How to work out the 'adiabatic lapse rate' using logic and maths (instead of relying on observations).

Some of my posts drag on a bit before the conclusion, so this time I will do bullet points first (with links to skip down to the fuller explanation).

1) The traditional explanation, that the sun warms the Earth's surface; the surface warms the air above it is largely nonsense, or is only a small part of the picture, if you bother thinking about it for a few minutes. Fuller explanation.

2) The actual explanation is that energy tries to spread out and equalise in all directions. The air at the surface has no potential energy but lots of heat. Further up, the air has more potential energy but less heat, all the way up to the top of the troposphere. The two forms of energy (potential plus heat per unit volume) is a constant all the way up. Fuller explanation.

3) Some constants and formulae we have to accept as a given; we plug these into the workings; and work backwards to get the lapse rate that balances Joules/m3 all the way up. Constants, formulae and workings.

4) Conclusion: to get the Joules/m3 to balance all the way up (I calculated up to 6km, half-way up), the local temperatures of the troposphere has to fall by 8C (or 8K) for every 1 km you go up. The extra potential energy = the energy no longer needed to warm the air in each m3.

That's a bit more than the observed lapse rate of 6.5C, and a bit less than the 9.8C which Wiki says is the theoretical dry lapse rate. So I'm pretty happy with it for now.

5) How to set up hyperlinks within one web page, from ComputerHope.com.
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1) The traditional explanation is (largely) nonsense.

You're seven years old and your science teacher tells you that the Sun warms the earth surface; this warms the air above it; that warms the air above that etc. So the bottom layer is warmest and it gets colder the higher you go. So kids, she trills, that's why it's so cold at the top of Mt Everest.

Sounds plausible until you relate it to real life...

- there is a lapse rate at night, when the surface is not being warmed and it's actually cooling down faster than the air. Which is why you should have more layers underneath you than above you when sleeping outside.

- there is a lapse rate, even when it's cloudy and the surface is not being warmed; the sun is hitting the clouds first. It's still colder above the clouds than below.

- the Sun (when it's shining) hits the top of Mt Everest with the same intensity as it hits land at sea level; but it doesn't warm it up nearly as much.

- the surface cools down from average 293K by day to average 273K at night (let's say for sake of example) and warms to 293 again the next day. So 3.5% of the heat is lost and regained each day, it has a 'half life' of about three weeks. Most of the heat is left over from previous days and that has had plenty of time to mix evenly (if it wanted to do so).

- it's like saying "Bath water is deeper at the tap end, because that's where the water goes in." Wrong, the surface quickly levels off. Bath water is indeed deeper at the tap end, but it's because that's where the plug hole is, and the plug hole end is lower.

- Yes, on a sunny day, you can see hot air rising from e.g. dark tarmac = 'thermals'. Vultures and hang glider pilots use these to get around. But this is a dynamic thing, it disturbs the equilibrium, which is constantly trying to re-establish itself, thermals are not the equilibrium or an explanation for it.

2) The actual explanation.

- I hope we all remember "Energy cannot be created or destroyed, it just transforms from one from to another". The sun sends us energy as light; plants use it to grow and they store chemical energy. You burn wood (stored chemical energy) in a power station and convert it to heat; that turns water to steam; steam gets converted to kinetic energy and water (again); the kinetic energy gets converted via magnetic to electrical energy. Which goes down the wires, through a filament in a bulb and becomes light energy and heat; the same as what we got from sun a long time ago.

- energy will choose whatever form (radiation/light, magnetism, electrical, heat, sound etc) enables it to spread out as evenly and as quickly as possible for a given medium.

- heat will flow from warm to cold (speed depends on how good the medium insulates); photons will go from light to dark (which is why it takes them so long to leave the sun; it's equally bright everywhere and they're not sure which way to go); once the photons have escaped the sun, they whizz off at the speed of light; sound waves travel as fast as they can from loud to quiet (760 mph through the air at sea level; faster through solids).

