Wednesday 10 June 2020

Why the 'moist lapse rate' is what it is.

As I pored over my master Greenhouse Effect spreadsheet after work today, another thing struck me (and which I have glossed over so far, it being a tricky topic).

We calculate the 'dry lapse rate' as 9.75 K/km, that's easy enough to understand. But the measured real-world 'moist lapse rate' is only 6.5 K/km. We know that water is a moderating influence, so we correctly put it down to water, water vapour and clouds with a hand wave, but how do we know it must be one-third lower than the 'dry lapse rate' for everything else to makes sense?

Let's do the numbers... (UPDATE: if you want to see the official equation, see www.theweatherprediction.com. Comes to the same thing, I just like to do stuff step-by-step).
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If you need a refresher...

Rule 1, energy cannot be created or destroyed, it just changes from one form to another.
Rule 2, energy tries to spread out as evenly as it can, taking into account the medium and the forms it can take within the restraints of that medium.

I don't like the 'parcel of air' analogy, but it will do for now. Air cools as it rises, so converts kinetic energy (aka 'heat' in layman's terms) into some other form of energy. What kind of energy does it convert into? Easy - it changes into potential energy (think about it for a minute or two).

1 kg of air higher up has the same total amount of energy (of all forms) as 1 kg at sea level. Kinetic energy (mass x specific heat capacity, referred to as Cp, dunno why) has changed into the same amount of potential energy (mass x height x acceleration due to gravity). The fall in temperature is called the 'lapse rate'.

Maths of the 'dry lapse rate':
Joules converted = mass x change in height x gravity = mass x change in K (temp) x Cp.
Cancel 'mass' on both sides.
g x change in h = change in K x Cp.
Reshuffle that, you get
Change in K ÷ change in height = g ÷ Cp.
9.807 m/s2 ÷ 1,006 J/kg = 9.75 K/km
(see Wiki for the hopelessly confusing explanation with the same end result).
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So far so good. The easy bit is PE gained = KE lost, but there's some energy unaccounted for.

PE gained is easy:
1 kg of air at sea level has no PE for these purposes. It can't fall any further.
1 kg of air 1,000 metres above sea level has 1 kg x 9.807 m/s2 x 1,000 metres = 9,807 Joules of PE.

So 9,807 Joules of energy, mainly of KE ('heat'), have changed into PE.
We know it has cooled by 6.5 K and that the specific heat capacity of air is 1,006 J/K/kg. So only up 6,539 Joules of KE have been converted into PE.
That leaves 3,268 Joules unaccounted for, doesn't it?

The missing figure is latent heat of evaporation of evaporation/condensation ('LHEC'). The ground and the ocean surface get radiation (energy) from the Sun, reflect a bit and convert two-thirds of the rest into KE and one-third into LHEC. I'm sure this is not a fixed ratio; in the desert, there's no water to evaporate; if it's colder, then not much water evaporates anyway etc.

1,000 metres higher up,  just under 1.5 grams of water vapour per kg of air (approx. one quarter of the water vapour per kg of air at 15C, sea level pressure and 50% relative humidity) condenses, freeing up 0.0015 kg x 2,264,705 J/kg =  3,268 Joules of LHEC in addition to the 6,539 Joules of KE that 1 kg or air already had, and all 9,807 Joules then end up as PE.

Which is why the 'moist lapse rate' observed in the real world is 6.5 K/km, not 9.75 K/km.

(6,539 Joules KE converted ÷ 9,807 total Joules converted) x 9.75 K/km dry lapse rate =
6.5 Joules KE converted per km = moist lapse rate.
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I know I'm reinventing the wheel here, and I'd hope that this is all old hat to people who've got a physics A-level or degree, but why don't they give this sort of fun stuff the same coverage as all the other Greenhouse Effect-related stuff?

And why does the IPPC's energy budget show that the hard surface only converts one-sixth of incoming radiation into LHEC and not one-third?

(Rhetorical questions, we know the answers, which I will cover soon).
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While checking my numbers, I stumbled across this fine article (ignore the ghastly typo in the title, I've made that same mistake many a time), well worth a read if you are into maths and are a bit dubious about the Consensus.

12 comments:

Bayard said...

Mark, something I've been wondering about, ever since you published that IPCC infogram, is the energy required to heat the Earth. Presumably you have the figures to hand, i.e., the mass of the atmoshere, its specific heat capacity, the mass of the oceans etc, to calculate how much heat the Earth has absorbed to warm the atmosphere and the oceans the number of degrees they have warmed since 1900 or whenever and then translate that into giga?joules of energy being absorbed by the Earth every day in order to do that and see how that figure compares with the figures in the infogram (which show no heat being absorbed by the Earth whatsoever). The consensus do a good job of "showing" how the GHE enable the Earth to be at a higher steady state temperature, but they do a very poor one, for all their fudges and handwaving, at showing how the Earth warms to that steady state. What you have produced so far has nicely skewered them on the steady state physics. It's time to attack them on the dynamic physics as well.
The essay you link to is all very fine, but the result of having air with a very low heat capacity and water with a very high one is simply to affect the rate at which heat can be transferred from air to water. It makes it slow, not, as the author is implying, impossible. It is only impossible within a given time frame, but he doesn't really make that point.

