Many people believe that because the geothermal heat flux is ~91 milliwatts/m², this would imply that, without the sun, the Earth’s surface would be ~36 Kelvin. This is calculated the following way:
The math is correct, but there are two problems. First, this is for the top-of-the-atmosphere, not for the surface. But I will not address this today. The second problem, I will address today, is that this flux is wrong for this type of calculation.
Imagine an internally heated object is radiating to space. It’s very warm on the inside, but it very rapidly cools as you approach the exterior surface. Then a much hotter object joins in and starts to heat the exterior surface. What happens? The surface is now much warmer then it would be, and the flux going from inside to outside is … reduced!
The famous [small] geothermal heat flux is already a product of this reduction! The question now becomes … what would the flux be without the sun?
This question is extremely difficult to answer given all the variable parameters, and as far as I can tell … NO ONE has attempted it. But I’ll give it a small shot 🙂
I made a very simplistic 1-dimensional heat transfer program (code below) to help me out. First let me tell you its shortcomings:
No Albedo. No emissivity. No seasons. One latitude. No accounting of variation of specific heat or k-value as a function of depth. m*Cp = 222 and k=1, for all depths.
The sun simulation: Sun rises at 6am and goes to a full 1360 W/m² (no albedo) at high noon and disappears again at 6pm (like equator on equinox). This is repeated for 10 years (3650 days).
Despite these limitations, my central point is correct and will become obvious.
Not knowing where to start, I chose 273.15 K (0 °C) as the inherent geothermal temperature at 200m depth. The result becomes:
$ cat plot.csv
0 253.63
2 253.82
4 254.02 ...
196 272.76
198 272.95
200 273.15The geothermal flux here is:
That is very close to the measured geothermal flux. Now let’s see what happens when we turn off the solar influence:
This result can also be achieved via a simple equation:
The geothermal heat flux now is 0.9462 W/m². That is … nearly 10 TIMES the flux when the sun is included. 9.695x to be more exact.
So the actual geothermal flux without the sun is very different than the flux with the sun. And this is not surprising.
Before my critics complain that I arbitrarily kept the temperature at 200m depth constant, they should be reminded that this doesn’t matter to my central point. The top (surface) cools much faster than the interior, and that this immediately raises the flux. Either way, the famous geothermal heat flux is irrelevant for no-sun calculations as the REAL “steady state” tendency [without the sun] is to cool from top to bottom to the external temperature from distant space objects (2.725K). Given infinite time … the heat flux goes to ZERO, and the surface temperature is obviously driven much colder than the 35.65 K figure calculated simply from today’s current geothermal heat flux with solar influence.
The 91 milliWatts/m² geothermal heat flux is worthless for no-sun predictions of surface termperature. QED
Enjoy 🙂 -Zoe
Code
# Zoe Phin, 2022/05/06
# File: hsim.sh
# Run: source hsim.sh; heat; plot
heat() {
awk 'func show(s,t,T,x) {
if (x=="AVG") printf "%*s AVG ",12," "
else printf "d%03d %02d:%02d |%04dW| ",t/1440+1,t%1440/60,t%60,s
for (i=0;i<=L/2;i++) { printf "%06.2f ",T[i]; A[i]+=T[i]/24 }; print ""
}
func reset(s,t,T) {
show(s,t-1440,A,"AVG")
for (i=0;i<=L/2;i++) { A[i]=0 }
}
BEGIN { C=222; d=3650; L=200
for (t=0;t<d*1440;t++) {
T[L/2] = 273.15 # Geothermal Base Temperature
if ((S=1360*cos(atan2(0,-1)*(t-720)/720))<0) S=0
if (t/1440>0&&t%1440==0) reset(S,0,A)
if (t%60==0) show(S,t,T,"")
for (i=0;i<L/2;i++) {
D = (T[i]-T[i+1])/2
T[i]-=D/C; T[i+1]+=D/C
}
T[0] += (S-5.67e-8*T[0]^4)/C
}
reset(S,0,A)
}' > data
}
plot() {
tail -1 data | tr -s ' ' '\n' | sed 1,2d | awk '{print (NR-1)*2" "$1}' > plot.csv
echo "set term png size 740,540; set nokey
set title 'Depth (m) vs Temperature(K)'
set yrange [200:0]
set xrange [240:290]
set ytics 20; set mytics 4
set xtics 5; set mxtics 5
set grid ytics xtics
plot 'plot.csv' u 2:1 w lines lw 2 lc rgb 'red'
" | gnuplot > plot.png
}
Zoe's Insights 
Excellent article Zoe, enlightening as usual. Thanks!
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Thank You. Appreciate it! Very much
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Yes. I agrees. An excellent article.
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Your hypothesis about CSR still alive?
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Yes, but it was never for steady state.
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Can you please explain what you mean? All heat fluxes are dynamical.
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Radiation ~ T^4, Conduction ~ T. Conduction can’t keep up with radiation loss.
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Does that mean that the only possible steady state for a “unirradiated” internally heated body is zero Kelvin?
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Sorry, that was not very well though through. Stars are internally heated and does of course have a surface temperature greater then zero Kelvin. You have previously suggested that Earth’s surface temperature would be 0-5 centigrades without the sun. Wouldn’t this at least be qasi-steady? I mean, the energy inside the earth wouldn’t drain so fast that it couldn’t be considered steady state in human perspective?
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My previous statements did not take time into consideration.
The surface will cool at a rate of ~T^4, while getting conductive support at ~T. It will cool at a rate of T^3. The inside will cool slower at just ~T.
Also, I can’t tell you if the 0-5C is at the surface. I think it is not. It’s at some depth below the surface. How else can we get evaporation?
Even though geothermal energy is enormous, it is blocked by matter above. That’s why we dig boreholes, so it can escape. The dirt “soaks up” the heat, so we get rid of it.
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2.725 K. The avg temperature in space. Although that too may go down to zero. The heat death of the universe.
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A the height of the last glaciation, 20k years ago, 30% of our planet was covered in ice. When only small perturbations in the Milankovitch cycles can cause this, one must really wonder what would happen if the sun went dead. A snowball earth undoubtedly. Perhaps the gases in the atmosphere would freeze. Permafrost would propably be very deep. Even today, the ground is continually frozen to a depth of 1.5km in a region of eastern Siberia, near the Lena River.
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“What would the flux be without the sun? This question is extremely difficult to answer given all the variable parameters, and as far as I can tell … NO ONE has attempted it.”
Here somebody calculated it. Note the comparison to solar energy reaching the surface, in Figure 6. Geothermal is only 0.025% of solar. https://www.mathscinotes.com/2013/07/geothermal-power-math/
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And here is a peer reviewed paper using 38,347 measurements to yield 47 TW: https://se.copernicus.org/articles/1/5/2010/se-1-5-2010.pdf
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I had a whole post about this Davies & Davies paper.
You seem to have a reading comprehension problem. This is the flux WITH the sun. The question is flux WITHOUT the sun.
Without the sun, the top would cool faster than the bottom. The flux would thus begin to increase. The deep level flux will be the Davies result for some time, but eventually it too will increase.
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