- gases, i.e. the atmosphere, are very compliant. Energy can travel through it by conduction; by radiation; or by convection i.e. converting heat to potential energy (air will carry sound and shock waves, and even conduct electricity if the potential difference is high enough = lightning). The precise method and speed are immaterial in an equilibrium situation. They all boil down to the same thing and the speed is to all intents and purposes instantaneous. If heat from the atmosphere hits the edge of the atmosphere, it doesn't care, it turns into light/radiation and keeps going, the only way for energy to travel through a vacuum.

- on a more mundane level, a hot air balloon is a way of converting heat energy to potential energy (altitude). If it were 100% efficient, you would gain exactly the same amount of Joules in potential energy as you use up in Joules of heat energy. The troposphere is as close to 100% efficient as you can get.

- so it seems reasonable to assume that the total energy per unit volume (like a m3) will be the same at all altitudes in the troposphere in an (idealised) equilibrium situation, let's call this 'energy equivalence' for the sake of calling it something.

3) The constants, formulae and workings.

The main steps:

1. Set up your spreadsheets with all the scientific constants and formulae.

2. Find out surface/sea level temperature and plug that into Barry to find the altitude at which pressure = 0.5 atm, which is 5.9 km.

3. Take 0.5^(1/5.9) to find the % by which pressure falls for each km altitude.

4. Work out total Joules/m3 at sea level.

5. By trial and error, find the temperatures/lapse rate which give you the closest amount of Joules/m3 all the way up. You get the best fit with 8K per 1,000 metres.

So the result of Barry feeds into this calculation and vice versa. Doesn't matter, it is not a circular calculation if there is an equilibrium and the results match up to observations.

Bottom row shows I'm pretty close up to 5 km and only 3% out at 6 km altitude. The measured temperature half-way up is apparently 250K, not 240K. But overall, I call it a good result for a couple of hours with a calculator in the back garden. Click to enlarge.

Is there life on Mars?

I doubt it, but what I do know is that it has an atmosphere:

Despite the high concentration of CO2 in the Martian atmosphere [95%], the greenhouse effect is relatively weak on Mars (about 5°C) because of the low concentration of water vapour and low atmospheric pressure. While water vapour in Earth's atmosphere has the largest contribution to greenhouse effect on modern Earth, it is present in only very low concentration in the Martian atmosphere.

So the CO2 makes little difference then? It's the water vapour all of a sudden? (Venus also has hardly any water vapour and is 95% CO2).

What makes the big difference is the low atmospheric pressure, but they only mention that in passing. On the surface of Mars this is only 0.61% of sea level atmospheric pressure on Earth (and on the surface of Venus it is 92 times as much).

The effective temperature of Mars, i.e. its surface temperature if it had no atmosphere, is 210K. It's actual average surface temperature is 215K. To get the Barometric Formula to balance and give you an average temperature of 210K for the whole atmosphere*, you have to put in a surface temperatures between 215 and 218K.

This is not far off the official figure and gives you a lapse rate of a bit less than -1K/km altitude. NASA say it is almost exactly -1K/km. And they've been there and measured it. Interestingly, the altitude/temperature chart in the Wiki article (first link) also shows a lapse rate of about -1K/km, even though the article itself says actual -2.5K/km and predicted -4.3K/km,

So the formula comes through for us yet again; all that matters is that a mole of CO2 has an atomic mass of 44, about one-and-a-half times the mass of a mole of the Earth's atmosphere (mix of N2 and O2 = 29). You don't need to know anything else about the gases' properties.

As an approximation, density and pressure are proportional, so Mars has about fourteen times as many CO2 molecules per unit volume as Earth does. Earth = 420 ppm; Mars = 1,000,000 x 0.61% x 95% = 5,800.

So this next bit is a real cop-out:

Moreover, under low atmospheric pressure, greenhouse gases cannot absorb infrared radiation effectively because the pressure-broadening effect is weak.

CO2 molecules are a lot closer to each other on Mars than on Earth!

* Earth's effective temperature is given as 255K, which is very close to the average temperature of the atmosphere, depending on how it is calculated. So let's assume that the the average temperature of a planet's atmosphere is equal to its effective temperature. The charts tell us that the surface temperature is higher than the average and the temperature at the top of the stratosphere is lower than the average.

Somebody please make it stop!!

