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Here is a picture of the usual vigorous Niagara Falls (in the winter). enter image description here

Here is the picture of Niagara Falls frozen in 1933 (in the very cold winter). Niagara Falls frozen in 1933

Here is the picture of Niagara Falls frozen in 1911 (in the extremely cold winter). enter image description here

Background knowledge: The Melting/Freezing point of the static water at 1 atm is 32°F (0.0°C or 273.15 K).

$\bullet$ I am curious to know under what conditions of the air pressure(atm), temperature, solute density in the water would cause the Niagara fall frozen?

Are other conditions like: winds, snowing or not, the strength of Van der Waals force, the moving speed/velocity of water and the pulling force of gravity (acceleration $g$), the height $H$ of waterfalls would be some important factors to predict the conditions to freeze the waterfalls?

Is there some simple qualitative relations or quantitative formulas describing the freezing point for waterfall? A $P$(pressure),$T$(temperature), etc, $\dots$ curve for the water-ice phase transition on the phase diagram? Perhaps the answer is very simple(?), but it is not entirely trivial to me.

ps. Happy winter/holiday times for those who reading the post, and especially those residing on the cold side of the north-hemisphere. Thanks for comments/replies!

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I suppose the waterfalls and the water in the river are NOT moving at a constant velocity (there are gravity, accelerations and also turbulence, etc), otherwise we can just choose another inertial frame of reference (with the same velocity) so the freezing point would be the same as the static one(?). :) –  Idear Dec 28 '13 at 23:27
    
[clarification]: consider the conditions of nearly completely frozen falls: which means there are water branches from the very top of the falls are frozen connected to the bottom of the falls. –  Idear Dec 29 '13 at 2:59
    
I suppose that the waterfall wont freeze easily; instead, I suspect the water will freeze somewhere upriver (as many rivers do), and then gradually work its way to the waterfall. In essence this means that when it actually freezes the quantity and velocity of the water are a lot less, making it easier to freeze. –  Ruben Jan 12 at 8:08
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3 Answers 3

I am curious to know under what conditions of the air pressure(atm), temperature, solute density in the water would cause the Niagara fall frozen?

In general, the answer is "a bit lower than 32 Fahrenheit". Here's two things which one might think would come into play, but actually do not to an appreciable extend.

  • Solute concentation

The major solutes which are present in the Great Lakes are sodium, chloride, and a host of other ions in lesser amounts. According to the government website here, the typical sodium concentration found in the Great Lakes is on the order of 10 mg/L. This is well within the regime of ideal solution behavior, and so we get a freezing point depression of $\Delta T\approx k_fbi=0.0016$ Kelvin (assuming NaCl). This is tiny compared to seasonal temperature variations, so solute concentration can be mostly ignored.

  • Air pressure

Air pressure is also almost completely negligible, for two reasons. One is that the pressure variation through the water depth of the water is much greater than typical atmospheric pressure variations; for example, atmospheric pressure ranges from 29.5 to 30.5 inHg, which is comparable to the pressure variation in a foot of water (and the river is considerably deeper than one foot at most places). Two is that even accounting for pressure variations of both the atmosphere and within the water itself, the linearized Clausius-Clapeyron relation predicts a negligible temperature dependence. For example, for the water-ice phase transition we obtain $$\frac{\Delta T}{\Delta P}=\frac{T\Delta v}{L}=0.00025\mbox{K/inHg}$$ which means that seasonal or daily variations in atmospheric pressure will only change the freezing point by something on the order of milliKelvin. Thus atmospheric pressure variation, and most likely water-depth variations in pressure, are negligible compared to seasonal variations in temperature.

