3 ms·
Considering multiple layers doesn’t provide more insight here. The atmosphere form one "thick" layer around the Earth and the air moves within this layer. Assum
by pixcavator 17y ago
Considering multiple layers doesn’t provide more insight here. The atmosphere form one "thick" layer around the Earth and the air moves within this layer. Assuming that this layer is closed (does not gradually comes to nothing), what we have is a "thick" sphere. Its Euler characteristic is still 2, thick or not. Now you can think of the wind as a continuous transformation of this solid. This map is homotopic to the identity map and, therefore, has Lefschetz number equal to the Euler characteristic, i.e., 2. Hence there is a fixed point. With some extra work you can show that this implies that the vector field has a stationary point. At that point the wind speed is 0, and not just the horizontal component. That point, certainly, does not have to be a cyclone. However, there is nothing wrong with the conclusion about the weather presented in the article; it’s just more idealized than this.
- Retric 17y agoThe point about layers is for each vertical cm around the earth you need to have that point with zero air speed. Same deal for any width you feel like talking about. You also need to have a line that connects all those points from earth to space. However, that line's path need not be strictly vertical it could wrap around the earth 3 times before reaching space. However, "In meteorology, a cyclone refers to an area of closed, circular fluid motion rotating in the same direction as the Earth[1][2]." which is not nessisarily true.
- pixcavator 17y agoAs I understand your point, the existence of a cyclone is proven only if the thickness of the atmosphere is ignored. I agree with that. On the second point, the hairy ball theorem does not guarantee circular motion. True.