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I was looking for ways to make gigantic mountains, and found Willk's answer to such a quandary, where a giant diamondberg forms in an ice giant and is incorporated into a terrestrial planet's crust, producing a mountain much stronger and less prone to subsidence than a rock mountain. As Willk noted, however, diamond has high thermal conductivity, and one side of this diamondberg is embedded in the mantle, meaning it'll be able to transmit a lot of heat from the mantle to itself/the surface. I did some more digging to figure out how much heat, precisely, and found several things.

Diamond's heat capacity is impressively low; i.e. a certain mass of diamond doesn't take much energy to heat to a certain temperature relative to other substances. Diamond also has a high thermal conductivity somewhere in the thousands of watts per meter per degree Kelvin. The Earth's crust has a thermal conductivity in the single digits of watts per meter per degree Kelvin. Therefore, even at higher temperatures where it's likely diamond's thermal conductivity lowers, it's still likely the diamond is on the order of a hundred times more effective of a heat conductor than the crust.

Note that it isn't easy to build things on bare diamond, and it's especially difficult to do so when said diamond is hundreds of degrees. On the other hand, a giant diamond partially ("partially" because it hasn't been sunk all the way under) subducted under a continental plate has a nice, thick layer of rock and dirt atop it to put things on and in, which can also act as a heat sink so you can walk on the surface without getting your shoes melted. Therefore, that's what I decided to do in order to make my giant mountain.

Imagine covering a sea buoy with a weighted tarp; the buoy will go somewhat lower in the water, but won't actually sink. This is the case here: the downward force from the diamond's mass and the mass of the crust covering it equates to the buoyant force caused by the diamond displacing the denser mantle/athenosphere. As such, the diamond floats in the lower mantle. There are two tectonic plates trying to push into it from either side, which is why it's covered in crust: each plate encounters it and tries to ride over it.

Here is a not-to-scale, side-on cross-section of a crust-covered diamond mountain made with Google Drawings:

enter image description here

For the sake of making this question answerable, several assumptions must be made:

  • The diamond is an octahedron approximately 56 kilometers to a side, with one vertex protruding straight up and one sunk directly down into the mantle. Roughly 9⅔ kilometers of the diamond are above sea level (albeit still buried in crust); the rest are either embedded in the crust or below it in the mantle.
  • Exactly half the diamond's roughly 10,863 square kilometers of surface area are in contact with a 1,400° Kelvin asthenosphere/upper mantle; the other half are in contact with the crust that sits atop it.
  • At 1,400° Kelvin, the diamond has a thermal conductivity of 500 W/m/K, and the crust has a constant thermal conductivity of 5 W/m/K.
  • The crust is 3.15 grams/cubic centimeter (presumed to be a 50-50 silica/magnesium oxide mixture), the diamond is 3.5 grams/cubic centimeter, and the mantle is ~3.6 grams/cubic centimeter (presumed to be pure magnesium oxide). The differentiation between all three is essentially total; this is, of course, unrealistic, but having distinct layers rather than a blurry transition between crust, asthenosphere, and mantle simplifies the math.
  • Radiogenic heat in the crust and mantle around the diamond-mountain is ignored.
  • The diamond has been in the crust for ~100 million years, and has had plenty of time to heat to permeate throughout it.
  • Gravity is 2.5G.

Given all the above: I need a rough idea of how the heat conducted by this diamond mountain will geologically affect its surroundings. Will it turn the area around it into a sea of lava? Will it cause hot springs and exploitable geothermal activity? Will the ground be only slightly warmer than it would otherwise? My greatest concern is that the sheer quantity of heat being transferred through the diamond may be enough to form a magma chamber in the crust above it before the crust can rupture enough to release it, resulting in a supervolcano.

Please show relevant math if at all possible, and if you need more data, please make a comment to that regard. If nothing else, I need to know what the temperature of the diamond would be so I can determine the heat it transfers into the crust.

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So, there are a couple points that make a hard-science answer ...hard (no pun intended).

While the diamond itself is a great thermal conductor, it is only being heated by the mantle. And given the low thermal conductivity of the mantle, it will never be heated to the same temperature as it. Therefore this diamond would be behaving more like throwing water on the mantle - it will cool the area directly underneath it, resulting in a thicker mantle locally, and a warmer-than-average mountain. But it will not get hot enough for lava flows (but it will definitely have a different local climate and vegetation).

In terms of this "mountain" actually floating on the surface, the diamond vs. the mantle is only 0.1g/cm^3 buoyancy force, while the diamond vs the crust is 0.35g/cm^3. The volume of half of the octahedron is 41,400km^3, while the volume of the peak is ~600km^3. So there is actually negative buoyancy force on the diamond - it would be trying to sink deeper through the crust.

Also, the normal resting attitude of a d8 (an octahedron) is actually flat-side-up, not point-up, as can be seen in this video.

So a hard science answer would say your crust has to be much thinner, and/or the diamond much smaller. It would also be trying to tilt , so it would probably look more like a triangular mesa, with one peak perhaps reaching this 9.6km height.

Story-wise, I think you could waive some of the math and it would make sense.

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It seems likely to me that if the diamond remained buoyant in the mantle and the crust, that the transmitted heat would melt the crust above the diamond, eventually exposing it to the air, at which point, if there was any oxygen in the air, the diamond would begin to burn away.

The phenomenon of a large octohedral diamond floating between the crust and the mantle would be transitory, especially if there were any oxygen around. It would tend to either sink deeper into the mantle if the diamond was denser, or melt and float to the surface of the crust if the crust was less dense, where it would burn in any oxygen.

Either way, such a diamond would be highly unlikely to simply stay put between crust and mantle for more than a few thousand years at most.

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