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In my world, physics works exactly like on Earth, except that there is a narrow, 400 foot tall magical column sticking vertically out of the ground, which is magically always extremely cold -- I'm thinking -100 ˚F (200 K), the temperature at which dry ice sublimates, but I'm curious about other temperatures as well. I haven't decided on the exact local environment, but I'm thinking a temperate climate.

How would a permanently cold column like this affect its surroundings?

What I have considered so far:

  • I suspect it would become encased in frost from the surrounding humidity.
    • How far would we expect that frost to extend?
  • Right at the edge of the freezing zone, I would expect there to be liquid water from the humidity, but I'm not sure how much.
    • Is there any limit?
    • Would a lake eventually form around it?
    • Would it flood the surrounding areas?
  • Beyond the edge of the freezing zone, it would be cool.
    • How far would that extend?
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    $\begingroup$ 400 tall, how wide and deep? $\endgroup$ – Raditz_35 Aug 9 '18 at 13:57
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    $\begingroup$ What happens to the thermal energy that would get passed to the column? Is it moved somewhere else or destroyed? $\endgroup$ – Kyyshak Aug 9 '18 at 14:04
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    $\begingroup$ the other important question is where is the tower - in a dry desert (not going to do much) is going to be very different to somewhere tropical (lots and interesting) to the Antarctic (nothing at all). $\endgroup$ – Windlepon Aug 9 '18 at 14:10
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    $\begingroup$ I concede that I'm not the best-versed in this site's conventions, but isn't the intention of reality-check, science-based, and hard-science to only use one of them? The descriptions of science-based and hard-science seem to directly contradict each other, so maybe this question should only be using one of those. $\endgroup$ – Zenon Aug 9 '18 at 15:57
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    $\begingroup$ Flick from A Christmas Story wants to put his tongue on that thing so bad. $\endgroup$ – Pink Sweetener Aug 9 '18 at 16:44
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For the column to permanently remain cold it would have to break the laws of thermodynamics.
I think it all depends on the how the column works but I can think of two possible outcomes.

The column is magically isolated
If the column never absorbs heat then it would remain cold.
That would also mean the surrounding environment would never dissipate heat to it and therefore would remain entirely unaffected.

The column is a void for heat
The column does in fact absorb heat, however all the heat is absorbs is magically destroyed.
If this were the case then it would absorb all the heat in the immediate area until it reached -100 also.
It would then presumably stop when the heat transfer rate of the air around the column was the same as the heat transfer rate of the sun over and around the area.

For the heat transfer rate of the column I need to take a couple of assumptions.

First I'm gonna assume the column is placed on the ground not into the ground - I know this is unrealistic but it makes it simpler.

The next assumption is a big one - size. 400ft (121.92m) tall and lets make it relatively thin with just a meter radius (3.28ft), also assuming its a perfect cylinder.

So given that the average temperature (obviously very variant) is 31 ºC or 304.15K that gives us a temperature difference of 104.15 K.

In addition the thermal transfer will be different in the ground and air.
There is lots of different thermal ratings dependant on soil.
Choosing a clay based soil the thermal coefficient is 1.1 W/mK.
For air this is 0.0262 W/mK

Plugging these into the equation Q/t = kAdT that gives us an initial transfer rate of approximately 360 W through the ground and 2.1 kW in the air.
Therefore total energy to counter balance this would be 2.46 kW

If the column is destroying heat the temperature difference will remain constant and therefore so will transfer rate.

Finally the sun gives us 1 kW/m^2 of thermal energy on the ground (assuming the surface is perpendicular the entire time).

This means the area would reach only 2.46 m^2 around the column, though this is a bit of a mean estimate. This area would 1.34m radius from the centre of the column or 0.34m around it.

Edit - The Day & Night Cycle
So I know it's been a few days but I've been thinking on it some more and realised I missed one pretty huge factor... the night.

For a quick recalculation the moon reflects 12% of the light of the sun.
I know this is subject to the moons phases but as with the rest of the question I'm taking large assumptions and averages.

The next of said assumptions is that the moon reflects light in the same proportions as the suns light (UV, visible, Infrared) and therefore its energy transfer is also proportional at 12% of 1kW/m^2 (120 W/m^2)

If this is correct then the 2.46kW transfer rate will need a much larger area to be countered.
This area would be 20.5m^2 around the column, 2.74m radius from centre or 1.74m out from the pole.

Obviously this would be hugely varying dependant on the cycle of the moon.
On a new moon night the area would reach as far as possible in the 12h (again avg) of no sunlight it has, with this being limited by the transfer rate.
On the flip side this area would be almost half the size on a almost full moon night, but not half on actual full moon as on a full moon the light of the sun is blocked by the earth and therefore wouldn't reach the same amount of light reflected back on the earth.

The cold area would fight a constant war between day and night expanding and shrinking.
This massive heating and cooling effect would have similar properties to large desert areas and over time I think the dry area would create a desert like effect around the cold zone.

You could include some species that would take advantage of this cycle and move into the area in the day for some of the nutrients or plants it creates.
Certain desert creatures and plants may like the area but would have to be introduced mostly.

However the area size wouldn't make much wiggle room.
If you were to also state your pillar is very thick then the area around it would increase in a square proportion to the radius of it, giving you a larger varying climate to go with it!

