Fun with numbers - where does the sunshine go?

To try and put the numbers into perspective, I did some workings.

Step 1. Multiply up the amount of incoming solar radiation per second (in Watts/m2, i.e. Joules/second/m2) to find out the total number of Joules each m2 gets in a 12-hour day (= 20.6 million of 'em).

Step 2. Look up mass/kg and specific heat capacity (Joules/kg/1 K) for air, wet soil, and water.

Step 3. Adjust/tweak the main variables until you get 'sensible answers' in the last column. The main variables are:

a) how those 20.6 million are split between air/soil and air/water. (answer 80/20 and 20/80 respectively, partly due to albedo and partly to do with how good soil and water are at moving heat downwards or back up into the air again)

b) the height of the column of air which noticeably warms during the day (answer 800 metres*)

c) how far down from the surface the soil warms up (answer 4.5 inches)

d) how far down from the surface the oceans warm up (answer 39 inches).

The 'sensible answers' are that the soil surface/the air above it warms by 16K during the day; the ocean surface/the air above it warms by 4K during the day - which is why in the day time you tend to get onshore breezes and at night you get offshore breezes. The Earth is two-thirds covered in oceans and that averages out to 8K.

* Clearly, there's not a clear cut-off of 800 metres altitude. So for example, maybe the lowest layer above the land warms by 16K, 800 metres up it warms by 8K etc, and at 1.6 km (about 1 mile up), the air barely changes temperature from day to night. Same applies to soil and water, going downwards. If you are building a sand castle, you don't have to dig very far down before the sand is noticeably colder than at the surface. Go for a swim in the afternoon, the top few inches are pleasantly warm; stand chest deep further out and your feet get cold.

At night, the reverse happens, and the lapse rate flattens again. In extreme cases, the land cools so far and so fast that it drags down the temperature above it so far and so fast that you get a temperature inversion, i.e. warmer air over colder air, that's like a negative lapse rate.



Which is all good fun, but what is the relevance?

Firstly, that you don't need to worry about quite how or why energy/heat is absorbed, transferred or distributed (conduction, convection/down drafts, mixing or wind/currents, radiation, latent heat of evaporation/condensation). The sun sends us a certain amount of total energy and it warms stuff up, and we can reconcile/estimate how much stuff is warmed up by how many degrees K. Sometimes the obvious answers are the correct ones and need little further investigation.

Secondly, what it reminds us that is the daily variation, based on incoming solar radiation, is relatively small compared to the absolute temperature. At its coldest (just before dawn), the surface is (say) 284K and at its warmest (mid/late afternoon) it's 292K.

Which, as ever, makes me question the Consensus obsession with this chart. That particular one is gloriously mislabelled as "Earth's annual and global average energy budget". It's not! That's the global average energy budget per second! They don't even understand their own propaganda.

What the Consensus is trying to do is explain that you can and should work out how many people are in a shop by looking at how many go in or come out every second (or in their terms, the average difference between the number people going in and coming out, which must be zero anyway, hence meaningless). Sure, it gives you a guide, but you'd also have to know roughly how long each person remains in there. If ten customers enter and exit a corner shop every hour, there will only be one or two customers in there at any one time. If ten customers enter and exit a large car show room every hour, there might be about ten customers in the show room at any one time. (Ignoring the lock down rules).

The "customers entering and exiting" are like the sunshine that arrives every day, which is sufficient to warm the soil/ocean surface and the air above it by 8K on an average day; it cools down again by 8K on an average night. The infamous chart gives you no clue whatsoever as to what the baseline average temperature is ("the number of customers actually in the shop").

So why not just count the actual number of people in the shop (the baseline average minimum temperature)?

The answer to this is not particularly difficult: stuff warms up and then it cools down again. Basic physics. The smaller the surface area relative to the volume/mass, the slower it is to warm up or cool down. Warmth from the sun can't get very deep into the soil or the ocean, so for each 1 m2 of surface, there's a column of troposphere with a volume of +/- 11,000 m3, which can only lose heat to space (counting the stratosphere and higher layers as 'space') via the 1 m2 at the top.

Most of the energy (kinetic energy, potential energy or latent heat of evaporation/condensation) in the air and top bit of land and oceans (which is effectively part of the atmosphere for these purposes) is left over from the previous day; and most of what was left over from the previous day was left over from the day before that ad infinitum. Mathematically, this energy has a half life of about 24 days.
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These ramblings have now reached full circle. My guess is that the Earth's atmosphere is set up to radiate a certain % of its energy every 24 hours. The equilibrium temperature is therefore where the extra sunlight that comes in during the 12-hour day is equal to the amount lost during the subsequent 12-hour night. If Earth's surface is radiating 2.78% (relative) every 24-hours, and is receives 8K's worth (absolute) when the sun is shining on it, then the equilibrium is 8K ÷ 0.0278 = 288K. Something like that.