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## What is the scale height of water vapour in the Earth’s atmosphere?

Someone recently asked me what was the scale height of water vapour in the Earth’s atmosphere, so I decided to see if I could find out. The scale height of water vapour is particularly important for infrared, sub-millimetre, millimetre and microwave astronomy, as it is the water vapour in the Earth’s atmosphere which prevents large fractions of these parts of the electromagnetic spectrum from reaching the ground. This is why we can, for example, only study the Cosmic Microwave Background from space or from a few particularly dry places on Earth such as Antarctica and the Atacama desert in Chile.

## What does the term ‘scale height’ mean?

First of all, let me explain what the term “scale height” means. It is the altitude by which one needs to go up for the quantity of something (water, nitrogen, oxygen, carbon dioxide) to go down by a factor of $1/e$, where $e$ is the base of natural logarithms, and $e=2.71828.....$. The scale height, usually written as $H$, is dependent upon the temperature of the gas, the mass of the molecules, and the gravity of the planet. We can write that $H = \frac{ kT }{ Mg } \text{ (1)}$

where $k$ is Boltzmann’s constant, $T$ is the temperature (in Kelvin), $M$ is the  mass of the molecule and $g$ is the value of the acceleration due to gravity. If we were to plot the atmospheric pressure as a function of altitude we find that it follows an exponential, this is because of the differential equation which produces Equation (1) above (I will go into the mathematics of how Equation (1) is derived in a separate blog).

In the case of air, which is some 80% nitrogen molecules and 20% oxygen molecules, the scale height has been well determined and is $7.64 \text{ kilometres}$ (or, to put it another way, it drops by a half every $5.6 \text{ km}$). So, if one were at an altitude of $5.6 \text{ km}$, half of the atmosphere would be below you. Go up another $5.6 \text{ km}$ and it drops by a half again, so at $11.2 \text{ km}$ 75% of the atmosphere is below you.

## What is the scale height of water vapour in the Earth’s atmosphere?

Determining the scale height of water vapour in the Earth’s atmosphere is, I have discovered, essentially impossible. Or, to put it better, it is a meaningless figure. This is because it varies too much. It depends on temperature, so even in a given place it can vary quite a bit. So, instead, we talk of precipitable water vapour (PWV) at a particular place (both location and altitude). PWV is the equivalent height of a column of water if we were to take all the water vapour in the atmosphere above a particular location and it were to precipitate as rain.

The Mount Abu Infrared Observatory in India, for example, is at an altitude of 1,680 metres, and quotes a PWV of 1-2mm in winter. The PWV would be higher in summer, as water sinks in the atmosphere when it is cold. For Kitt Peak in Arizona, which is at an altitude of 2,090 metres, the PWV varies between about 15mm and 25mm. This is why very little infrared astronomy is done at Kitt Peak. For Mauna Kea in Hawaii, which is at an altitude of just over 4,000 metres, it varies between 0.5mm and 2mm. This is why there are a number of infrared, sub-mm and millimetre wave telescopes there.

At the South Pole, which is at an altitude of 2,835 metres, the PWV is measured to be between 0.25 and 0.4 in the middle of the Austral winter (June/July/August). Why is this so much lower than Mauna Kea, even though it it is at a lower altitude? It is because it is so much colder. The average Precipitable Water Vapour at the South Pole averaged over a 50-year period from 1961 to 2010. Even in the Austral summer it is low, but in the Austral winter (June/July/August) it drops to as low as 0.25 to 0.35mm, one of the lowest values found anywhere on Earth.

High in the Atacama desert, on the Llano de Chajnantor (the Chajnantor plateau), which is at an altitude of 5,000 metres and where ALMA and other millimetre and microwave telescopes are being located, the PWV is typically about 1mm, and drops to as low as 0.25mm some 25% of the time (see e.g. this website). This is why Antarctica and the Atacama desert (in particular the Chajnantor plateau) have become places to study the Cosmic Microwave Background from the Earth’s surface; we need exceptionally dry air for the microwaves to reach the ground.