  • Stuff this leaves out

This obviously leaves out a lot of kinetic and rate information. For example, dissipative heating at the foot of the waterfall will heat up the water there by approximately $$\Delta T=\frac{mgh}{C_p}=0.12\mbox{K}$$ which easily swamps the contribution due to air pressure and solute concentration. This also ignores nucleation effects, and it ignores the rates of coolings from evaporative cooling and air cooling, so it says nothing about how long one would have to wait for the entire waterfall to freeze in so-and-so temperature. However, I think it's safe to say that the temperature needs to be a good bit lower than 32F for the falls to freeze. It may also be worthwhile to look through historical archives and see what the mean temperature was in the years for the photographs you posted.

I'm not quite sure what you mean by Van der Waals force, as that's pretty much going to be constant within a given substance.

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Interesting, thanks DumpsterDoofus for the post, let me see. –  Idear Dec 29 '13 at 2:45
    
Thanks for the nice post. Actually we concern about the (nearly) completely frozen falls. In that case, I believe the temperature cannot be just slightly below 0$^o$C. I have seen the nearby falls of my working place which do not freeze (completely) at -5$^o$C or below, for example(?). –  Idear Dec 29 '13 at 2:53
    
let us say, nearly completely frozen falls means there are water branches from the very top of the falls are frozen connected to the bottom of the falls. –  Idear Dec 29 '13 at 2:58
    
I definitely agree that it'd have to be much lower than 0C for freezing to occur. Another thing to keep in mind is that the temperature one sees in weather reports is air temperature; the ground and bedrock underneath is a vast heat reservoir and is probably going to be slightly above the temperature of the air, so there is going to be warming of the river due to the ground itself. hwlau also makes a good point about the water flow rate, with streams of slow flow freezing more easily than streams of fast flow. –  DumpsterDoofus Dec 29 '13 at 4:22
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consider the conditions of nearly completely frozen falls: which means there are water branches from the very top of the falls are frozen connected to the bottom of the falls.

The picture you shown suggests the stuffs connecting from top of waterfall to the bottom is icicle, which can only be formed with not so fast flowing of water, otherwise high speed water flow can melt the icicle.

Is there some simple qualitative relations or quantitative formulas describing the freezing point for waterfall?

The first thing to know is that waterfalls are not a static system, so it cannot be specified solely by thermaldynamic variables $P$, $V$, $T$. Also, the temperature of air can be different from the temperature of river. Since water is flowing under gravity, it is hard to be freezed since the gravitation energy is kept converting to heat due to friction and viscosity (otherwise, the water would flow faster and faster, which is not the case). Moreover, when the temperature is cold enough, ice is formed on top of the river. Ice is not good conductor so that it takes even longer time for the river to freeze completely. It is similar to the "frozen" lake that often have thick layer of ice on top and water under it. In such cases, the temperature of water is usually close to or a bit higher than $0^oC$, which is higher than the atmospheric temperature.

It means that there is no simple single condition of "frozen" waterfall. It also depends on the shape of river, and "external" factor other than the river itself. Instead, it can be achieved with many different conditions.

That siad, the most important factor should be a slow flow of water. The width and depth of the river is a important factor. Also, the circumference of waterfall. Since we are consider how long does it take for icicle to connect to the bottom, the height of the waterfall is also a factor. The most important thing should be that there is long enough time of no rain fall and cold weather to freeze the source part of river so the water volume is low. With combination of these, we should be able to see the waterfall being frozen after a long time.

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@ hwlau, Firstly thanks for the answer. +1 from my support. Let me see in the next. –  Idear Dec 29 '13 at 7:35
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consider the conditions of nearly completely frozen falls: which means there are water branches from the very top of the falls are frozen connected to the bottom of the falls.

as long as water meet the zero deg C...it freezes...no matter what the situation...position..location...etc.

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@ user37046, hi thanks for the thought. Although one may not complete agree, for example, how to exactly measure the "temperature" at "accelerating" (non-equilibrium) fluid. To be honest, I have recently just do some REAL experiments on recording a nearby flow; so the room temperature recently was -20^C at my city, but the nearby waterfalls with a height of 1~2 meters do freeze but NOT completely freeze! I plan to show the Phys SE community the result and post it as a partial answer/thought. Update soon. –  Idear Jan 10 at 5:07
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