In fluid dynamics vorticity is defined as the circulation per unit area at a point in the flow field. It is a vector quantity, whose direction is (roughly speaking) along the axis of the swirl. For a two-dimensional flow in the x-y plane, the vorticity vector points out of the plane, in the z direction. One way to visualize what vorticity means is this: consider a fluid flowing. Imagine that some tiny part of the fluid is instantaneously rendered solid, and the rest of the flow removed. If that tiny new solid particle would be rotating about itself, then there is vorticity in the flow. In fact, the magnitude of the vorticity would be twice the magnitude of the angular momentum of the new solid.
The barotropic vorticity equation is the simplest way for forecasting the movement of Rossby waves (that is, the troughs and ridges of 500 mb geopotential) over a limited amount of time (a few days). In the 1950s, the first successful programs for numerical weather forecasting thus utilized that equation.
In modern numerical weather forecasting models and GCMs, vorticity is just one of many prognostic variables.
Vorticity is important in many other areas of fluid dynamics. For instance, the lift distribution over a finite wing may be approximated by assuming that each segment of the wing has a semi-infinite trailing vortex behind it. It is then possible to solve for the strength of the vortices using the criterion that there be no flow induced through the surface of the wing.
See also: vortical motion.