A general vortex has the velocity field tangential to the concentric circles about some point of origin and the magnitude of the velocity is is constant on any circle and is a function of the radius of the circle, say V(r).
In natural or flow-following coordinates the formula for the point vorticity ζ is:
For the case of counterclockwise flow Rs=r and ∂/∂n
The above equations reduce to
If a fluid is turning as a disk so that V(r) = ar for some constant a then the vorticity ζ is equal to 2a. If the velocity field were to have some constant component, say V(r) = ar+b, then the vorticity would be
It could be presumed therefore that V(0)=0. It might also be presumed that beyond some rmax the velocity would go to zero. If beyond some radius the velocity function were c/r then the vorticity would be zero.
A velocity field such that:
is sometimes called a Rankine vortex. For this vortex
Since the circumference of a circle of radius r is 2πr and the time τ taken to traverse this circle when the wind speed is V is 2πr/V the vorticity of a Rankine vortex is given by:
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