A potential vorticity equation may be derived from eq.(63) by taking the scalar product with .
It is easy to see that the baroclinic term vanishes under this operation since:
and and so that both triple scalar products are zero. Therefore, using integration by parts in the material derivative term:
followed by use of the identity (A9):
(for which the first term on the RHS is zero for adiabatic flow), gives:
Finally, using eq.(40) in the form:
to eliminate from eq.(64) gives the AQB potential vorticity equation:
This is identical in form to the Ertel potential vorticity except that the density is approximated with its basic state value.