Exhibit A, from the BBC:

There are natural fluctuations in the climate but scientists say temperatures are now rising faster than at many other times.

This is linked to the greenhouse effect, which describes how the Earth's atmosphere traps some of the Sun's energy. Solar energy radiating back to space from the Earth's surface is absorbed by greenhouse gases and re-emitted in all directions.

This heats both the lower atmosphere and the surface of the planet. Without this effect, the Earth would be about 30C colder and hostile to life.

1. As ever, they say "the Earth would be about 30C colder" instead of "the Earth's surface would be about 30C colder".

2. The Earth's atmosphere is, on average, about 30C cooler than the Earth's surface. They always gloss over that bit. The lowest part (where it touches the surface) is 30C warmer than average and the highest part is 30C cooler than the average.

3. The Barometric Formula explains that there has to be a temperature gradient going up ('adiabatic lapse rate' in Newspeak).

4. After we'd done out Physics O Level, the teacher said, let's stretch our mental legs a bit, and did some clever workings to show that if you know atmospheric pressure and density of air at sea level, you can extrapolate upwards to see where the top of the troposphere is, which contains 99% of the atmosphere (it gets a bit mad further up). I can't remember how he did it, but it all stacked up. To cut a long story short, pressure falls at a fairly constant rate, so it's not difficult to estimate pressure at a given height if you know pressure at sea level and that it's effectively zero 11 km up. And temperature also falls at a fairly constant rate. That's two of the variables you need for the formula without breaking a sweat. Expected temperature then just pops out of the formula, and is surprisingly close to observed results.

5. If the facts fitted the BBC's "really simple guide", then not only would the surface be 30C warmer, the top of Mt Everest would also be warmer - but it's not, it's a lot colder. They hedge their bets and fudge it with this claim: "This heats both the lower atmosphere and the surface of the planet." Are they trying to say that the top of Mt Everest would be warmer if there were no CO2?

6. To use a crude analogy, a perfectly efficient fridge doesn't change the average temperature of your kitchen. It causes the inside to be colder and warms up the rest of the kitchen to balance. The atmosphere does the same sort of thing. It can't change its own average temperature (how?), that's dictated by the Sun, but it cools the top half and warms the bottom half in equal and opposite measure.
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Exhibit B, from Wikipedia:

The temperature of the troposphere generally decreases as altitude increases. The rate at which the temperature decreases, -dT/dz, is called the environmental lapse rate (ELR). The ELR is nothing more than the difference in temperature between the surface and the tropopause divided by the height. The ELR assumes that the air is perfectly still, i.e. that there is no mixing of the layers of air from vertical convection, nor winds that would create turbulence and hence mixing of the layers of air.

The table below gives an ELR of 6.5C/km. You can measure this directly or work it out by plugging height and pressure into the formula. Rather infuriatingly, they give a variant of the formula in the section above, but don't draw any conclusions from it.

The reason for this temperature difference is that the ground absorbs most of the sun's energy, which then heats the lower levels of the atmosphere with which it is in contact. Meanwhile, the radiation of heat at the top of the atmosphere results in the cooling of that part of the atmosphere.

Sure, direct sunlight heats up the ground and that must warm the air directly above it. So the equilibrium is disturbed and tries to re-establish itself. But that is moving the goal posts. How much of the earth's surface is getting direct sunlight at any one point in time? It can't be more than half (day-night) and in the day time, maybe there's direct sunlight half the time (clouds). So overall, one-quarter? And two-thirds of the Earth's surface is water, which warms (and cools) much more slowly that the land.

Nonetheless, the temperature gradient is much the same by day or by night, when it could be argued that the surface cools the air above it. It is the same above the oceans (fairly constant surface temperature, day or night); it is the same above land (warms faster by day; cools faster by night); it is the same whether it's sunny or cloudy.
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We know all this stuff, or can easily find out. Why do they keep trotting out explanations that contradict each other and are completely at odds with known facts and basic physics?

(As ever, for sure, we also know that CO2 blocks/reflects infra red at a few frequencies, but that is a very marginal effect, no more than one degree here or there. There is twenty times as much C02 in the Martian atmosphere, but it is still bloody cold because it is further from the Sun, obviously, but also because the Martian atmosphere is only one-hundredth as thick as Earth's. The fact that it is 95% CO2 is largely irrelevant).

