What's the difference between vortex, vorticity, swirl and helicity? I want use plot3d function swirl to analyse the flow field, who can tell me it's meaning?

Vortex itself is a description of a special flow phenomenon. In fluid mechanics, it is strictly defined by vorticity. The helicity was originally defined as the intergation of scalar product of velocity vector and the vorticity vector. The momentum, the energy and the helicity are there main conservation variables of fluids. These things can be found in Frisch's `Turbulence'. I never come to strict definiation of swirl. It sounds like the vortex. Anyone knows?

Swirl and Helicity are the same thing: a scalar property field defined as vorticity-dot-velocity. It defines the component of vorticity which is parallel to the local velocity vector.