Watery Wheels

From The Philosopher, Volume. LXXXXVIII No. 2 Autumn 2010

Lorenz’s Water Wheel

How do cycles in nature, chaos mathematics and physical science all combine to create apparently paradoxical effects, as discussed by Red Rachtagáin in his article ‘The Paradox Principle’. But consider that reliable old fashioned source of power - the water wheel. A simple but effective way to generate power from nothing more than a little bit of rainfall arriving via a water course. And here then a fine cast iron waterwheel, designed (mentally) by the weatherman-cum-physicist-of-chaos, Edward Lorenz.

It has two states:

1. The wheel is in a steady and predictable state - it is motionless.

2. Another steady state, as the buckets steadily fill and drain, it turns briskly.

As water pours into the first bucket, the bucket becomes heavy and the wheel starts turning clockwise. As soon as the first bucket has moved, as second bucket begins to fill up, and its weight too now adds to the force turning the mighty waterwheel. A third and fourth bucket arrive in turn, by which time the first bucket will have completed its descent and begun to rise up the other side of the wheel. This could slow the wheel down, but fortunately, or rather, by good design, each bucket has a hole in the bottom to let the water slowly out again, so by now as it ascends it is less heavy than when it started going down, and indeed by the time it reaches the top and the water supply it may be completely empty again.

Naturally, the waterwheel works very well and after a rather hesitant start even begins to accelerate. And so now the question: how much effort is required to stop this wheel spinning clockwise - and send it spinning in the other direction?

And no matter how huge and heave the water wheel is, at the crucial moment, the answer is just a single drop of water in the right place is enough.

The explanation for those who find this counter-intuitive (which it is) is that as the wheel spins faster and faster, the buckets going down will pass under the spout too quickly to properly fill up with water, whilst at the same time, the buckets on the other side of the wheel heading upwards may cease to properly empty. At some point, the poor wheel will be trying to descend with half empty buckets going down and half full buckets going up. That is to say, it will rotate ever more slowly and painfully, until one bucket find itself poised motionless under the water spout with the buckets underneath on each side exactly balancing.

At which point, that single drop of water is enough to decide the wheel in its decision whether to carry on turning clockwise, or to change direction and start going anti-clockwise.

Edward Lorenz offered the imaginary waterwheel example to demonstrate how physical systems can be steady (and that is to say predictable) for long periods but unpredictable under certain, perhaps rare, conditions. That a waterwheel like this will flip direction can be anticipated. but exactly when it will happen cannot be known in advance.

The wheel is now turning too fast for the buckets to either fill up properly or to empty reliably. It will eventually grind to a halt at which point a single drop of water will be enough to start it spinning in either direction.

- Martin Cohen

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