Wednesday, 11 March 2020

Vertical pressure variation - In the context of Earth's atmosphere


Main article: Barometric formula

If one is to analyze the vertical pressure variation of the atmosphere of Earth, the length scale is very significant (troposphere alone being several kilometres tall; thermosphere being several hundred kilometres) and the involved fluid (air) is compressible. Gravity can still be reasonably approximated as constant, because length scales on the order of kilometres are still small in comparison to Earth's radius, which is on average about 6371 km,[7] and gravity is a function of distance from Earth's core.[8]

Density, on the other hand, varies more significantly with height. It follows from the ideal gas law that

{\displaystyle \rho ={\frac {mP}{kT}},}


m is average mass per air molecule,
P is pressure at a given point,
k is the Boltzmann constant,
T is the temperature in kelvins.

Put more simply, air density depends on air pressure. Given that air pressure also depends on air density, it would be easy to get the impression that this was circular definition, but it is simply interdependency of different variables. This then yields a more accurate formula, of the form

{\displaystyle P_{h}=P_{0}e^{-{\frac {mgh}{kT}}},}


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.

Therefore, instead of pressure being a linear function of height as one might expect from the more simple formula given in the "basic formula" section, it is more accurately represented as an exponential function of height.

Note that in this simplification, the temperature is treated as constant, even though temperature also varies with height. However, the temperature variation within the lower layers of the atmosphere (tropospherestratosphere) is only in the dozens of degrees, as opposed to their thermodynamic temperature, which is in the hundreds, so the temperature variation is reasonably small and is thus ignored. 

For smaller height differences, including those from top to bottom of even the tallest of buildings, (like the CN tower) or for mountains of comparable size, the temperature variation will easily be within the single-digits. (See also lapse rate.)


James Higham said...

Phew, found that out just in time.

Mark Wadsworth said...

JH, I'm here to inform and entertain in equa measure.

Dinero said...

That only relates correlation not causation. As I have said before this is not a debatable subject. Air pressure and air temperature are not mated together. That fact is not debatable because the alternative has no physical meaning. But if you do not like physical description there is also the mathematical description.

The gas equation PV=nRT

For an increase in n there is an increase of P and so T remains the same.

Once again this is upper high school science and trivial . not something that requires improving by discussion.

Bayard said...

Do try reading the post again and you will find that it is not talking about the relationship between temperature and press, nor temperature and mass, nor yet temperature and volume, but the relationship between atmospheric pressure and the height above sea level at a constant temperature. I can't find a variable in PV=nRT that refers to the height above sea level, can you?

Dinero said...

> Bayard . Yes it is a cut and paste that temperature and pressure decrease wish allude. No doubt about that.

But Mark is trying to make some conclusion that violates PV=nRT .

Dinero said...

He is saying that lower atmosphere is ageater pressure an so he concludes that because of the higher pressure it is a higher temperature.. That is wrong.

It is wrong

I hope that settles it once and for all.

Mark Wadsworth said...

Din, it's correct. There is a clear link.

It is correct. You're the only nutter who doesn't see it.

I hope that settles it once and for all.