Re: [Phys-L] endless vortex lines

[John Denker <jsd@av8n.com>]

On 12/06/2014 02:10 PM, Chuck Britton wrote:

> Does this Mathematical Result apply to tornadoes, hurricanes, bathtub
> drains, coffee cups etc.? 

Yes.

> (or only to rings?)

Not just rings.  It's not rocket science.  Pick any
(or all) of the following:

    ∇ • (∇ × velocity) = 0                            [1]

Actually I'm pretty sure that 

    ∇ • (∇ × almost_any_vector_field) = 0             [2]

To say the same thing without cross products:

    ∇ ∧ ∇ ∧ almost_anything_whatsoever = 0            [3]

We can also write

    d d almost_anything_whatsoever = 0                [4]

which is equivalent to [3] by definition (i.e. the definition
of "d", the exterior derivative).

That simplifies to the operator equation

    d d = 0                                           [5]

In words, that says "the boundary of a boundary is zero".  For
example, the topographic contour lines on a landscape do not
end.  You can make contours on a map "seem" to end by cutting 
the map, but the map is only a representation of the physical 
landscape, which is unaffected by such chicanery.

The result depends on the equality of mixed partial derivatives
and not much else.

I say "almost" anything to avoid quibbles about stuff that 
wasn't differentiable to begin with.

In cases where it's not immediately obvious where the
vortex lines went, it is almost always a worthwhile
exercise to figure it out.

Hint:  For the tornado:  Consider the no-slip condition.
There is always a layer of non-moving air right next to
the ground.  How is that related to the next layer up?

=================

As a general rule, people don't understand fluids nearly so
well as they think they do.  Just because folks have seen 
lots of fluids doesn't mean they have any real grasp of
what's going on.  Feynman once said fluid dynamics is at
least as complicated as quantum electrodynamics.  He's not 
the only one to think so:
  http://www.claymath.org/millennium-problems/millennium-prize-problems
  http://www.claymath.org/millenium-problems/navier%E2%80%93stokes-equation
  http://www.claymath.org/sites/default/files/navierstokes.pdf