5 ms·

I think it would be better to delete the eintire paragraph above your addition, as the notion that this theorem guarantees at least one cyclone somewhere on ear

I think it would be better to delete the eintire paragraph above your addition, as the notion that this theorem guarantees at least one cyclone somewhere on earth is ridiculous.

byrneseyeview 17y ago
Is it wrong because the earth is not a perfect sphere, or for some other reason?
- tesseract 17y ago
  I suspect it's more to do with the fact that (as Retric alludes to) the atmosphere is not a monolayer.
  - byrneseyeview 17y ago
    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?
    - RiderOfGiraffes 17y ago
      "Creating Cyclones" is a mis-leading phrase. 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. Down that path lies chaos.
      - byrneseyeview 17y ago
        How do the particles of air immediately adjacent to the zero-horizontal particles move?
        RiderOfGiraffes 17y ago
        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.
- tierack 17y ago
  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).
- ars 17y ago
  It's wrong because this theorem only applies to continuous vectors. However wind can simply fade (i.e. become slower), and is thus non-continuous. Edit: I'm getting beaten up in the downmods, but no one has shown me that wind is continuous.
  - WilliamLP 17y ago
    Do you know what continuous means?
    - ars 17y ago
      Yes, a function which has no large jumps. Wind has no such constraint. On top of that this only applies to "nonvanishing vectors", and again wind has no such constraint. -------| | | | | |------ -------| | | | | |------ -------| | | | | |------ -------| | | | | |------ -------| | | | | |------ Assume the lines wrap left/right and up/down. Wind can do this, and you have no pole. (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 http://en.wikipedia.org/wiki/Banach-Tarski_paradox to a real life object, even if mathematically it's correct.
      - deleted 17y ago
        [deleted]
  - RiderOfGiraffes 17y ago
    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.
- tome 17y ago
  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.
russell 17y ago
> Note that the eye can be arbitrarily large or small and the magnitude of the wind surrounding it is irrelevant. Given that the eye can be arbitrarily small, I would accept the statement that the earth would always have at least one circular wind pattern. The paragraph is not talking about just hurricane size.
- pixcavator 17y ago
  >the earth would always have at least one circular wind pattern Not necessarily. There may be two stationary points and all winds blowing from the first point to the second. Nothing circular...
  - DougBTX 17y ago
    Hopefully that won't last too long, or all the air on the planet will end up at point two! (A constant amount of non-compressible air was assumed in the example.)

I think it would be better to delete the eintire paragraph above your addition, as the notion that this theorem guarantees at least one cyclone somewhere on earth is ridiculous.

byrneseyeview 17y ago

Is it wrong because the earth is not a perfect sphere, or for some other reason?

tesseract 17y ago
I suspect it's more to do with the fact that (as Retric alludes to) the atmosphere is not a monolayer.
- byrneseyeview 17y ago
  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?
  - RiderOfGiraffes 17y ago
    "Creating Cyclones" is a mis-leading phrase. 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. Down that path lies chaos.
    - byrneseyeview 17y ago
      How do the particles of air immediately adjacent to the zero-horizontal particles move?
      - RiderOfGiraffes 17y ago
        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.
tierack 17y ago
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).
ars 17y ago
It's wrong because this theorem only applies to continuous vectors. However wind can simply fade (i.e. become slower), and is thus non-continuous. Edit: I'm getting beaten up in the downmods, but no one has shown me that wind is continuous.
- WilliamLP 17y ago
  Do you know what continuous means?
  - ars 17y ago
    Yes, a function which has no large jumps. Wind has no such constraint. On top of that this only applies to "nonvanishing vectors", and again wind has no such constraint. -------| | | | | |------ -------| | | | | |------ -------| | | | | |------ -------| | | | | |------ -------| | | | | |------ Assume the lines wrap left/right and up/down. Wind can do this, and you have no pole. (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 http://en.wikipedia.org/wiki/Banach-Tarski_paradox to a real life object, even if mathematically it's correct.
    - deleted 17y ago
      [deleted]
- RiderOfGiraffes 17y ago
  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.
tome 17y ago
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.

russell 17y ago

> Note that the eye can be arbitrarily large or small and the magnitude of the wind surrounding it is irrelevant. Given that the eye can be arbitrarily small, I would accept the statement that the earth would always have at least one circular wind pattern. The paragraph is not talking about just hurricane size.

pixcavator 17y ago
>the earth would always have at least one circular wind pattern Not necessarily. There may be two stationary points and all winds blowing from the first point to the second. Nothing circular...
- DougBTX 17y ago
  Hopefully that won't last too long, or all the air on the planet will end up at point two! (A constant amount of non-compressible air was assumed in the example.)