One of the first things they teach you in topology lectures is Brouwer's Fixed Point Theorem.
What this says, very roughly, is that you can twist and deform a rubber sheet as much as you like, but as long as it stays in the same plane, and you don't change its boundaries, and you don't tear it, then however much bending you do, there will always be at least one point that is in the same place as it started.
This has two important results. One is the implication that the skin on the top of your bedtime cocoa will, if it remains unbroken, always have one point in the same place as when you last stirred it. The other is the celebrated Hairy Ball Theorem (a corollary of the Fixed Point Theorem) which says that if you have a hairy ball - or a spherical, orifice-free dog - there is no way of combing it so that the fur lies flat everywhere, without a parting or a tuft.
Or, to express it in another way, there must always be at least one point on the earth's surface where the wind is not blowing. (To a topologist, of course, a combed dog and a map of the earth's winds are the same thing: they are both no more than vector fields defined over topologically equivalent surfaces).
This means that if you are looking for a calm spot for a picnic, you can be certain that somewhere on earth, one such exists. The trouble is that by the time you arrive, the point of windlessness will probably have moved somewhere else, so for all practical purposes the theorem is quite useless.
While you are sitting in the wind, trying to stop your picnic blowing away, try staring up at the clouds and see whether they are blowing in the same direction as your sandwiches. You may even notice that different patches of cloud at different levels are blowing in different directions to one another. This can make life difficult for both weather forecasters and topologists.
There are three basic features to bear in mind when considering winds: the temperature differential between the equator and the poles; the spin of the earth; and the friction of the earth's surface. It is the last of these that makes such a difference between low-level and high-level winds.
Since the sun gives most heat to areas near the equator, and the earth and sea heat the air, and cold air is denser than warm air, there is a constant movement of air between poles and Equator. This would flow in a north-south direction, were it not for the rotation of the earth, which (thanks to the Coriolis effect) gives an eastwards component to air flowing polewards in the northern hemisphere.
Near the ground, variations in the Earth's topography, the friction of the land itself, and differences between the heat-retaining properties of land and sea, all have their own effects on wind speed and direction.
At heights above half-a-kilometre or so, the winds are better behaved and often more predictable - as Richard Branson's team discovered to their cost when their balloon blew away at ground level.Reuse content