More climate change fun with the BBC

From the BBC:

More than three billion people will be living in places with "near un-liveable" temperatures by 2070, according to a new study.

Unless greenhouse gas emissions fall, large numbers of people will experience average temperatures hotter than 29C. This is considered outside the climate "niche" in which humans have thrived for the past 6,000 years...

Researchers used data from United Nations population projections and a 3C warming scenario based on the expected global rise in temperature. A UN report found that even with countries keeping to the Paris climate agreement, the world was on course for a 3C rise.

According to the study, human populations are concentrated into narrow climate bands with most people residing in places where the average temperature is about 11-15C. A smaller number of people live in areas with an average temperature of 20-25C.

Ho hum.

From National Geographic:

The tropics are regions of the Earth that lie roughly in the middle of the globe. The tropics between [sic] the latitude lines of the Tropic of Cancer and the Tropic of Capricorn. The tropics include the Equator and parts of North America, South America, Africa, Asia, and Australia.

The tropics account for 36 percent of the Earth's landmass and are home to about a third of the world's people.

So the hot areas between the tropics have much the same population density as north and south thereof. They are not concentrated into "narrow climate bands", they are at all latitudes (except the Arctic circles and deserts. And half of the world's population (3.5 billion people) live within this circle, which is almost entirely between the tropics (the emptiest bit of that circle is the half of PR China, which is north of the Tropic of Cancer).

The tropics are warm all year, averaging 25 to 28 degrees Celsius (77 to 82 degrees Fahrenheit). This is because the tropics get more exposure to the sun. Because of all that sun, the tropics don't experience the kind of seasons the rest of the Earth does.

So one source says "A smaller number of people live in areas with an average temperature of 20-25C" and the other says that one-third of the world's population lives in areas with average temperatures of 25 - 28C. Which source looks more reliable?

If you add an arbitrary increase, like 3C, you can predict that average temperatures will be about 29C. Easy. I'm sure there are already plenty of inhabited areas with average temperatures of 29C, if it's so terrible, why do people live there?

And so on.

They own land! Give them money!

Apropos my comment on the last post "They own land! Give them money!", from The Guardian:

London NHS Nightingale hospital will shut next week

No 10 says decision is due to limited demand, with no coronavirus admissions expected in coming days

The showpiece Nightingale hospital in London will shut next week after treating a small number of patients but will be kept “in hibernation” in case a second wave of Covid-19 infections emerges.

No further patients will be admitted to the facility, which was created amid much acclaim in just 10 days, and the 12 patients being treated there at the moment are being transferred to other London hospitals...

Originally planned to have 4,000 beds, the Nightingale has treated just 54 patients since it was opened by Prince Charles on 3 April and received its first patient on 7 April. It has not admitted a new patient for a week as London hospitals have had spare capacity in their own intensive care units.

The four other Nightingales that were opened to stop hospitals being overwhelmed – in Manchester, Birmingham, Bristol and Harrogate – will also be wound down, though the London hospital will shut first. All were conceived in March, when ministers and health service bosses were concerned that NHS hospitals risked being overwhelmed by significant numbers of people needing to be ventilated to keep them alive, as Italy was confronting at the time.

But while the Manchester hospital has taken some patients, its sister facilities in Birmingham, Bristol and Harrogate have not admitted anyone.

Give that man some medals!

From The Daily Mail:

A son has managed to nurse his coronavirus-stricken father back to health at home despite the 81-year-old being released from hospital to die.

Raj Nathwani revealed his Suri [sic] went into Watford General Hospital on March 26 with suspected coronavirus after struggling on his daily walk. Doctors said they were 95 per cent sure Suri, who has chronic obstructive pulmonary disease, had the deadly bug. But they wanted to discharge him.

Raj used a Google spreadsheet to monitor Suri's vital signs, kept his house clean and isolated him from the rest of the family in Watford. The 55-year-old relied on a continuous positive airway pressure, also called 'black boxes', which are used to tackle sleep apnoea.

