(Solved): a. (40) Supercell mesocyclones and tornadoes can be approximated with a highly ideal ized model cal ...
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
fis 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
Rfrom 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.
