# Rotarium

Posted on August 14, 2010

Here’s my video!  It’s far from flawless, but I’ve decided to stick a fork in it.  Thank you to all who gave me some excellent constructive criticism.  I’ll be sure to apply it to my next video!

… and tornadoes.  My dad, Stanley Schleifer, happens to be the chair of the Department of Earth and Physical Sciences at York College, so he had a few things to say about the role of the Coriolis effect in weather systems.  He prefers to think of it in terms of conservation of angular momentum.  You know how figure skaters often start themselves spinning and then pull their arms inward to make themselves suddenly spin faster?  That’s conservation of angular momentum.  Angular momentum is defined as angular velocity times moment of inertia.  Moment of inertia is sort of the angular equivalent of mass.  “Conservation of angular momentum” means that the angular momentum of an an object won’t change unless it is subjected to a torque (angular equivalent of force).  You can’t change your body’s mass without taking in or expelling material – an inconvenient prospect while figure skating – but you can easily change your body’s moment of inertia.  When a figure skater pulls their arms inward, that decreases their moment of inertia.  Since their angular momentum (angular velocity times moment of inertia) must stay the same, their angular velocity increases!

The same thing happens with the formation of hurricanes.  When a very large region of low pressure air is surrounded by higher pressure air, the system will equalize the pressure – the high pressure air will flow inward.  In other words, the weather system will “pull its arms inward” like a figure skater.  So if it already has some angular velocity, it will be increased to conserve angular momentum.  But why would the air already have significant angular velocity?  Because the whole planet is spinning!  In the case of a hurricane, the change in moment of inertia is so huge, that you end up with, well … a hurricane.  Regarding the direction of rotation: when you view an analog clock from the front, its hands move clockwise of course.  But if the clock face is transparent and you could see the hands from behind, you’d see them moving counterclockwise.  It is for the same simple reason that hurricanes in the northern hemisphere rotate counterclockwise while their southern brethren rotate clockwise.

This same phenomenon can be understood from an Earth-fixed frame of reference by examining the Coriolis forces at work in the rotating environment that is the planet Earth.  So now we see our low pressure air surrounded by high pressure air, and the air rushes inward.  “Rushes inward”?  That sounds like high velocity.  And Coriolis forces are velocity dependent, right?  The air will “feel” a Coriolis force in a direction orthogonal to the air’s velocity and orthogonal to the environment’s (Earth’s) axis of rotation.  Since the air is moving inward, all those right angles push the air around in a circle and a hurricane is born.

If the scale is reduced, say to the size of a tornado, then the velocities involved are smaller and the distances over which the Coriolis force can accelerate the air are also smaller.  For that reason, only about 70% of tornadoes in the United States rotate counterclockwise.  That means that about 30% of the time, the air already has clockwise angular momentum strong enough to overpower the Coriolis force.  Still, the Coriolis force can claim responsibility for the fact that there is a bias at all.

If the scale is reduced even further, say to the size of a toilet bowl, the Coriolis forces are even weaker.  Contrary to popular belief, at this scale, Earth-based Coriolis forces are dwarfed by angular momentum introduced in the flushing process by asymmetries in the toilet itself – we see toilet water vortex in both directions in both hemispheres.  But if the effect of these asymmetries is so variable, then why does the toilet water always vortex on its way down?  Why doesn’t it sometimes go down without spinning at all (crazy toilets that flush super-fast notwithstanding)?  Because even the tiniest angular momentum in the water will undergo a self-reinforcing amplification process enabled by the water’s viscosity and fueled by the gravitational energy released as the water descends.  If that made no sense to you, please read it as, “There really is a good reason, but I won’t be fully explaining it here.”

YOUR HOMEWORK: Next time you use your toilet, take note of which way the water vortexes.  Then include that information in a comment on this post along with the geographic region in which the aforementioned flush occurred.  Let’s see for ourselves whether or not there is a latitude-based bias here!

And now… MATH!

But how do we know all this stuff about the Coriolis and centrifugal forces?  Where did all this “right angles” and “distance from the center” business come from?  It can all be derived mathematically!  All we have to do is transform Newton’s famous equation, $\vec{F} = m\vec{a}$, into our rotating reference frame and the Coriolis and centrifugal force terms pop right out.

Rather than painstakingly entering my own derivation here in LaTeX, I’ll direct those who want to prove this to themselves to the derivation supplied by Wikipedia.  The “Euler force” shown there does not apply to the situations I’ve described here, since I’ve only discussed environments rotating at a constant rate.

Finally, I’ll leave you with a link to an XKCD comic: http://xkcd.com/123/.