta-channel dynamics it is proved that, for any q_i(y) such that dq_i/dy > 0 throughout |y| < L, the change in M is always negative. This theorem holds even when "mixing" is understood in the most general possible sense. Arbitrary stirring or advective rearrangement is included, combined to an arbitrary extent with spatially inhomogeneous diffusion. The theorem holds whether or not the PV distribution is zonally symmetric at the later time. The same theorem governs Boussinesq potential-energy changes due to buoyancy mixing in the vertical. For the standard quasigeostrophic beta-channel dynamics to be valid the Rossby deformation length L_D >> \epsilon L where \epsilon is the Rossby number; when L_D = \infty the theorem applies not only to the beta-channel, but also to a single barotropic layer on the full sphere, as considered in the recent work of Dunkerton and Scott on "PV staircases". It follows that the M-conserving PV reconfigurations studied by those authors must involve processes describable as PV unmixing, or anti-diffusion, in the sense of time-reversed diffusion. Ordinary jet self-sharpening and jet-core acceleration do not, by contrast, require unmixing, as is shown here by detailed analysis. Mixing in the jet flanks suffices. The theorem extends to multiple layers and continuous stratification. A corollary is a new nonlinear stability theorem for shear flows.