What is the relationship between tornadoes and dew point?

[vmedica]

Homework Statement

I am currently stuck on parts 2b,c and d of this problem.
http://www.wopho.org/download/Theoretical_3_Tornado.pdf

Homework Equations

The Attempt at a Solution

For part 2b, I use the the equation from part 2a and just integrate. I know that as we exit the tornado, the pressure should be atmospheric pressure so I can solve for the constant. but the problem that I have is the ln(r) term. I can choose a sufficiently small r such that the pressure is negative which does not make sense.

2c) I'm assuming it related to part 2b) so I have left that out for now.

For part 2d) I think that the solution would involve something about the dew point in air but when I type for information regarding the dew point into Google, I just get complicated formulae. Could someone perhaps suggest a website where I can get more information about this.

Any help would be appreciated.

 

[haruspex]

For part 2b, I use the the equation from part 2a and just integrate.

Since v = v(r), I don't understand how you can "just integrate" that equation. Seems to me you need another equation from somewhere. E.g. if we assume irrotational flow except at the centre, v ~ 1/r; or if we treat it as rigid body, v ~ r. Either way, the DE gives you a formula for P=P(r), and doesn't seem to contradict any given info. Maybe I'm missing something.
Oh, and please post your working.

 

[vmedica]

For my working, P=∫ρ_airv²/r dr and then I got a ln(r) term which I can now see is wrong. For a later part (3b inside the tornado) they assume that it is a rigid body so so v~r. Would it be sensible to assume irrotational flow and what other things exhibit this kind of flow? Also, would the method then become:

v=k/r
dP=ρ_airk²/r³dr
Now I integrate and use the contidion that when r=r_c, v=v_c to solve for k. Then I use that as P=P_atm as r→∞ to eliminate the constant of integration.

 

[haruspex]

For my working, P=∫ρ_airv²/r dr and then I got a ln(r) term which I can now see is wrong. For a later part (3b inside the tornado) they assume that it is a rigid body so so v~r. Would it be sensible to assume irrotational flow and what other things exhibit this kind of flow?

The standard treatment of a vortex is that it is irrotational flow, except in a core region, where it is more like rigid body. See http://en.wikipedia.org/wiki/Vortex. I was not at all sure which you were expected to use, or maybe some fancier relationship that encompasses both.

Also, would the method then become:

v=k/r
dP=ρ_airk²/r³dr
Now I integrate and use the contidion that when r=r_c, v=v_c to solve for k. Then I use that as P=P_atm as r→∞ to eliminate the constant of integration.

Looks right.

 

[vmedica]

Thanks for your help. I have been researching the tornado and wikipedia says the water condense due to adiabatic cooling. I know that means P^1-γT^γ=constant and they also give γ suggesting this is perhaps the correct approach to part d). I understand that with the equation for pressure, i need to equate this to the pressure at which water condenses but how can I do this?