I was reading about IRMA Hurricane and I passed over some algebric theorem called The hairy ball Theorem. And at least, there is one hurricane in the world each day but of course, it could go from small wind storm to huge level 5 hurricane.
I watched a video on youtube about the theorem to understand it more. And I read a post on reddit saying:
Be aware that this doesn't mean a hurricane level storm or even a storm at all due to the added complexity of different layers of the atmosphere.
Anyway, as an engineer, I never passed with this similar theorem and I am not understanding it well. All vectors will be null at some point, like in the pole. And different wind map vectors in any area will have a null point where there is contrary wind vectors.
Can someone explain this theorem to me and why is related to say that storms and specifically wind storms are related to him ?
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$\begingroup$I'll confess. I didn't read the reddit post or watch the video. Hopeful this will answer your question though...
However, if we fix a vertical high above earth and treat wind as a continuous vector field at this height (meaning wind restricted to this layer has no up or down velocity, just horizontal), then the theorem states that there will be at least 1 point, at the given altitude, that has zero wind speed. This can be said for any fixed altitude.
Someone who misunderstands the last sentence might think that this implies there is a column of air somewhere on earth with zero wind speed (meaning its the center of a cyclone).
However, it doesn't really say this at all. All it says is that at each fixed altitude, there is 1 point with no wind. These points between altitudes do not necessarily coincide at all. Plus, in reality there are vertical wind speeds. So its not clear if this theorem is useful at all in talking about storms.
On the other hand, from an atmospheric modeling point of view, it is often common to consider so-called stable layers where this is little transport in and out of the layer, and on large enough spatial scales this layer 'looks' 2D. This has implications in the stability of coherent structures, turbulence, etc. Perhaps the theorem may be more useful in this setting, but I've never heard an atmosphere scientist mention hairy balls. Maybe I shouldn't say never...
$\endgroup$ $\begingroup$This is not true. Imagine the vector field were you just project one coordinate axis to the tangent spaces.
So say the wind only points to the north pole. You maybe also want to vanish it at some disc around the pole. Then there is no hurricane at any point.
But there is a "circle" around north and south pole where you have up and down winds.
Hairy ball tells you that you need to have an cyclone or an anticyclone(which I gave an example above). But since a hurricane is cyclone you can't conclude there is always one.
What is a vector field on a sphere? It is just a collection of vectors (arrows) which lie "flat" on the sphere. Now you can identify its direction with the direction of wind at a point and its length with the velocity of wind. Continuity means that if you move slow enough on the surface the wind speed/ direction does not make any jumps. So you could identify wind with a continuous vector field on a sphere. Then hairy ball theorem tells you there is spot where is no wind in direction to the surface.
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