Whirlpools and Tornados

If you take a body of fluid having some angular momentum, like air or water in a bathtub, and draw it toward a center, such as by draining it down a hole, or draining it up with heat, it's going to concentrate its angular momentum in a smaller volume. That's how you get a vortex.

Tornadoes happen when angular momentum of air is concentrated sufficiently in a vortex vertical to the ground. It depends on various factors like updraft and wind shear between blocks of air. People are working hard to understand how they form. Here's some current knowledge.

I think this is a good question, for two reasons. First, I believe that whirlpools in the bath and tornadoes have much in common. Second, while whirlpools can be easily observed in detail, tornadoes cannot.

One important difference between whirlpools and tornadoes is that the water drains out of the bottom of a whirlpool while tornado air drains upward through the low pressure centre of the vortex. This is interesting because both phenomena must ultimately be driven by energy released as the overall system relaxes from its initial state to one with a lower energy.

For a whirlpool the energy flow looks obvious, the water falls in gravity and gains kinetic energy as it does so. The energy flow for tornadoes is less clear, to me at least: after all, the air is draining upward against the force of gravity. The Wikipedia entry for 'dust devils' says that the pre-condition for their formation is a layer of warm air lying below a layer of cool air. The air is in pressure equilibrium, but the situation is unstable because the opposite configuration - cool air below warm air - would have less gravitational potential, the cool air being denser than the warm air at the same pressure. If a dust devil or tornado provides a way for such layers to swap places then that could explain the energetics.

Apart from the energy consideration, there are two other physical principles involved:

First, angular momentum is conserved: at any time during the draining of the bath or unstable layer the total angular momentum of the system stays the same.

Second, water and air are viscous fluids. Viscosity works to eliminate velocity shear, I'll come back to this later.

Whirlpools start when the bath plug is pulled. At first the water immmediately above the drain hole just falls out. Water rushes in from the surroundings to take its place. As it does so it will tend to spin - actually it will just be spinning faster than it was, as the conservation of angular momentum acts to amplify any initial angular velocity. As more water drains away, water will be pulled in from greater distances from the hole axis, and its final angular velocity will increase, until a point is reached where the pressure needed to balance its centrifugal force equals that of the water in the bath. At this point the vortex has formed.

The basic physics of the vortex are fairly straightforward. Along the central axis there is a low pressure (or empty) volume, maintained by the outward centrifugal force exerted by the spinning air/water. In fact the vortex contains layers of fluid, spinning less slowly with increasing distance from the centre, each in force balance with the layers on either side of it.

If water was not viscous there would be no friction between the 'layers' spinning at different speeds. In that case, once the inner surface of the vortex moved outside the radius of the hole, no more water would flow out and we would be left with a steady vortex - the bath would stop emptying.

However, water is viscous, so the friction between layers slows down the inner parts of the vortex and transfers the momentum to layers further out. So, as water moves towards the hole its angular speed increases and so does its angular energy, but its angular momentum decreases as this is transferred away from the vortex. Water falling through the hole has high angular energy but low angular momentum. The angular momentum it loses is left in the water still in the bath. I think that's why as the bath is nearly empty, you often notice that the whole body of water is starting to visibly rotate.

I don't know enough about tornadoes to say definitely how they start. However, the initial conditions are unstable and it only needs something to start a body of warm air moving upward for the instability to develop - perhaps vapour condensing to form rain and releasing heat would be enough.

Once some air is moving upward, surrounding air will be pulled in, and a vortex should begin to form, with warm air draining upward. The initial vortex will be a weak affair, with slow moving winds, but as more air is moved upward so the effect will become stronger. Once the vortex has formed its behaviour should be very similar to a whirlpool.

Observers often comment that a thundercloud is ripe for a tornado to form when the circulating motion becomes more obvious to the eye. The whirlpool comparison would say that the increasing circulation is more likely caused by angular momentum left by air that has already drained upward - in other words a vortex has already formed, and been working for some time, you just can't see it.