[ExI] Diminishing influence of increasing carbon dioxide on temperature

[Anders Sandberg]

Tomaz Kristan <protokol2020 at gmail.com> , 15/8/2014 9:26 AM:
Venus "surface" is in fact deep down in the stuff. Sun has nearly nothing to do with the temperature there, it's geothermic.

Hmm. Let's see: if it is geothermic dominated, then the planet will radiate way more energy than it gets from the sun. Assuming it to be a blackbody it will radiate A*sigma*T^4 Watts, where A is the total surface area (~5e14 m^2) , sigma is 5.67e-8 W/m^2K^4 and T is the average atmosphere temp (737 K). I get 7.85e18 Watts. The total solar input hitting it is I*a, where I is the solar irradiance (2613.9 W/m^2) and a is the cross sectional area (1.17e14 m^2): 3.07e17 W.
That is a factor of 25. So clearly this is missing something. Or I made an error.
But if we estimate the geothermal flux as what is missing, we get 15,000 W/m^2! Earth has 0.087 W/m^2. Now, I can imagine more geothermal on Venus due to a bit stronger tidal forces, but most geothermal heat is due to radioactive decay and the compositions are not too different. But 5 orders of magnitude?15 kW/m^2 is not sustainable for long - if Venus was a lava ocean it would cool off within a geologically short timescale. 
What is going on? The most likely answer is that the naive blackbody model above is too simple. If we instead refine it so that there is an atmosphere layer that lets through sunlight to deep levels/the surface, and then acts opaque for infrared (which we do know is true for CO2 from lab experiments), then the thermal balance starts looking saner. 
Let Ts be the surface/troposphere temperature and Ta be the high altitude atmosphere temperature. Then energy balance requires S + sigma Ta^4 =sigma Ts^4 (solar + atmosphere downwards = surface upwards) and sigma Ts^4 = 2 sigma Ta^4 (surface upwards = atmosphere up + down radiation). The last equation makes Ts = 2^(1/4) Ta = 1.19 Ta: adding the layer increases surface temperatures by nearly 20% *regardless of solar input*.
And given that the optical depth of Venus' atmosphere is about 80, one could argue that there should be many such layers! If we have multiple layers, then the outermost could have the equilibrium Venusian blackbody temperature 238 K and energy balance would work, while the average temperature below could be way higher. Each layer would make the stuff beneath warmer. In practice the layers would be grey and there would be some messy interactions (i.e. time to write simulator code), but it is not hard to see that this kind of insulation could produce a high temperature at the surface/troposphere even for a modest solar influx.
So I would be surprised if geothermal had a huge influence on Venusian surface temperatures. The solar input is on the order of 2613 W/m^2: to measure up you need a geothermal flow higher than Io's 2 W/m^2 - by a factor of a thousand! I think we would notice those eruptions. Also, that would assume an optically thin atmosphere, which we know is not true given observations. One can of course tone down the geothermal and make the atmosphere more opaque, but it seems that when we start to reach scenarios compatible with a solid surface we will already have added enough greenhouse to make geothermal rather moot.

(Thanks Tomaz, this was fun to work through - at first I thought a mere paragraph would be enough to dismiss your idea, but it actually demonstrates just how quickly even fairly simple atmosphere models get nontrivial. )


Anders Sandberg, Future of Humanity Institute Philosophy Faculty of Oxford University
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