  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.
