On a recent trip to Niagara Falls we noticed that the pool below had a
rising mist or haze above it. We were told that this is always the
case. Later, we learned that all waterfalls exhibit this.

The mist apparently consists of droplets of water (like fog or steam).
We would like to know how the energy of the dropping water produces it
and what causes it to rise. An answer should trace the contributions
made by (and relationships among) heating, changes in vapor pressure,
condensation, mechanical "splashing," and so forth. How these are
influenced by the height of the falls, the ambient termperature and
humidity, and possibly other relevant variables are also of interest.

The more quantitative the result, the better.

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Request for Question Clarification by lmnop-ga on 23 May 2003 15:12 PDT

A truly inspired question! Many of us would love to tackle it, but
your requirements make it very demanding. Are you looking for short
answers to each of those points?? It's hard to push it too hard for
the category you've agreed to. You certainly have the general points
to consider mapped out well, and that's a help.
Any clarification would be appreciated! 
LMNOP-ga

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Clarification of Question by redzinonly-ga on 23 May 2003 19:04 PDT

We are most interested in the relative contributions of warming (motion at the
molecular level) versus mechanical action (motion at the particle level). Our
calculation indicates that warming does not supply so much change in temperature
as we had expected.

We are also especially interested in the role of the fact that water vapor is
lighter than air. To what extent does that play a part?

For this price, we certainly don't expect anybody to explore the phenomenon from
scratch. And it's unlikely that there is a suitable explanation on the web. On
the other hand, this certainly can't be the first time the question has been
posed. We're hoping that somebody (probably well-grounded in physics) can locate
a quantitative explanation in the literature, perhaps in a Meteorology textbook.

A satisfactory answer is *not* simply a recounting of isolated known physical
facts. We already have that.

Thanks for your interest.

The phenomena you describe are powered by the conversion of potential
energy to heat as the water falls:

"In 1847, when [James] Joule married Amelia Grimes, he took a large
thermometer on their honeymoon in order to check the temperature of
water falling down the face of waterfalls--he was looking for a
temperature change as an indication of the presence of kinetic
energy--and there is a wonderful photo of Amelia, sitting in a
carriage with a parasol, while James is checking the waterfall. One
wonders what she thought!"
http://america3.pcs.cnu.edu/~gwebb/103_l12.joule.htm

I like this question.  I don't know the answer, but here are some
thoughts:

The increase in temperature may be important for other reasons, but I
don't think it is enough to create unstable conditions just by the
thermal expansion of the air.

Take a waterfall 40m high.  Each kg of water that falls loses 400 J of
potential energy.  Some of that goes into kinetic energy on a
macroscopic scale, but most gets converted to thermal energy.  So the
increase in temperature could be as much as almost 0.1 degree C. 
(heat capacity (cp) is about 4200 J/kg/deg). But the atmosphere only
becomes unstable when the rate of decrease of temperature with height
is greater than the adiabatic lapse rate (that is, when the potential
temperature decreases with height).  The adiabatic lapse rate is g/cp,
and over 40m, it amounts to a difference of about 0.1 degree.
Presumably, the water was in thermal equilibrium with the air at the
top of the falls.  So the falling water may just be able to create
neutrally stable conditions, but not unstable ones.

It may be that the slight increase in temperature raises the vapor
pressure of the water, and that the relative lightness of water vapor
is important.  But vapor pressure of water only increases by about 1%
for a temperature increase of 0.1 deg (from the Clausius-Clapeyron
equation).  Water vapor makes up at most a few percent of the air, so
a 1 percent change in something that's already a small fraction of the
total seems negligible.  Especially because the difference in density
is not drastic: a water molecule weighs about 2/3 of what air weighs
per molecule on average.

Could it be just that the moving water creates turbulence in the air,
which carries the mist upward from its generation point at the bottom?

DISCLAIMER: This is an unprofessional, half-educated guess from
someone who has only very limited knowledge (but a great interest) in
this subject. I do hope that you are able to find a documented answer
for this using Google Answers!

Even at low temperatures, water is evaporating (also below room
temperature). The water-water hydrgen bonds (being unstable) and
constantly reforming is partially what makes water polar. The two
hydrogen atoms in H20 have electrons that are whizzing by at fast
speeds and even when water is sitting at room temperature, and below,
it will slowly evaporate because the fast (warmer=faster) molecules
"whiz" into the air leaving (escaping) the cooler molecules of water
on the surface. Another property of water is that it tends to go where
there is less water to where there is more water, so the droplets that
have escaped in essence, PULL more water into the droplets
(adhesion/cohesion - same as water being pulled up against gravity in
a plant goes faster in dry conditions), while others are "splashed
there" by the fast moving water.

In addition, when water falls down a waterfall, the speed of the
molecules moving is heightened so that more of them manage to escape
and form water vapour, which returns to liquid form shortly, hence
water droplets dropping.

Wouldn't it be cool if this was at least partially right? :)

tisme-ga