## Thursday, 12 March 2020

### Fun with numbers: Gravity, altitude, pressure and temperature.

I hope we've all learned something new this week, it has been a slog!

If you don't understand why this relationship holds on the basis of commonsense/observation; or you can't imagine how a big cloud of "air molecules" all repelling each other but attracted by Earth's gravity behave, then you can revert to the "modified gas/pressure laws as they apply in a strong gravitational field over long vertical distances" (I don't know if there is a snappier term).

This is a maths thing, so you don't really need to "understand" it. Clever physicists have worked it all out for us.
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Key concept - there is a difference between "density" and "pressure". If you pump a gas into a smaller fixed volume (such as a diver's oxygen tank), the density (number of molecules per unit volume) goes up (obviously).

Initially, it will warm up - pressure and temperature will be higher. The tank will then cool down to the temperature of the surroundings, and the pressure will fall again slightly. But the density of oxygen in the oxygen tank (number of molecules per unit volume) won't change.
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The Barometric formula give the relationship between height (i.e. above sea level); pressure; and temperature T (in Kelvin), it describes the equilibrium state where all opposing forces cancel out:

where
Ph is the pressure at height h,
P0 is the pressure at reference point 0 (typically referring to sea level),
m is the mass per air molecule,
g is gravity,
h is height from reference point 0,
k is the Boltzmann constant,
T is the temperature in kelvins.

It's an equation. So you plug in as many known figures or constants as you can and see how many "unknowns" there are.

Pressure is easily estimated; it's highest at sea level (about 1,000 mbar) and half that (about 500 mbar) when you are about halfway up through the atmosphere, which (we are told) is on average 5.5 km.

There are three constants - "m", "g" and "k". Whatever they are, and whatever the units, you can multiply them together to get one constant. If you define height in kilometres and do the numbers, "mg/k" boils down to approx. 31.

[UPDATE, you are supposed to define "m" in kg and "h" in metres, but it saves decimal places if you use grams and km (the 1,000s cancel out). Either way it comes out to 34.2 (not 31), but that doesn't seem to make much difference overall]

That only leaves one "unknown", the temperature - so if you know height, you can work out temperature.

We know, or are told, that the expected average temperature of earth's surface would be 255K in the absence of an absence of an atmosphere.

So it's reasonable to assume that the overall average temperature of the atmosphere is 255K, and that (average) halfway up, 5.5 km, it's 255K. Which turns out to be broadly true.

Plug all that into the formula and it all checks out:

1,000 x e^-((31 x 5.5)/255) = +/- 500

In Excel, you do =1000*EXP(-(31*5.5)/255), which comes out at 512 mbar, close enough to expected 500 mbar.

You can't plug height = zero into the formula, because e^0 = 1, so the answer would come out right whatever temperature you assume i.e. you can't use it to calculate temperature at sea level, but you can substitute, say 1 km altitude. T will come out at about 282K

This is 27K warmer for a 4.5km fall in height = 6K per kilometre, which is pretty close to typical "moist adiabatic lapse rate".

Extrapolate that down to sea level, and you get 288K, which, we are told, is the average temperature of the surface of the earth.

Extrapolate that up to the top of Mount Everest, and you know why it's so cold up there, i.e. colder than it would be if we had no atmosphere at all.

That's where the normal "greenhouse gas" explanation falls flat on its face - it's one possible explanation for why it's 33C warmer at sea level, but it doesn't explain why it's so cold up there. That's the thing that has been bugging me for a decade and nobody has ever even tried to tweak it.

So there we have it!

The extra 33C surface temperature is not down to back radiation from "greenhouse gases", it is just how the "modified gas/pressure laws as they apply in a strong gravitational field over long vertical distances" work.

Dinero said...

" at1 km altitude. T will come out at about 282K"
"Extrapolate that down to sea level, and you get 288K, which, we are told, is the average temperature of the surface of the earth."
"The extra 33C surface temperature is not down to back radiation from "greenhouse gases", it is just how the "modified gas/pressure laws as they apply in a strong gravitational field over long vertical distances" work."

In that Formula P is the dependant term . The causality runs from the terms on the right hand side of the equation to the dependant term on the left hand side of the equation. You cannot conclude that T is caused by P after rearranging the equation with T on the Left. The causality is not rearranged when the equation is rearranged.
The P is dependant on T.

As far as I am aware the reason temp drops with altitude is because the ground warms up over the coarse of the day and heats the atmosphere over the coarse of the day, and so the air nearest the ground is warmer.

Din, the penny drops. Having shown that you don't understand basic physics...

"In that Formula P is the dependant term . The causality runs from the terms on the right hand side of the equation"

In maths, if you have an equation where all terms but one are known, the other one is the "unknown" and you calculate it.

X = 2Y is exactly the same as Y = 1/2 X. If you know one, you can work out the other. No need to read from left to right or right to left.

So maths fail, just to start the proceedings.

"the reason temp drops with altitude is because the ground warms up over the coarse of the day and heats the atmosphere over the coarse of the day"

1. It's spelled "course". Spelling fail.

2. That is superficially convincing if we had 24 hour sunshine. But the basic vertical gradient holds at night time (when the ground is cooler than the air above) or in cloudy conditions.

