<p>You are on track to answering this yourself. Temperature is a continuous field, so near any surface it has to follow the boundary conditions, with the effect of any surface topography decaying away with depth.</p>
<p>For a steady-state heat flow in a homogeneous media, the <a href="https://en.wikipedia.org/wiki/Heat_equation" rel="noreferrer">Heat equation</a> reduces to the <a href="https://en.wikipedia.org/wiki/Laplace%27s_equation" rel="noreferrer">Laplace Equation</a>, which is easy to solve numerically. For example, here are the steady-state temperature contours for a 4 km tall square &quot;mountain&quot; with a 10°C surface temperature being heated from below by a 40 km deep 1000°C mantle boundary, assuming homogeneous rock.
<a href="https://i.sstatic.net/RZkfM.jpg" rel="noreferrer"><img src="https://i.sstatic.net/RZkfM.jpg" alt="enter image description here" /></a></p>
<p>The real profile, of course, also depends on the details of the subsurface geology and history.</p>
<ul>
<li>Different rocks have different thermal conductivities, so even if the surface is flat, changes in rock types along a transit will change the underground temperature profiles.  An example of this is the temperature dip at kilometre 135 in the Gotthard Base Tunnel that is due to porous rocks allowing cold surface water to penetrate deep underground.</li>
<li>The time it takes for heat to diffuse into topography can be comparable to the time it takes for the topography to form, so the geological history matters as well. For example, a typical rock thermal diffusivity is <span class="math-container">$\kappa~\sim 1.2\times 10^{-6}\,\mathrm{m^2/s}$</span>, so the thermal diffusion time (<span class="math-container">$d^2/\kappa$</span>) into a 4 km mountain is about a half-million years.  According to <a href="https://doi.org/10.1007/BF01020206" rel="noreferrer" title="Rybach, L., Pfister, M. Rock Mech Rock Engng 27, 77–88 (1994).">&quot;Temperature Predictions and Predictive Temperatures in Deep Tunnels&quot;</a>, the time of topology evolution in the Alps is about two million years. (This seems reasonable given their current <a href="https://en.wikipedia.org/wiki/Geology_of_the_Alps#Quaternary" rel="noreferrer">uplift</a> and <a href="https://www.mic.com/impact/somehow-the-swiss-alps-are-still-growing-shocking-scientists-47686827" rel="noreferrer">erosion</a> rates are on the order of millimetres per year.)</li>
</ul>

CC BY-SA 4.0
2024-01-26T19:43:55.630Z
Thanks a lot. What puzzled me was also that some sources blame pressure in the mountain base (with logic that large pressure means large temperature).
Leos Ondra
CC BY-SA 4.0
2024-01-26T19:07:49.467Z
2024-01-26T19:07:49.467Z
David Bailey
<p>Wikipedia <a href="https://en.wikipedia.org/wiki/Gotthard_Base_Tunnel" rel="nofollow noreferrer">article</a> says about the Gotthard Base Tunnel:</p>
<p>&quot;It is the deepest railway tunnel in the world, with a maximum depth of 2,450 m (8,040 ft), comparable to that of the deepest mines on Earth. Without ventilation, the temperature inside the mountain reaches 46 °C (115 °F).&quot;</p>
<p>More authoritative source with detailed temperature profile, a paper by L. Rybach and A. Busslinger (<a href="https://www.hbi.ch/fileadmin/user_upload/unternehmen/publikationen/04_Verification..._WTC-2013_Genf.pdf" rel="nofollow noreferrer">pdf</a>) gives the following diagram:</p>
<p><a href="https://i.sstatic.net/87M4l.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/87M4l.jpg" alt="enter image description here" /></a></p>
<p>What is the cause of the elevated temperature? Why does temperature follow depth below surface despite virtually uniform height above sea level of the tunnel?</p>
<p><strong>Edit</strong></p>
<p>In quite simplified model I have homogeneous (the same thermal properties, geothermal gradient etc.) rock with heat flow (conduction only) from the bottom kept at some high temperature (say 1000 °C) to the surface with average annual temperature say 10 °C. Isotherm 50 °C is at some depth below horizontal surface (left diagram, not to scale). Does the isotherm follows surface even in mountains?</p>
<p><a href="https://i.sstatic.net/TqHfB.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/TqHfB.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-01-23T08:25:20.100Z
Have a Look at the geothermal Gradient
Weiss
<mountains><geothermal-heat>
What determines rock temperature inside mountains?
Leos Ondra
# What determines rock temperature inside mountains?

