If we look at a two-dimensional sphere and draw a continuous vector field on it, you can compare this with a tennis ball whose hairs you comb in a continuous way. Two points on the ball will have an undefined direction of the hair. If you comb them from up-down, or from the right to the left, or whatever way you try to comb it, two points arise where there is no direction of the balls hair.

The wind field in the atmosphere is a continuous vector field (which can be compared with the directions of the hair on the ball). The difference is that the atmosphere in which the wind field is situated is three-dimensional. But I can't imagine that this one extra degree of freedom makes any difference for the same two points.

So my question is: are there in the atmosphere of the earth (at least?) two points in which the wind field is not defined, and does this mean the same as saying that there are two positions in the atmosphere where there is no wind?

  • $\begingroup$ Related and possibly answered, but I am not capable of judgement Cyclones on Earth $\endgroup$ – user154420 Jun 20 '17 at 12:09

You are right that the degree of freedom in the altitude / height in the atmosphere does not change things greatly for the implications of the hairy ball theorem for meteorology. All you need to do is apply the theorem to every constant altitude surface or any continuous deformation thereof contained in the atmosphere. The theorem then dictates that there must be at least one point somewhere on that surface where the wind vector vanishes - i.e. a cyclonic / anticyclonic eye - or a source / sink (where air flows upwards / downwards and leaves a sink / source in each constant altitude layer). The hairy ball theorem applied to the 2-sphere $\mathbb{S}^2$ simply says that there must be at least one point where the wind vector field, assuming that the field is continuous, vanishes.

You don't need two singular points as you seem to think: consider the flow lines resulting from the projection of the lines $\mathrm{Re}(z)=const$ of constant real part (or any family of parallel straight lines) on the complex plane onto the Riemann sphere through stereographic projection. There is one singular point at the North pole.

Now I'm not knowledgeable enough about fluid dynamics or meteorology to deal properly with the proviso in the theorem that the vector field must be continuous. What physical atmospheric conditions correspond to discontinuities? Presumably these could be physical insofar that one could imagine regions corresponding to a discontinuity where the air motion were so chaotic and varying with position that one couldn't really assign a wind field there properly.

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  • $\begingroup$ Thanks for your informative answer, I was going to write the word "Turbulence?" in the comments section. $\endgroup$ – user154420 Jun 20 '17 at 12:17
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    $\begingroup$ There certainly are apparent discontinuities in the atmosphere at a macroscale and larger (or what meteorologists call microscale and larger). They're called weather fronts, squall lines, dry lines, etc. $\endgroup$ – David Hammen Jun 20 '17 at 12:26
  • $\begingroup$ @DavidHammen Of course! $\endgroup$ – Selene Routley Jun 20 '17 at 13:32

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