## Summary

To summarise, it is meaningless to talk about a scale height for water vapour in the Earth’s atmosphere, as the vertical distribution of water vapour not only varies from location to location, but varies at a given location. So, instead, we talk about Precipitable Water Vapour (PWV); the lower this number the drier the air is above our location. To be able to do infrared, sub-millimetre, millimetre and microwave astronomy we need the PWV to be as low as possible, the best sites (Antarctica and the Atacama) get as low as 0.25mm and are usually below 1mm. The exceptionally dry air above Antarctica and the Atacama desert enable us to study the Cosmic Microwave Background from the ground, something we usually have to do from space.

### 10 Responses

1. on 27/06/2016 at 13:31 | Reply Brian Clegg

Thanks, Rhodri – you explain what the scale height is, but not specifically why it should be the height to get the value to drop by 1/e as opposed to any other reduction – that would be interesting to clarify. (I can guess, but it’s nice to be explicit.)

• on 27/06/2016 at 13:32 | Reply RhEvans

Ok – if you want I can add the maths for that. 🙂

• on 27/06/2016 at 13:44 | Reply RhEvans

I’ve added a little bit more maths, but to properly explain why it drops exponentially I would need to derive the equation for the scale height $H$, which means setting up the differential equation from which this equation comes. It is because $H$ is the solution to a differential equation that it has an exponential profile.

• on 27/06/2016 at 15:12 Brian Clegg

I’m not sure my question was clear enough. I wasn’t asking for maths, I was asking why we should care about 1/e more than 1/2 or 1/my age in days…

• on 27/06/2016 at 15:50 RhEvans

Because the variation of pressure with altitude is exponential. So, on a logarithmic plot (altitude plotted linearly on the x-axis and the natural log of pressure on y-axis) it would be a straight line with the gradient being $-1/H$, where $H$ is the scale height. So, for each increase in altitude of $H$, the pressure drops to $1/e$ of the previous value. The scale height is expressed in terms of $1/e$ because it’s a straight line on a logarithmic plot, and $e$ is the base of natural logarithms.

Is this a better explanation?

2. on 27/06/2016 at 14:00 | Reply S Schultz

Perhaps this link will be of interest. Forrest Mims III, a well-known name in electronics and also a staunch supporter of citizen science, has been monitoring atmospheric water vapor for 25 years. The link describes his efforts and results:

https://fmims.wordpress.com/2016/03/07/25-years-of-atmospheric-measurements/

• on 27/06/2016 at 14:03 | Reply RhEvans

Very interesting. Thanks. I took a quick look at the PWV measurements, the average is about 3 centimetres, which is why one would not try and do infrared, millimetre or microwave astronomy from this particular part of Texas! 😉

3. […] « What is the scale height of water vapour in the Earth’s atmosphere? […]

4. on 05/09/2017 at 00:19 | Reply elfyhead

Sorry to necropost on this thread, but I ran across it when googling for something else and would like to clarify for anybody that runs across it again.

It does actually make sense to speak of the scale height of water vapor in the atmosphere of the Earth, because it turns out that the distribution of water vapor is (mostly) exponential with height. This is known both from radiosonde data and from sounding (both from Earth surface and from orbit). (As an aside this also answers Brian Clegg’s question – it is natural to speak of a scale height as being the 1/e distance because the distribution is exponential.) Now, it does not follow a strict hydrostatic equilibrium definition (you can’t determine the scale height from simply plugging the mass of the water molecule into H = kT/mg) because that only applies to the neutral atmosphere. The scale height does vary significantly with location (latitude, altitude, whether you’re in a humid or arid location, etc.) – it typically has values around 1-3 km. See for instance Paramaswaran and Krishna Murthy, J Appl Met, 29, 665, 679, 1990 and references therein. There are more recent ones but I have that one to hand.

• on 05/09/2017 at 00:29 | Reply RhEvans

That is essentially what I say, that one cannot speak of the scale height of water vapour in the atmosphere. Did you actually read the post?