a.
(40) Supercell mesocyclones and tornadoes can be approximated with a highly ideal ized model called a Rankine vortex. The Rankine vortex has an inner portion that is in solid-body rotation and an outer portion where the tangential velocity is inversely prope tional to the radial distance
r, as shown in the figure below:
In polar coordinatios, the velocity field can he expressext as
V=v_(r)r^(˙)+v_(v)hat(\theta )
where
f is the unit vector is the radial dirextion and 0 is the unit vector in the direction of increasing angle
\theta .
Is the Rankine vortex, of -0 and
\psi _(\theta )={(V_(n)(r)/(R) if 0<=r=R):}
where
V_(R) is the maximum flow speod, which oscurs at radias
R from the wortex centen, Take the curl of this velocity field to find the wrticity in the two megions of the Rankine vortex. Big hint: the formula for the vertical component of worticity in cylindrical onondinates is
\zeta =(1del)/(rdelr)(1\theta r)-(1delv_(r))/(rdel\theta )
Here
\pi _(r) is the radial component of the flow, and
v_(0) is the tangental component of the flow. Note in this case that the wortex flow is axially symmerric with only a tangental component, so one of these terms will be zero.
b. Find expressions for the circulation at
r=0.5R,r=1.0R_(1)r=2.0R, and
r=4.0R.