(Asked by: username_0 on 2024-01-22)

Wikipedia [article](https://en.wikipedia.org/wiki/Gotthard_Base_Tunnel) says about the Gotthard Base Tunnel:

"It is the deepest railway tunnel in the world, with a maximum depth of 2,450 m (8,040 ft), comparable to that of the deepest mines on Earth. Without ventilation, the temperature inside the mountain reaches 46 °C (115 °F)."

More authoritative source with detailed temperature profile, a paper by L. Rybach and A. Busslinger ([pdf](https://www.hbi.ch/fileadmin/user_upload/unternehmen/publikationen/04_Verification..._WTC-2013_Genf.pdf)) gives the following diagram:

[![enter image description here](https://i.sstatic.net/87M4l.jpg)](https://i.sstatic.net/87M4l.jpg)

What is the cause of the elevated temperature? Why does temperature follow depth below surface despite virtually uniform height above sea level of the tunnel?

**Edit**

In quite simplified model I have homogeneous (the same thermal properties, geothermal gradient etc.) rock with heat flow (conduction only) from the bottom kept at some high temperature (say 1000 °C) to the surface with average annual temperature say 10 °C. Isotherm 50 °C is at some depth below horizontal surface (left diagram, not to scale). Does the isotherm follows surface even in mountains?

[![enter image description here](https://i.sstatic.net/TqHfB.jpg)](https://i.sstatic.net/TqHfB.jpg)

**Tags:** mountains, geothermal-heat

> **Comments:**
> **username_1**: Have a Look at the geothermal Gradient  

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## 1 answers

### Answer by username_2 on 2024-01-26. Score: 5 | Accepted answer

You are on track to answering this yourself. Temperature is a continuous field, so near any surface it has to follow the boundary conditions, with the effect of any surface topography decaying away with depth.

For a steady-state heat flow in a homogeneous media, the [Heat equation](https://en.wikipedia.org/wiki/Heat_equation) reduces to the [Laplace Equation](https://en.wikipedia.org/wiki/Laplace%27s_equation), which is easy to solve numerically. For example, here are the steady-state temperature contours for a 4 km tall square "mountain" with a 10°C surface temperature being heated from below by a 40 km deep 1000°C mantle boundary, assuming homogeneous rock.
[![enter image description here](https://i.sstatic.net/RZkfM.jpg)](https://i.sstatic.net/RZkfM.jpg)

The real profile, of course, also depends on the details of the subsurface geology and history.

* Different rocks have different thermal conductivities, so even if the surface is flat, changes in rock types along a transit will change the underground temperature profiles. An example of this is the temperature dip at kilometre 135 in the Gotthard Base Tunnel that is due to porous rocks allowing cold surface water to penetrate deep underground.
* The time it takes for heat to diffuse into topography can be comparable to the time it takes for the topography to form, so the geological history matters as well. For example, a typical rock thermal diffusivity is $\kappa~\sim 1.2\times 10^{-6}\,\mathrm{m^2/s}$, so the thermal diffusion time ($d^2/\kappa$) into a 4 km mountain is about a half-million years. According to ["Temperature Predictions and Predictive Temperatures in Deep Tunnels"](https://doi.org/10.1007/BF01020206 "Rybach, L., Pfister, M. Rock Mech Rock Engng 27, 77–88 (1994)."), the time of topology evolution in the Alps is about two million years. (This seems reasonable given their current [uplift](https://en.wikipedia.org/wiki/Geology_of_the_Alps#Quaternary) and [erosion](https://www.mic.com/impact/somehow-the-swiss-alps-are-still-growing-shocking-scientists-47686827) rates are on the order of millimetres per year.)

> **Comments:**
> **username_0**: Thanks a lot. What puzzled me was also that some sources blame pressure in the mountain base (with logic that large pressure means large temperature).  

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