The dynamics of the gradient of potential vorticity

The transport of the potential vorticity gradient along surfaces of constant potential temperature θ is investigated for the stratified Euler, Navier–Stokes and hydrostatic primitive equations of the oceans and atmosphere, in terms of the divergence-less flux vector , for any smooth function Q of the potential vorticity q . The flux vector is shown to satisfy a transport equation reminiscent of that for magnetic field flux in magnetohydrodynamics. The result may apply to satellite observations of potential vorticity and potential temperature at the tropopause.