What would happen if a volcanic cave was connected by a tunnel to a frozen cave?

Question

Subangea is a giantic cave that stretches below the surface through endless cavities. From oil oceans to volcanic sinkholes and even upside-down forests, the environment is ever changing from one cavity to the other.

So is the fauna except for one small detail, it is always trying to eat, dismember you or simply ram into you. All kinds of species of monsters living in all kinds of conditions and they are all trying to kill you to survive.

But no matter how big or venomous or fire-breathing, men and women will stand to slay those monsters and gather their valuable parts. They are hunters.

In the shining light of the glass roofs or within the blackest obscurity of the tunnels, they draw their powerful swords to... Wait, where are the swords ? The ore shipment was supposed to arrive yesterday ! The trading route with the dwarfs from the volcanic sinkhole is too long and dangerous and without new weapons the fields won't be safe from the monsters for long.

Legends tell that past the ice cave in the north lies a tunnel that leads directly to the volcanic sinkhole. The city scientists say it is pure folly, the wind would engulf in and blow anyone trying to get through or the tunnel would simply not handle the heat difference or- WHATEVER ! We need the dwarf's ore if we are to keep the beasts away !

Setting with a group of adventurer through out the caves to the frozen cave, you finally catch a glimpse of it through the blizzard: The legendary Mifrost.

Excitation runs through your veins and you run towards it to see... What exactly ? A half-frozen half-molten path in the rock ? A collapsed pan of the cave ? A gale so strong you get sucked in ?

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The frozen cave is a 5km x 10km cavity with ice ceiling in which temperatures go from -10°C to -40°C. It runs under permafrost ground.

The volcanic sinkhole is a 5km radius cavity with high volcanic activity in which temperatures go from 45°C to 60°C. It is sitting on a hot spot and has vents on the ceiling from which gas and hot air can evacuate.

The tunnel is 65km long and connects the two caves. The first half is subject to volcanic activity and the second half is cooled down by water streams from the ocean.

Answer 1

To know what happens in the tunnel, knowing why these areas separated by a 5km tunnel have these climates. Apparently, there is a sinkhole connected to a volcano at one end of the tunnel, but why the cave at the other end is frozen ? Subzero temperatures may happen on our planet surface in a variety of places and times of the year (or even of the day/night) but temperatures rises the deeper you are under the surface.

Unless there is some climatic reason for such a discontinuity, we are assuming it's magical. Therefore, virtually anything can happen wetween both caves, including nothing.

Now let's assume that even if the reason the frozen cave is, well, frozen is magic, sciences rules apply otherwise. Since you don't mention a vertical opening above the volcanic cave/sinkhole, I believe it has a roof. Since heat means higher pressure, I envision an air flow going from the volcanic pit toward the frozen cave through the tunnel until the pressure difference between both its ends is null.

Since the frozen cave seems to be meant to stay that way, it means that the freezing temperatures will be maintained somehow and now it begins to become hard to apply sciences without Newton getting mad at us.

So I need to also assume that the frozen cave absorbs as much energy as the volcanic sinkhole radiates, through magical or whatever means, that's not ht epoint I guess.

What we would have, then, would be a hot air flow constantly traveling from the volcano to the frozen cave. If no magic makes the incomming hot air to disappear, the system will reach an equilibrium regarding the flow but the pressure will still be higher the closer you are to the volcanic end of the tunnel. How its thermic energy decreases as it approaches the cave is hard to guess in part because it involves many factors (the tunnel's width and height, for one) and mostly because of the critical factors be handwaved or purely negated.

I'm inclined to conclude that this hard to guess part mentioned above corresponds to the range in which you have your saying as the world's Creator. Partly because wiggle room comes in a handy, and mostly because you are the mad world creator who made physics to allow an ice cave neighbouring a volcanic pit.

If that seems quite quiet - say it 15 times quickly - compared to what you envisionned, keep in mind that the difference between both caves regarding temperature isn't in the apocalyptic range, far from it. You can see it as someone pointing a hairdryer toward an unmeltable ice cube, except that both are 10km wide and standing only 5km from each other.

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EDIT : Since the tunnel is now 65km long, I guess the effects will be less a lot less violent. Visiting the hot cave is likely to be fatal without proper training and especially gear, but most of the travel through the tunnel would be okay, assuming there is dioxygen in it.

Answer 2

What you need is a cold trap, similar to those used for igloo entrances by the Inuit. So the entrance of an igloo is actually down below where the rest of the igloo is and then travels low until you finally come up to be in the open. The reason is that the warm air stays in the igloo and doesn't travel down because warm air rises and doesn't fall to escape through the entrance. So the way to make this work, or at least be remotely feasible, you'd need the connecting tunnel to travel downward so that the temperatures don't mix and try and create an equilibrium.

Though as mentioned natural occuring variations like this are pretty difficult to justify, maybe a natural hot spring in only one spot, like Iceland, near a cold area like Greenland, just closer together without an ocean between them.

Answer 3

From what I read in comments I see a bit of confusion, but I believe this setup can be made workable without resorting to magic.

• Being under permafrost doesn't really help because it is (as most of the ground) a very good heat insulator; for reference: a liar dug just 15cm (6") underground is enough to save an animal when a huge fire is roaring above (assuming it can avoid suffocation, which is another matter).
• You will need sizeable openings in both caves, allowing high air exchange with external atmosphere.
• Volcanic activity will heat air in sinkhole, which will rise as a thermal current and escape from sinkhole openings.
• This will suck air to replace what escaped; physics says vents on the lower part of the cave will be preferred.
• One of the preferred ways will be your tunnel, so you'll get an airflow from ice-cave to volcanic sinkhole.
• Air escaping from tunnel will be replaced with very cold air incoming from ice-cave openings helping to keep it frozen.
• Airflow in the tunnel is governed by Bernoulli Equation and it's behavior derived from Reynold's law; in a nutshell:
  • larger openings will have greater airflow
  • air speed will be highest in the place where cross-section is minimal
  • if tunnel becomes too thin or convoluted airflow will switch from laminar to turbulent with a much reduced airflow (but also high turbulence in the zone with minimal cross-section).
  • I cannot give precise figures because too much depends on specific geometry, but if you are looking for high speed you should opt for a linear tunnel with reasonably smooth surfaces (polished by perpetual wind action); a relatively short (<1km) shrinkage of the tunnel would not impede flow too much, but it would boost air speed probably well beyond 100km/h.
• Where cold air enters sinkhole you'll have a relatively inhabitable zone where your dwarves can refrigerate before returning to forges.
• If I know dwarves they would find a good way to use the constant power of the air current, possibly in the form of "interesting" windmills. ;)