Thursday, 28 May 2020

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.
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.


Dinero said...

An object subject to an certain w/m2 of heat radiation is itself emitting IR as it is a warm object. If the emitted IR from the warm object is reflected back towards the object this heats the object further, to a higher temperature. That is how the temperature/heat calculation is done, the degrees temp per w/m2 is for an object emitting IR and IR not being reflected back. Reflect the IR back and that is an addition to the w/m2 calculation and so a higher temperature. Does that makes sense...

Bayard said...

You don't need any fancy maths to show that that diagram disproves the Consensus. Incoming radiation = 342 W/m2, outgoing = 107 + 235 W/m2 = 342 Wm2. Therefore there is no net energy gain, therefore the Earth is not getting any warmer, unless the first law of thermodynamics no longer applies.

Mark Wadsworth said...

D, not sure.

B, the chart is deliciously wrong on so many levels.

Dinero said...

If you direct sunlight onto the side of a object its temperature rises, place a canopy of aluminium foil above it and the temperature of the object rises to a higher temperature.

Mark Wadsworth said...

Din, yes, I thought you meant something like that. Which is why paramedics give people metal foil blankets.

But if you follow that logic through, and imagine 'greenhouse gases' as a foil blanket half-way up (which reflects precisely half back down and lets precisely half through), then the air above it would be cooler than predicted using T/h = g/Cp, which it isn't.

And don't forget, infra red does not have a temperature in itself, the same as you can't see visible light unless it goes into your eye.

Mark Wadsworth said...

Din, although what happens is that the body is giving off LESS heat while receiving the same additional heat, it is not getting MORE heat.

Dinero said...

With the IR reflected back it is actually getting more heat than if there is not the Back reflected IR. That is what the diagram is showing at the bottom right. It adds up to more than the incoming solar radiation. I thought this was strange at first but if you think about it there is no other way of doing the thermal calculation for specific heat capacity. The specific heat capacity formula is for a an object emitting IR and not receiving it back and so an object that is receiving back that IR is getting more heat. The atmosphere absorbing IR and emitting IR is doing this back reflecting action. I think they call it IR downwelling,

Mark Wadsworth said...

Din, sure, the numbers add up but don't make sense.

Let's accept that there's IR pinging about in the atmosphere.

This can go three ways

1. Some increases kinetic energy of N2 and O2. We are solemnly told that these can't emit IR, so that part of the IR is now out of the picture.

The rest, you would assume is split between
2. Back to hard surface
3. Out into space.

How is it possible that the 'greenhouse gases' emit more IR downwards than upwards? That is implausible.

Bayard said...

D, you have to move out a little. Once you are outside the "greenhouse gas" layer, it doesn't matter what is going on below that level, what you are concerned with is the amount of heat going in and the amount of heat coming out, which, according to the chart, are the same.

AFAICS, the Consensus view is that the greenhouse gases act as a one-way filter, allowing more energy into the atmosphere and hence to the surface than they allow out and this is warming the planet. This is the only way that the planet can be warming, unless the first law of thermodynamics is wrong. However, if that were the case, then the amount of energy emitted by the earth would be less than the amount of energy received by the Earth, which is not what is says on the chart.

Dinero said...

-Mark , the flux between surface and atmosphere is more than the flux between atmosphere and space because the surface is heating the atmosphere and the atmosphere is heating the surface.

-Bayard . The flux into a heated object is equal to the flux out. The temperature is a result of the particular thermodynamics.

Mark Wadsworth said...

Din, simply repeating the same thing doesn't make it true.

It is all quite simple. There is a lapse rate because of the kinetic energy-potential energy trade off. So lower layers are warmer so have more of a warming effect (i.e. more than colder upper layers). As far as the surface is concerned, it couldn't care less whether that warming is via conduction or down drafts or radiation.

This can be twisted to show that 'greenhouse gases reflect heat back to the surface' which is arse backwards.

Mark Wadsworth said...

Din, you have helped me clarify my thoughts, see next post!

Bayard said...

"The flux into a heated object is equal to the flux out. The temperature is a result of the particular thermodynamics"

Yes, for a steady state, but we're not talking about a steady state here, we're talking about a planet where the temperature is increasing. If the temperature is increasing and the mass remains the same, the amount of energy has to be increasing, too. The only way the Earth can receive an input of energy is from the sun, therefore, for the temperature to be increasing, the amount of energy incoming has to be greater than the amount outgoing.

Mark Wadsworth said...

B, "for the temperature to be increasing, the amount of energy incoming has to be greater than the amount outgoing"

Yes, but temperatures have only increased by 1C over a century (or whatever), so each day they have increased by 0.00003C.

This relates to my last paragraph, it depends how long the atmosphere can hold on to energy before it radiates it again.