Edit if you can...
If you think my calculations are wrong or I missed something then suggest an edit.
I will double check it against sources to confirm it, if you name the source all the better.
However I know I've probably made a mistake or overlooked something so please do suggest!

The area outside of the absolute cold area would be very dry as water may be frozen out the air. It'd gain moisture for a while then lose it again as it became water vapour.

I doubt it'd be enough to form a lake but in the middle of the cool zone you'd get a lot of snow/sleet/rain.

Also can recalculate if you give more values.

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I'll assume a cylindrical tower 120m high and with 3m of diameter.

I suspect it would become encased in frost from the surrounding humidity. How far would we expect that frost to extend?

If the surrounding air is not definitely dry it will be surrounded by ice without doubt. Ice won't extend a lot because ice is a thermal insulant:

  • copper: 401 [W·m−1·K−1]
  • ice: 1.6 - 2.2 [W·m−1·K−1]

Ice will transfer on average 211 times less heat (the higher number, the better heat transfer).

Because of this property igloos are actually a thing.

During sunny day Sun gives us max 1'413W per square meter, assuming surrounding air at 25°C, the ice creation/melt balance will be at no more than 130mm of ice. (assuming a flat surface, the math for a cylindrical tower is way more complicated). To estimate the ice thickness i've matched the two energy flux, the sun's one and the one that would pass through the balanced ice insulation.

enter image description here

At night? Well, it's really hard to estimate something in general since it will depend by a lot of unkown factor (estimated air temp?)

In any case, since ice is thermal insulant, the ice creation process is a strong self-limiting process, it won't extend a lot.

Right at the edge of the freezing zone, I would expect there to be liquid water from the humidity, but I'm not sure how much. Is there any limit?

Yes, the same aforementioned limit. At that limit you would have an interface of melting/frozing water because you would have a perfect balance between the air melting "power" and the tower "freezing" power.

Would a lake eventually form around it? Would it flood the surrounding areas?

Not really, and in any case this would drastically depends on the soil water permeability and the air humidity. 1138 square meters (tower esposed surface) of frozen surface won't create a lake even in a tropical environment. You have 28 grams of water per cubic meter of air at 30 °C . Lake Ontario, for instance, has 1,640 km^3 of water, 1.64*10^12 m^3, 1.64*10^18 grams of water.

You need to collect the water from 5.86 * 10^16 m^3 of basically water saturated air to create such a lake.

5.86 * 10^7 km^3 are 58.6 MILLION of square kilometers can't be really affected by your tower. Imagine a cube 400km wide (not to mention that at 400km high we already are in space, but this is conservative) full of saturated air. Can you "feel" a snowy mountain -way bigger than your tower- 200km away? Is the mountain drastically influencing the air condensation near you? Then you have the answer about the lake in our world.

Beyond the edge of the freezing zone, it would be cool. How far would that extend?

"Cool" doesn't mean a lot, 10°C are cool? Do we have winds near the tower? In that case you could be upwind (who cares?) or down wind (nice air conditioner during summer, annoying colder air stream during winter). In any case it won't be worst than a common central europe winter, and only in a very specific unlucky zone near the tower.

No wind: i'm actually guessing how could you effectively feel if that tower exist or not once you are only 100m away. Obviously you can't have conduction heat transfer.

Convection is quite difficult to justify at this distance (the tower itself will create convection vortexes that will "circulate" the air in the surrounding zone, but we are talking about meters).

Irradiance, despite the fact that -against the common sense- ice is a near black-body in the infrared wavelenght and humans mostly emit in the infrared spectrum, won't be appreciable by humans because of the little difference in temperature (37°C at maximum?), and the infinitesimal solid angle (steradian). I know that this question is marked as hard-science but i guess that's not really important for the OP to know decimal figures of the irradiated power if it's something negligible in real life. Moreover, a simple 1.2m high bush 1m far from the person will shield him from any possible effect.

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Make the air cold at twice the tower's height, but not decreasing until it is past the 1xheight of the tower, unless there is high wind, in which case make it cold air to 1x -wind speed, in 2xMph.

The ground would also be cold according to the diameter of the tower, according to how much contact it has with the ground. give the ground the coldness of the tower, out to the radius, decreasing out to the diameter.

Close enough. You wouldn't want to try to do this by mass and diffusion. Keep it simple. No need to go crazy.

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A short answer:

The effect will roughly be this: It will absorb heat from adjacent matter, that matter will also absorb heat from adjacent matter. And the tower will keep its temperature even though it's behaving as if it's quite cold. If everything on the planet would remain somewhat "constant", it could freeze the entire planet down to its own temperature. It could even eternally prevent the heat death of the universe... at least "officially."

However, our planet is not constant. Our sun will provide us with continuous energy, and the relatively hot winds will counteract that effect. The sunlight will also affect the area itself, and further diminish the freezing effect. The effect will have a gradual range, and it will depend on its heat conductivity, fixed temperature and mass. You can't go much into the "negative" heat as you can go into "positive" heat though (basically no limits to that) - however: Magic.

It wouldn't have a large effect on the surrounding environment, except maybe occasionally causing tornadoes and having snow instead of rainfall.

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