Vorticity and Potential Vorticity Equations
The AQB form of the vorticity equation may be obtained by taking the Curl of eq.(39). To achieve this, the material derivative is expanded and the advective term is re-expressed using the vector identity eq.(A7) (with ) giving:
where is the vorticity; the momentum equation may then be written as:
where is the absolute vorticity given by:
Now taking the Curl of eq.(62) and using vector identity (A3) gives:
but
from identity (A6), as so one may write the vorticity equation as:
This states that the Lagrangian rate of change of vorticity (term 1) is due to three processes : volume expansion of fluid parcels (term 2); stretching and tilting of vortex lines (term 3) and baroclinic generation (term 4 on RHS). Note that vorticity generated by baroclinity is horizontal i.e. there is no effect on the vertical component of vorticity in this term. The only difference between the AQB vorticity equation and the unapproximated vorticity equation is the simplified form of the baroclinic generation term.