**SAN JOSÉ STATE UNIVERSITY**

*Thayer Watkins*

**The Vorticity of a Vortex**
**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:
**

**ζ = V/R**_{s} - ∂V/∂n

**
**For the case of counterclockwise flow R_{s}=r and ∂/∂n
is -∂/∂r

thus

####
ζ = V(r)/r + ∂V/∂r

Now consider

####
∂(rV)/∂r,

This is equal to
####
V + r∂V/∂r

= r(V/r + ∂V/∂r)

The above equations reduce to

####
∂(rV)/∂r = rζ

or, equivalently

####
ζ = (∂(rV)/∂r)/r

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

#### ζ = + 2a + b/r

and thus ζ would go to infinity as r goes to zero.
It could be presumed therefore that V(0)=0. It might also be presumed
that beyond some r_{max} 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:

### **V**(r,θ) = V_{max}(r/r_{max})__θ__ for r≤r_{max}

= V_{max}(r_{max}/r)__θ__ for r>r_{max}

where __θ__ is the unit tangential vector

is sometimes called a Rankine vortex. For this vortex

###
ζ = 2V_{max}/r_{max} for r≤r_{max}

= 0 for r>r_{max}

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:

####
ζ = 4π/τ.

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