Hats off for his devotion, initiative, courage etc, but the key is the last bit (in bold).

I read an article somewhere or other, probably in a covid-conspiracy Whatsapp group, by a doctor who said that using ventilators was overkill and actually made matters worse. The gist of it was that it's not the virus that kills you, it's your immune response. Injuring the airways and lungs by intubating and then overloading them with oxygen exacerbates things. The survival rate of people who are put on ventilators is apparently quite appalling (although maybe it is not a fair comparison - they only put people on ventilators as a last resort).

The article concluded that the common-or-garden face masks (which people use to treat sleep apnoea) which increase the air pressure slightly was were much more appropriate (they are also a lot cheaper, easier to use, safer etc).

In this instance, it looks like the author of that article is on to something, and Mr Nathwani did exactly the right thing.

They own land! Give them money!

It had to happen, I suppose...

From the FT:

In a letter to Rishi Sunak, the chancellor, a coalition of the UK’s biggest retail chains and property owners warned that “many viable companies” will file for administration if the government does not intervene further, causing job losses and a “devastating impact” on high streets.

“We have come together, as voices of both commercial tenants and landlords, to propose that the government introduces a scheme of rental support for the space that has in effect been furloughed, just as staff have been,” said the letter, from the chiefs of the British Retail Consortium and the British Property Federation...

The two industries want the government to support a “furloughed space grant scheme” where the state would cover the fixed costs of businesses that have experienced dramatic falls in turnover. Similar schemes have been set up in Denmark and other European countries.

The groups have proposed a sliding scale of partial payments to cover property costs, which would still leave some of the burden on the tenant and landlord.

Fun Online Polls: cars and garages

Suggested by Bayard.

Vote below; or here; or use the widget in the sidebar.

Which of the following apply to you (multiple answers allowed)?

I keep my everyday car in my garage.

I keep my second car in my garage

I use my garage as a workshop.

I use my garage to store junk

My garage has been converted to living space

I use my garage for some other purpose (please specify).

I have no garage

pollcode.com free polls

Killer Arguments for Raising the Rent, Not

Down in the West Country, a railway company is trying to bring in more income by raising the rent:

Their reasons include these gems:

"This rent is considerably below what the PLC (the railway company) feel is its value, and as a reference (1) the rent for Williton/Sherrings Yard is £17000 pa; (2) WSRA at Bishops Lydeard (BL) pay a rent of £4,500.00 pa for their office space; (3) the WSSRT pay a rent for the BL Museum of £100.00 pa paid in 4 instalments and (4) for the Blue Anchor Museum it is £741.60 pa paid in 4 instalments, however this has been waived until Oct 21 to assist the refurbishment. (5) Quantock House at BL was set at £22,000 pa, but then withdrawn by the landlord. (6) 5542Ltd pay £10 per foot of siding space pa."

As we know, what sets the rentable value of land is two things. By far the greater one is location. The other is what stands on the land. The site in question is in a small village with access only over a working railway off a tight bend in a major road. The tenants originally rented a bare site, with no buildings, rails or services on it and have paid for all the infrastructure themselves. To compare this site with a yard in an industrial estate (1) in a large village/small town with good level access is a classic diagonal comparison.

As is it to compare the rentable value of an empty site with that of a building, (items 2-5). Even in example (6), 5542 Ltd. didn't install the rails themselves. Moreover, as we also know, what the landlord feels is the value of the rent is neither here nor there, the value of the rent is what someone else would be prepared to pay for it.

The railway company also think that the needs of the landlord have a bearing on the rentable value. A large part of the statement from it is along the lines of "the company needs money, therefore the rent must go up".

"The PLC met the Trust and made clear it needed to increase its income wherever possible. It was suggested that the Trust should review its finances, that a rent increase was needed and that as this would take some time to manage, the PLC would accept a staged increase over time, but that an increase was needed."

The final piece of wishful thinking is the suggestion that the rent was originally set at below market levels out of benevolence, however there is no attempt at justifying this statement apart from the slightly ludicrous :

"At that time the PLC agreed the original lease it was trading profitably and could afford to be more generous than now; the level set is also an indication of the historic weak financial management which has been part the PLC’s current financial situation."

