The dynamics of the gradient of potential vorticity
arXiv:0911.1476
Abstract
The transport of the potential vorticity gradient $\bnabla{q}$ along surfaces of constant temperature is investigated for the stratified Euler, Navier-Stokes and hydrostatic primitive equations of the oceans and atmosphere using the divergenceless flux vector $\bdB = \bnabla Q(q)\times\bnablaθ$, for any smooth function . The flux $\bdB$ is shown to satisfy $$ \partial_t\bdB - {curl} (\bU\times\bdB) = - \bnabla\big[qQ'(q) {div} \bU\big]\times\bnablaθ, $$ 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.
8 pages, 1 figure. Comments welcome! Accepted at J Phys A