Some other reason. It's not the geometry ("perfect sphereness") of the earth that's the issue, it's the topology (that it's a ball).
The other reason is something like "wind velocity is not a continuous vector field". When you get down to the microscopic level, what does wind velocity mean?
"There's a cyclone somewhere on earth" is a pretty good approximation though.
Topologically speaking, the Earth is a 2-sphere. It's irregularities don't change that. On the other hand, a hairy coffee mug must be combable (since it's a doughnut).
I'm having trouble visualizing how a weather system with multiple layers would allow some points to have a horizontal speed of zero without creating cyclones. Is the implication here that there could be some permanent vertical flow of air, iff the atmosphere is layered?
Consider air flowing from North Pole to South Pole at the surface, rising from the South Pole to an upper layer, returning to the North Pole, descending, and repeating. There are no cyclones, the flow is constant, and there are places with zero horizontal velocity.
In reality the Earth rotates and the Coriolis force would create cyclonic circulations, but that's another matter.
Well, first problem is to define "immediately adjacent". In analysis, there is no point "next to" the point 0. I assume you mean a small neighborhood.
There are several ways to define these things. For example, in a small neighborhood, the particles could move at a speed proportional (ish) to the distance from the point. When you take 3D into account, some move tangentially, some move radially (ish) and it can all be smoothed out. It's easiest to imagine the entirety, then try to extend the macro movement smoothly into the corners. By using tricks such as asymptotic slowing it can all be done.
Are you having trouble visualising? Or are you worrying about the precise details. This is not a good medium for discussing either. A good text or tutorial on fluid mechanics will cover the various techniques for mapping an obvious flow into a less obvious flow.
It doesn't apply to continuous vectors, it applies to continuous vector fields. There is a difference, and it does matter.
And I would be interested to see how you can construct a non-continuous vector field from an incompressible (which air at these speeds effectively is) fluid. Your graphic and description do not make sense - they do not allow that, macroscopically, air is a fluid. If it travels, it has to go somewhere. It can't simply stop at a boundary, it has to change direction, and such changes of direction cannot be instantaneous. This is why in electronics we need to deal with signal reflection, over-voltages, and similar phenomena.
Additionally, continuous technically does not mean "no large jumps", although that's how the technical definition was inspired, and how most people visualise it. In particular, it's possible to create a function that's continuous at every irrational, and discontinuous at every rational.
It's certainly true that the theorem is dealing with a theoretical approximation to a messy, physical situation, but broadly speaking it's applicable. It says that at the Earth's surface there is always at least one place where the horizontal component of the air movement (wind velocity) is zero. Errors are often made when trying to make folksey explanations, and it's the interpretations that often have errors. The theorem is true, applicable and in some cases, useful.
I thank you for an explanation and not just more downmods.
To explain the wind graph - imagine very low air pressure at the points where it stops. You wrote incompressible, but that's not actually the case for wind (although yes the wind doesn't cause compression, but rather the reverse).
"one place where the horizontal component of the air movement (wind velocity) is zero"
If that's what it says, then yes, I agree that is true for wind. It didn't seem to be what it was saying though, but I guess I misunderstood it.
It said cyclone, i.e. wind moving in a circle, and that is just not correct. Wind can simply move out radially in all directions in straight lines from that point, without making a circle.
That point of course is where the velocity is 0.
To quote "(Like the swirled hairs on the tennis ball, the wind will spiral around this zero-wind point - under our assumptions it cannot flow into or out of the point.)". This is not true. Air can flow out of a zero point - just heat it up, and air will flow out of it.
> This is not true. Air can flow out of a zero point - just heat it up, and air will flow out of it.
I'm having a lot of trouble visualising this. There is no air at a point. A point, by definition, has no volume. Are you manufacturing air?
I think you are using some definitions that are completely at odds with what everyone else, including me, are using. No doubt if we stood in front of a whiteboard you could make yourself clear quickly, but almost everything you have said is, according to my model of how the world works, wrong.
I'd like to understand you, but I suspect that's never going to happen.
I know the theorem - I proved a generalised version of it as a base case for a much bigger result. I haven't bothered to read most of the comments because usually the whole thing is mis-quoted or mis-interpreted, but I just had to say something to try to understand you.
It's true that the theorem does not require a cyclone. The theorem can imply a zero point with the vector field radiating from it, but in a conservative 2D fluid flow that can't happen. In a 3D flow you can get that effect on the surface as the fluid descends to that point and then spreads, but that's different.
Perhaps you're responding to incorrect "interpretations", perhaps you're right and I just don't understand you, but you're really not making yourself clear. Either that, or you're wrong.
"There is no air at a point. A point, by definition, has no volume. Are you manufacturing air?"
Sorry. Assume an area, heat it up and wind flows out of it. However the air in the area itself is moving, so conceptually all the air is moving out of the point in the center of it. (It's not really, it's moving out of the area, but all the vectors point away from the point, so that's what it looks like.)
"It's true that the theorem does not require a cyclone."
Thanks. That's really all I was arguing about.
The thing with continuous and cyclone: I was assuming, that people were saying, that the wind _always_ has to move - even if in a circle. And I was saying, no, it doesn't have to move, you can have a still area, and wind radiating out of it (or into it).
If I am correct about that, then please edit the wikipedia article to remove mention of cyclones.
Why do you say that can't happen in a 2d fluid flow? Why does it have to be a cyclone? My understanding of weather is you have a large area, you heat it up, and wind flows out of it - but there is no cyclone. (I guess with fluid flow you are assuming there is no way to manufacture fluid, but with wind you can since heat will "create" more of it.)
Tell me if I'm wrong here:
The hairy ball theorem assumes there is hair everywhere, so you have to have a cyclone at the poles. But with wind there are spots without hair, so the theorem just doesn't apply to wind.
Wind can not move out radially in all direction from a point, unless air is manufactured at that point. If you thind of the wind as a divergenceless vector field, that situation is not possible.
If you heat it up, the air around the point will move outward. It will still not move out of the point, because there's no volume of air in the point that can be heated up, and (b) that's not a time-independent field either. It only works for an instant.
(This graph is really bad, the lines are actually curved where the points meet the lines. Imagine drawing circles around the earth for 2/3 of it. Then perpendicular half circles on the rest.)
Applying a pure function like this to real life wind which is not so constrained is about as right as applying the http://en.wikipedia.org/wiki/Banach-Tarski_paradox to a real life object, even if mathematically it's correct.