Mark Wadsworth said...

B, I've pondered that as well, but we are stuck with Hansen's leaky bucket analogy (which is a good analogy for the maths and an appalling analogy for the physics).

Mass of atmosphere per m2 surface = 10,335 kg, call it 10,000.

Specific heat capacity of air, call it 1,000 J/K/kg

Joules required to heat up atmosphere by 1 K = 10 million/m2.

That's built up over a period of whatever the Consensus have currently decided, say 40 years.

So that's 685 Joules of energy per day being 'trapped' and never released.

Earth gets 342 Joules of radiation from the Sun every second.

So to get 1 K of warming over 40 years, the mismatch between Sun in and radiation out only has to be about two seconds' worth per day. That is such a stupid tiny figure that you can't prove or disprove it.

Bayard said...

Surely it's more than 1K. 1K is within statistical error/natural variations.

Dinero said...

Bayard -

Yes that is the figure 1 Degree C.

Dinero said...

I think analogies should not be used in science. An analogy is used for a theory and then the mechanism of the analogy itself is shown to be correct, but it does not follow from achieving that, that the theory is correct or not, as it is not the same thing. And so analogies obfuscate they do not clarify.

- Bayard
I too noticed that about the ocean paper , the author used a single stock of heat, not a flow over time.

Mark Wadsworth said...

B, what's 1C between friends? That's difference between 7 am and 7.30 am, it's like starting on the south coast and driving 100 miles north etc.

D, analogies should not be used in science. But in terms of maths, Hansen has a point. Water dripping into at a certain rate, so water must be dripping out at the same rate. If the leak is closed a bit, the water level must go up to increase pressure on the hole to force the same amount of water out.

Dinero said...

That is a case in point , by showing how the bucket is sufficient to explain the result of the maths , he therefore bamboozles the reader and possibly himself into thinking that the atmosphere slows down the transit of infra red, which it does not. What it actually does is remit it, thus adding to the incident radiation on the surface, and that extra is why the temperature is higher. Granted you are disputing the significance of that bit , but that is the process that he is writing about.

Bayard said...

So the amount of heat leaving the Earth only has to have decreased by 0.0017% for this warming to have occurred.

Bayard said...

Useful graph here,https://www.skepticalscience.com/solar-activity-sunspots-global-warming.htm from the motherlode of shite Quite rightly they conclude, "If the sun's energy is decreasing while the Earth is warming, then the sun can't be the main control of the temperature." but, of course, being Alarmists, they fail to follow through. Global warming can only be caused by a mismatch between the energy incident on the Earth and the energy leaving it. The Consensus's GHE is proportional, i.e. the greater the energy flows in and out, the greater the mismatch and the more the warming. To use the leaky bucket analogy, for the same hole, if the amount going into the bucket goes up, the water level has to rise in order for the amount going up to match, so insolation change must result in an equivalent temperature change, which it doesn't. The foot gun is fired again.

Mark Wadsworth said...

Din, common sense tells us that this "atmosphere reflects radiation back to surface and heats surface" is nonsense. Or else, tell me why during the day time, the GHW is NEGATIVE and only POSITIVE during the night?

B, have you transposed one word?

" Global warming can only be caused by a mismatch between the energy incident on the Earth and the energy leaving it. The Consensus's GHE is proportional, i.e. the greater the energy flows in and out, the greater the mismatch and the more the warming. To use the leaky bucket analogy, for the same hole, if the amount going into the bucket goes up, the water level has to rise in order for the amount going up to match..."

Shouldn't that last bit say "To use the leaky bucket analogy, for the same hole, if the amount going into the bucket goes up, the water level has to rise in order for the amount going OUT to match..."?

Anyway, they are arguing smaller hole and higher water level = less radiation emitted to space and higher surface temp.

They are not arguing same hole and same water coming in = higher surface temp and same amount emitted to space.

(Or are they? It's difficult to tell what the hell they are arguing.)

Mark Wadsworth said...

Din, I am very much disputing the significance of this bit:

"What it actually does is re-emit it, thus adding to the incident radiation on the surface, and that extra is why the temperature is higher."

It's either bollocks or impossible to prove or disprove, I've covered that one. The air near sea level is a lot warmer than the air at the top of the troposphere. So the bottom layer emits more radiation downwards than the top layer emits out to space, full stop.

If you start with radiation emitted from surface, it is easy to say "Oh look, the troposphere reflects much more back to the surface than it lets through to space!" and/or "Oh look, the troposphere prevents radiation from the surface reaching space!".

The real explanation - mine - is far easier and much more mundane but does not fit The Narrative etc.

Bayard said...

Mark, yes it was a braino. However, even if we take Hansen's leaky bucket, with its smaller hole and higher water level and then alter the amount coming in (remember, one variable at a time), the level will change until the amount going out is the same. This means that a change in radiation coming from the Sun, must, according to the Gospel according to James, result in a corresponding change in the Earth's temperature, which even the Alarmists agree is not the case. I am tempted to point this out on the Motherlode of Shite, but I am afraid I will be suffocated by the ensuing outflow of bullshit, they having failed to stun me with science.