There are few times that I can have been so genuinely surprised at what is such a stunning connection as this. It all started with a tweet by @SteveSGoddard (Tony Heller)
Earth’s interior is hot because of low thermal conductivity and high viscosity of the rock. You can’t compare that to a gaseous planet like Jupiter.
— Steve Goddard (@SteveSGoddard) December 28, 2017
The problem is that you can. So, let’s first recap how global warming really works again. The temperature of a black body the size of the earth is about 255k. It therefore follows that for thermal stability, the average temperature, or more accurately the average of T4, the average heat energy escaping, must be the same as a black body with temperature of 255k. Thus if we imagine a view of the atmosphere were we could pick out individual molecules as shown below, the average T4 of this planet at the position of the earth must be 255k.
That is, the average temperature of the highest molecules emitting IR to space must be 255k. From this we can easily work out the temperature of the ground. Because most of the density of the atmosphere is below ~5km and the lapse rate from this to the ground is 6.5C/km, so the temperature of the ground will be:
255 + 6.5 x 5 = ~288k
Of course, the 5km level I use here is not a hard and fast layer, but is a range of levels, except where the IR is being emitted from cloud tops. So, the 5km figure is an average that reflects the density of the atmosphere and a small variation in height for an increase/decrease in “greenhouse gases”. But for out purposes here, the greenhouse effect can be seen has being an emission from a nominal “top of atmosphere” level which averages at around 5km, which has an average temperature of 255, and that the lapse rate causes the thermal gradient that causes the earth’s surface to be about 32C warmer at 288C.
For a more detailed explanation see: The Greenhouse effect
Heating in the earth Core.
And this is where Tony Heller‘s comment comes in. Because whilst I’ve only ever thought about the heating that occurs when a gas is compressed, or to put it another way, the reduction in temperature as gas expands when it rises, which is what causes the lapse rate of air, there is the same effect also in liquids.
This is the point where my knowledge ends, so I’m going to have to quote a few articles. The first one I found “Determining the Thermal Expansion Coefficient of Liquids by Observing the Onset of Convection“, confirmed that the concept of an adiabatic lapse rate that I understand as the key to atmospheric greenhouse effect is also present in liquids but the calculations are much more complex than g/cp (gravitational acceleration over specific heat capacity of air at constant pressure) :
The adiabatic gradient (≡Tυβ/Cp), also called the adiabatic lapse rate, is the temperature increase caused by adiabatically compressing a fluid, e.g., if a bit of water sinks in the ocean quickly enough so it cannot lose heat to surrounding water, it will become warmer—the adiabatic gradient is its rate of change of temperature with respect to depth.
The Brunt‐Väisälä frequency, N[N2≡−g(g/c2−βT′)], is the frequency of buoyant oscillation of the bit of fluid mentioned in Ref. 2 if N is real. If N2 is negative the fluid is unstable. Here T′ is the vertical gradient of temperature in the ocean, c the velocity of sound.
The next article Convection in the Earth’s mantle is well worth reading and gives a lapse rate for the mantle of 0.4 and interior of about 0.3C/km. Thus if the earth is 6371km to the centre, the temperature at the centre should be about:
255 + 6.5 x 5 + 6371 x 0.3 = ~2000k
That is nowhere near the actual predicted temperature, but since the earth has many layers, it will be far more complex that this simple calculation. But the principle is sound: at least part of the temperature rise to the core is due to the same cause as the atmospheric greenhouse effect, the lapse rate.
As Oldbrew quite rightly pointed out I got Centigrade and Kelvin mixed up. What a daft Pillock I was!