Just because it could, doesn't mean that it did and just because the Plc is in financial trouble now because of "weak financial management" (even if that really is the cause) doesn't mean that was the case back in 1991.

3/10 - must try harder

Killer Arguments Against Citizen's Income, Not (28)

I'm sure I've covered this before, but it has raised its ugly head again (@CentreThinkTank on Twitter, who are otherwise pretty sound on all sorts of issues).

For some reason, they (like a lot of centre-right people) think Negative Income Tax is better than Citizen's Income because it can be 'targeted' at lower earners and so it is cheaper than paying it to everybody, including people 'who don't need it'.

As Sam Bowman at the Adam Smith Institute explains, The Negative Income Tax and Basic Income are pretty much the same thing.

His article is logically and mathematically correct, but I find it helpful to look at a real-life example...

A real life example

Ireland used to have* a hybrid system as follows, which I think is the best combination of the two:

1. Unemployment benefit was €300 a month.

2. There was, officially, no income-tax free personal allowance.

3. People in work received a 'tax credit' of €300 per month against their PAYE liabilities.

4. PAYE was calculated as follows:

step 1: multiply gross pay by 30% on all earnings (up to the higher rate tax threshold) = PAYE due.

step 2: reduce each person's PAYE by €300 each month (but not below zero)

5. If an unemployed person took on some some part-time work, he continued to receive the unemployment benefit but didn't get the tax credit as well. He had PAYE deducted at the flat 30% on all his earnings.

Worked examples:

The unemployed person gets net income €300 a month.

An unemployed person takes on some part-time work, €500 gross a month. Net of 30% tax he earns €350 and keeps his €300 unemployment benefit. Net income €650.

A low earner on exactly €1,000 a month looks at his payslip and sees PAYE deducted zero (the 30% x €1,000 and the tax credit exactly cancel out), net pay €1,000.

A medium earner on €2,500 a month looks at his payslip and sees PAYE deducted €450 (30% x €2,500 = €750, less €300 tax credit).

Conclusion

The final outcome (in terms of net income) is exactly the same whether you call the €300 'unemployment benefit' or 'tax credit' or 'negative income tax', or 'citizen's income'.

The 30% is the same whether you call it 'means-testing' or 'claw-back of unemployment benefit' or 'income tax'.

The majority earning more than €1,000 a month couldn't care less if you call it a €300 'tax credit' or a €1,000 'tax-free personal allowance'.

The part-timer on €500 a month should be indifferent whether he gets net pay €350 and €300 unemployment benefit; or whether he gets €500 gross pay and €150 negative income tax (being 30% x the excess of the notional tax-free personal allowance over his gross pay).

The difference here is psychological. We want the unemployed to take on low-paid or part-time work. If we say "You can keep your €300 unemployment benefit and keep 70% of your gross earnings", they won't miss the 30% tax deducted because they never had it. That is more of an incentive than saying "You can keep all the €500 you earn, but we will make you fill in some forms and will reduce your unemployment benefit by half".

It is impossible to say whether the Irish system was more like Citizen's Income or more like Negative Income Tax. Because in terms of the final outcome, they are identical and indistinguishable!!

It's just a question of administrative simplicity and minimising fraud and error. With NIT, you need a separate system for calculating NIT and clawing it back; with CI, the claimant just gets a BR tax code, so pays tax on all his earnings - the clawing back is done via the normal PAYE system. So the CI system makes life easier for the minority at the margins of employment, and those are the people we should be concerned with. For the long term-unemployed and the majority earning more than €1,000 a month, NIT and CI are the same.

With the Irish system, there would have been some people in work who continued to receive the unemployment benefit and get the tax credit. Boo. And there would have been some people who didn't get unemployment benefit and earned less than €1,000 a month, who didn't get the full value of the tax credit. Also boo.

Well, sure, but there will always be fraud and error in a welfare system and there will always be tax evasion if you try to tax output and earnings. And payroll departments and the tax office will always make a few mistakes.
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* I've simplified this a bit and am doing this from memory. It's now more complicated, but you can recognise the basic model.