The dynamics of the gradient of potential vorticity

TL;DR

This paper investigates how the potential vorticity gradient evolves in stratified fluids like oceans and atmospheres, revealing conditions under which it behaves similarly to magnetic flux in MHD and when it does not.

Contribution

It derives a new equation describing the dynamics of the potential vorticity flux vector, highlighting when it is frozen into the flow and when it is not, extending understanding of PV transport.

Findings

The flux vector satisfies a modified curl equation with a source term.

When divergence of the flow is zero, the flux behaves like a frozen-in field.

The results have implications for measuring PV and temperature at the tropopause.

Abstract

The transport of the potential vorticity gradient $\bnabla{q}$ along surfaces of constant temperature $\theta$ is investigated for the stratified Euler, Navier-Stokes and hydrostatic primitive equations of the oceans and atmosphere using the divergenceless flux vector $\bdB = \bnabla Q(q)\times\bnabla\theta$, for any smooth function $Q(q)$. The flux $\bdB$ is shown to satisfy $$ \partial_t\bdB - {curl} (\bU\times\bdB) = - \bnabla\big[qQ'(q) {div} \bU\big]\times\bnabla\theta, $$ where $\bU$ is a formal transport velocity of PV flux. While the left hand side of this expression is reminiscent of the frozen-in magnetic field flux in magnetohydrodynamics, the non-zero right hand side means that $\bdB$ is not frozen into the flow of $\bU$ when ${div} \bU \neq 0$. The result may apply to measurements of potential vorticity and potential temperature at the tropopause.