3. For sure, on earth in summer when the sun is shining, the ground is noticeably warmer if it's in sunshine than if it's in the shade. But the vertical temperature gradient holds even if the ground gets no sunshine (night time, cloudy, or indeed on Venus, where the atmosphere is so thick and cloudy that it's very dark at surface level, see also "why is it hot in Gas Giants? How to stars ignite?")

Failure to look at evidence and counter-examples.

Dinero said...

Physics is not the same as maths. Causality of physical relationships in nature cannot be changed by a man moving terms around on a piece of paper. It is opposite statements from yourself of that magnitude of stupidity that make me think this is one massive troll of your own blog. But you are adamant that this series of posts is not a joke.

You are stuck on the notion that pressure causes temperature. Here is a thought experiment that you could perform with yourself that could rid yourself of that erroneous notion.

It is Monday 12 mid day and your car tyres are 20 psi. You at 12pm mid day pump them up to 30 psi.
Then you wait . Now on Tuesday at 12 mid day the car tyres are 30psi you touch the car tyres. Are they hot or cold or the same temperature.

Din,

1. At such a small scale, we can ignore gravity.

2. There is no exchange of air molecules between inside and outside the tyre.

3. You have missedd the key concept that although pressure in tyres has fallen, air density has not!

Toodle pip.

.. do you not grasp the difference between a car tyre and an atmosphere in a strong gravitational field, without which there would be no atmosphere anyway?

Dinero said...

" You have missed the key concept that although pressure in tyres has fallen, air density has not"

No that is wrong on both parts, the pressurw in the tyre has not fallen it has risen, and the air density has risen.

"do you not grasp the difference between a car tyre and an atmosphere in a strong gravitational field, without which there would be no atmosphere anyway?"

In the thought experiment there is no task of grasping for me to perform. The task is for you to comprehend that the car tyre is the same temperature for different pressures. And so to rid yourself of the notion you have been repeating over many posts since a week Monday before last, that temperature is inherent with pressure.

Din, future comments will be marked as spam, sorry, but I can't be bothered any more, arguing with somebody who keeps contradicting themself and shifting the goal posts.

Bye bye!

Lola said...

"...Physics is not the same as maths. Causality of physical relationships in nature cannot be changed by a man moving terms around on a piece of paper..." Eh? I thought that that was exactly the point of maths. To move terms around on a piece of paper just exactly so they can be used a tool to explain physics? Or am I missing something?

L, exactly.

And these is not about "causality", it's about the equilibrium position where everything causes, and is caused by, everything else.

ThomasBHall said...

When racing, the friction causes the tyres to heat and the pressure in the tire to rise- you need to let air out to get back to the required pressure. Once the tires cool down, the pressure has dropped so you need to reinflate them. Basically, you need to make sure the quantity of air is correct for temperature to give the right pressure.
In Dins example, the pressure would have fallen if the temperature had dropped in the intervening time.

Bayard said...

Now, Mark, your next task is to get this on Sceptical Science. What will they come up with to refute it?

TBH, Din's example keeps changing.

B, ooh, now there's a challenge. This formula is the basis of the Nikolov Zeller explanation (I asked him on Twitter and he said yes), who are much hated in agw. circles

B, update, I have tweeted them to ask whether it's on their list of "myths".

Physiocrat said...

The atmosphere functions like an engine. If you quickly compress the air in a bicycle pump it becomes warmer. The energy for this warming comes from your work. The same principle is applied in a diesel engine, where the temperature rises to the point where a small amount of fuel will burn if it is forced into the hot cylinder. This happens explosively and the piston is forcibly moved as the gas expands. This force can be transferred to a drive system and the energy can thereby be utilised in an engine. This expansion happens adiabatically ie the temperature falls as the gas expands.

This is more easily seen in a steam engine, where a puff of hot steam under pressure is admitted to the cylinder, and the steam expands and pushes the piston. What happens to the steam? It expands to several times its original volume and its pressure and temperature falls, so that it is on the verge of turning into water, which is what you see coming out of the chimney. This is an adiabatic process ie no energy is absorbed or released through the walls of the cylinder, due to the speed with which the expansion occurs. The energy in the hot steam, which was originally chemical energy in the fuel, has been used to make the wheels go round.

The atmosphere is an ocean of compressible gas sitting on the solid surface of the ground. There is a constant supply of energy from the sun. At the top of the atmosphere, the pressure can be considered to be zero. At ground level, the pressure is that due to the mass of a layer of air approximately 10 km tall, which is the same as a layer of mercury 75 cm tall, which adds up to quite a lot. The air is, as a consequence, relatively compressed - 15 lbs per square inch in old money.

I cannot quite picture this myself but I assume that the radiation energy flux arriving at the edge of the atmosphere, passes through the atmosphere to the surface, where it is absorbed and heats the surface. This effectively sets up a heat engine, resulting in the lapse rate described.

The carbon dioxide effect on Venus might be attributable not to the IR absorption effect but to the higher density of CO2, which has a molecular weight of 44 compared to the average molecular weight of air which is about 29 ie the molecules of CO2 are 1.5 times heavier.

Which makes the war on carbon an expensive and futile diversion.