What is the expression for pressure distribution in a rotating tornado?

[Raptor112]

Homework Statement

A tornado rotates with constant angular velocity $\omega$ (i.e. like a solid object would rotate) and has
a uniform temperature T. Find an expression for the outward pressure distribution in terms of the central pressure $p_0$. Use this to calculate $p_0$ given that T = 300 K and that at 0.1 km from the centre the atmospheric pressure p = 100 kPa and wind speed V = 100 m s-1.2. The attempt at a solution
Pressure gradient force =$-\frac{1}{\rho} \nabla p$
PV=NRT

 

[Nathanael]

PV=NRT

In my incomplete understanding, an 'equation of state' (such as PV=NRT) describes the properties of a system near equilibrium. I don't believe it can be (at least not naively) applied to more dynamic systems (such as a tornado). If anyone has more to say on this, please do; I like to learn as well.

Pressure gradient force =$-\frac{1}{\rho} \nabla p$

Perhaps you mean "acceleration due to pressure gradient force" (check dimensions).
Anyway you want to model the tornado as having uniform angular velocity, so can you figure what the acceleration field (i.e. the left side of this quoted equation) is? (Use symmetry when choosing coordinates.)

 

[Raptor112]

Perhaps you mean "acceleration due to pressure gradient force" (check dimensions).

Pressure gradient (PG) = rate of change of pressure with distance which is from low to high pressure.PGF is from high to low pressure. But still don't see how you get the pressure distribution?

 

[Chestermiller]

If a body is rotating with radial acceleration $\omega^2r$, it is analogous to a gravitational force acting in the negative radial direction. So, from the hydrostatic equation,

$$\frac{dp}{dr}=\rho (\omega ^2r)$$

Of course the air density $\rho$ is a function of the pressure, so you need to take that into consideration using the ideal gas law. What is the equation for the density of an ideal gas in terms of the pressure, molecular weight, and temperature?