The Barometric Formula

UPDATE: I realised later on that I didn't fully explain how you can use the formula to calculate likely temperatures at different altitudes, which I thought was fairly obvious. Clearly, it isn't, so I have done a full explanation is here.
--------------------------------------------------------------
There is a good explanation of how this can be derived on Math24.net. It boils down to:

Pressure at height h = Pressure at sea level x e^-(mgh/RT)

This looks slightly different to the one I posted from Wiki two days ago. In that version "m" is the mass per molecule, which is the mass of a mole of gas divided by the Avogadro Constant (6 x 10^23), and "k" is the Boltzmann Constant (1.38 x 10^-23). So you are dividing a tiny number by a tiny number.

In this version, "m" is the mass of a mole of 'air' (0.029 kg), and it uses "R" instead of "k". "R" is the Gas Constant, which is just Avogadro Constant x Boltzmann Constant = 8.314. So this saves you a lot of decimal places.

"g" is the same in both version, 9.8m/s^2, which falls only imperceptibly as you go up through the troposphere, so can be assumed to be constant.

If you have three constants ("m", "g" and "R"), you can just multiply/divide them to give a single constant, which is 34.2. You can save more digits and decimal places if you express "m" in grams rather than kilograms and "h" in kilometres rather than metres.
----------------------
The Barometric Formula tells you the equilibrium/typical relationship* between height above sea level; pressure at that height; and temperature at that height (in K). There's no need to worry about what causes what, they all affect and are affected by each other. It's like balancing two playing cards edge to edge to form an inverted "V". Card A causes Card B to stay up and vice versa. "Equilibrium", for these purposes means that if you know two of the three variables (height, pressure, temperature), you can work out the third.

Height above sea level is easy; pressure declines at more or less the same rate all the way up to the top of the troposphere (average 11 km), so at sea level it's 1 atm and for each km you go up, it falls by about 0.9 atm; that just leaves temperature, which is either a known figure or you work it out using the formula.

Or if you know all three variables, you just plug them in and it should balance.

NASA tells us "Without naturally occurring greenhouse gases, Earth's average temperature would be near 0°F (or -18°C) instead of the much warmer 59°F (15°C)".

The claim is misleading - it just says "Earth's average temperature" and not "Earth's average surface temperature".

Does the Barometric Formula predict this anyway?

Let's start in the middle and work outwards.

The temperature half way up the atmosphere is - as a matter of fact - the 255K expected from solar radiation alone. If you plug in the known figures for half-way up the atmosphere (5.5 km height; 0.5 atm pressure; 255K), the formula balances. You can rework it for expected surface temperature at the surface and at the top of Mt Everest, and you get +/- 288K and +/- 230K respectively (255K plus or minus the adiabatic lapse rate x lower or higher altitudes). These are pretty close to the actual values, bearing in mind that the formula is based on the Ideal Gas Laws and maths - the formula stacks up in real life.

To sum up, the average temperature of the atmosphere is exactly what you would expect from solar radiation alone. The surface is 33K (or 33C) warmer than the average and the top of the troposphere is 33K cooler because of the barometric effect.

This is entirely independent of how much CO2 there is (apart from the fact that the extra 0.01% of CO2 in the atmosphere increases the mass of air by 0.005%, so "m" is not 29g, it is 29.00145g), and this reminds us that it is only Earth's surface that is 33K warmer, not the whole atmosphere.

I tweeted Den Nikolov (@NikolovScience) and pointed out that the formula makes the same predictions as the much maligned Nikolov-Zeller Hypothesis. (If you Google it, you'll get a dozen articles claiming to refute it for every article that agrees with it). Therefore, I continued, if his theory is wrong, the formula must be wrong and vice versa (the fact that one is correct does not necessarily mean the other is also correct).

He agreed with both of those points, but told me that the Barometric Formula is standard physics text book stuff, they won't be able to pretend it isn't widely accepted (like air brushing the Roman and Mediæval Warm Periods out of the history books).

So no longer shall I use obscure terms like Gravity-Thermal-Effect or Atmospheric-Thermal-Effect or Nikolov-Zeller Hypothesis, I'm sticking with "Barometric Formula"!
----------------------
Next time I can be bothered to discuss this with somebody who claims that Greenhouse Gases cause 33C of warming, I shall refer them to this post.

The Greenhouse theory is one possible explanation for higher-than-expected surface temperatures generally but it offers no explanation for the lower-than-expected surface temperatures at high altitudes, which is the Barometric Formula's trump card.

And as back up, I shall point out that the Barometric Formula applies to, and its effects can be observed on, all planets with an atmosphere, including Gas Giants, and that the vertical gradients apply at day (when surface is being warmed) and at night (when it isn't), over the sea (stable temperature) and over land (warms/cools).
---------------------
Of course, neither the Barometric Formula NOR the Greenhouse theory explain other climate changes over the past two thousand years - Roman Warm Period, Dark Ages Cold Period, Mediæval Warm Period, Little Ice Age, early 20th C warming, or mid/late 20th C cooling, recent warming. The Barometric Formula does not purport to do so, and the Greenhouse theory has to be modified endlessly to get the facts to fit the explanation.

To be fair, you can accept the validity of the Barometric Formula, and still believe that the Greenhouse theory explains the correlation between the higher CO2 and the slightly higher temperatures we have seen in the past few decades, maybe it does, maybe it doesn't. That's about it. I'm agnostic on that.
----------------------
Hopefully this will be last time I ever have to post about this! I'm sure I've lost most of my audience by now, but I do like to get to the bottom of things :-)
----------------------
* Yes, I am perfectly aware that there is loads more stuff that influence conditions at any one place or point in time; at the Equator it will always be warmer than at the Poles; the seasons; sun spots and solar activity; cold or warm winds; high or low pressure; the jet stream; clouds; Hadley Cells; the coriolis effect; updrafts, downdrafts; weather inversions; the fact that most landmass is in the Northern Hemisphere; Pacific and Atlantic oscillations; volcanic eruptions; large forest fires; oceans warming and cooling more slowly than dry land does; the list is endless.

But they skew the results in all directions and on the whole, their net effect is close to zero. These things also affect local results if you subscribe to the Greenhouse Gas explanation, so aren't really relevant when comparing the merits of Barometric and Greenhouse Gas explanations.