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**Unformatted text preview: **VORTEX r A A
Tic: haw“? M PhYS- 761H395/Hassam/Midterm Take Home #2,
Wa%»Nav«-l¥)~$—Mm The objective of this problem is to study a non-stationary (if % 0) equilibrium. Consider a. vortex, as shown. , A
( 9 =0) Um“ A e
‘- 1'“) ‘9) E-
6 1,_ 95
M We are given that 11' = Hg(r)fi, 6/62 2 0 and 3/33 = 0, Where (739,2) are cylindrical
coordinates. The vortex is of an ideal ﬂuid but the problem is studied in a frame which
itself is rotating with angular frequency 53 = (25 with respect to an inertial frame.
Consequently, in the frame of the lab, the equation of motion may be written as
3" . -. -. .. _, _
nM(€:-+ﬁ'-Vﬁ)=—Vp—2nMQ><u (9—0)
where 1? is referred to the lab frame and the Coriolis force appears in this frame (we ignore
the centrifugal force, proportional to {22, for convenience).
1. Write down the equilibrium equation involving a relationship between p, w, and n.
Beware: 7" - (if - 611') at {I - 6%]. For simplicity, assume that T = constant = To, Where p E 711". If ug(r) is given to be of the form is it possible, in principle, to solve for 110*) [and p(r)] assuming that p(r —> 00) ———> constant = noTo? . Suppose IQ is due to the Earth’s rotation about its axis. Consider a large scale
(continental) vortex of radius a ~ 3000Km. Suppose a?” 5 no N 10m.p.h. What is
the equilibrium equation to lowest order? What is p(r) to lowest order [i.e., when. a_ll
small parameters are neglected. Very simple but nonetheless signiﬁcant answers are
acceptable]? What is the equation for 33(7') to ﬁrst order (i.e., What is the ﬁrst most
significant terrn in the equilibrium equation)? Make a sketch of p(r) correct to ﬁrst order. ls p(0) < p(oo) or the other way around?
On what does the central pressure, “L912” or “High”, depend? Apply this to large scale weather patterns on the Earth’s surface — how would you expect pressure
lows/ highs to behave in Northern/ Southern Hemisphere for a given sense of the vor- tex? . Consider a tornado —-u Suppose (1 ~ 1km and no N 30mph. Again, ﬁnd p(r) to lowest
and ﬁrst order and comment on the central pressure, i.e., Whether a “low” or a “high”. Explain why, physically? Make a sketch of p(r). . Consider a hurricane, (1 ~ 1000km, uo ~ lOOmph. Can any reasonable approxima- tions be made to solving for 30(7)? ...

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