Atmospheric Lapse Rates

The question place is serving its purposes, one of which is to bring up points that warrant some discussion in a fuller post. At hand is the atmospheric lapse rate, which Bart brought up by way of his question:

2) I read elsewhere (I can only research what I read, I don't really have the ability to check much of this for myself) that models assume a constant lapse rate. Chris said the lapse rate is required for the greenhouse effect, but from everything I look at people only catgorize in "Dry" or "Moist" cases, but doesn't it vary everywhere over the globe?

There are models, somewhere, that assume anything one could mention, so I suppose there are some which assume the lapse rate. As you correctly notice, though, lapse rates depend on conditions, and those conditions vary over the globe. A serious climate model couldn't assume the lapse rate. And, in truth, they don't. More in a moment, but something to look back at is my description of the 16 climate models

Let's start with the lapse rate itself. What it is, is the change in temperature with elevation. Through the troposphere, the lapse rate is a negative number (cooling with elevation). In the stratosphere, it turns to zero and then positive (warming with elevation). In the mesosphere, we go back to cooling with elevation. This is a strictly observational issue. You can find temperature profiles, say from the Standard Atmosphere (a specific thing; Project: take a web look for it and see what they look like; they've changed through time, by the way). And then find the temperature difference between two levels, and divide by the elevation difference. That'll give you the average lapse rate. You can also find, radiosonde soundings of temperature. (I'd start my search for this project at the National Climatic Data Center.) This will let you see how the lapse rates vary day to day at a location, and between locations.

On the theoretical side, we go back to Conservation of Energy. We start with a completely dry (meaning no water vapor) blob of air, in an insulating bag that prevents it from radiating, conducting, or convecting energy to or from the surroundings. Then we lift it through the atmosphere. As we do so we'll find that its temperature drops. This happens because our blob does work in expanding. The energy for that work comes from its own thermal energy store. We can compute exactly how much the air would cool under this circumstance. It is about 10 K per km near the surface of the earth. This is what we are referring to in talking about the Dry Adiabatic Lapse Rate. The 'adiabatic' refers to our insulating bag around the air blob.

The polar regions, particularly the Antarctic plateau, are not bad approximations to that situation. But most of the atmosphere has fairly significant amounts of water vapor. We start, now, with a slightly different bag. It still prevents heat to be added or lost to the bag from outside. But now there's a second energy source inside the bag. Water vapor can condense, and when it does, it will release energy. We take the approximation that all the heat energy goes to the gases in the bag, and that the newly-formed liquid water is immediately moved outside the bag.

Now when we lift the bag, things go a bit differently. Let's start with air at 70% relative humidity, a typical global mean value. As we lift the air, it first acts 'dry', so cools at the about 10 K per km rate. But after a while, we have cooled to the point of being at 100% relative humidity. When we start to lift any further, water starts condensing and releasing heat. The condensation only happens if we're still cooling, so it can't reverse that tendency. But it can greatly slow the rate of cooling. This gives us a Moist Lapse Rate. Note that I dropped 'adiabatic' from the description. Since material is leaving the bag, it isn't an adiabatic process any more. It is pseudoadiabatic (a term you'll see) -- almost adiabatic, as the loss of mass isn't large. But not entirely adiabatic.

As a typical ballpark value, we take 6.5 K per km as the moist lapse rate. But this obviously will depend a lot on how much water was in the bag to begin with, and the temperature. If we start with a very warm, saturated, bag of air, then the lapse rate can be even lower than the 6.5 K per km. If we start, though, with a cold blob of air, even if it is saturated, we are still close to 10 K per km lapse rate. The thing is as we get colder, there's less water vapor present, which gives less condensation, then less heating. Consequently even in the tropics, the lapse rate heads towards the dry adiabatic value as you get high above the surface.

Whether moist or dry, the lapse rate computed this way is an idealization. In the real atmosphere, radiation does move energy around, and blobs of air do mix with each other (even when rising). Still, it's derived from a strong scientific principle (conservation of energy), and it turns out to give us good ideas (in reasonable accord with observation) about what the atmosphere should look like in the vertical.


For the modelling, let's think back to the 16 models. First, many of them are never used, so we'll ignore the longitude-primarily models. That leaves us with the 0 dimensional model I've already given an example of, and there's not even the opportunity to impose or even make use of a lapse rate in that. The 4 dimensional model definitely doesn't assume a lapse rate -- doing so would force violations of conservation of energy. Radiative-convective models can't force the lapse rate for the same reason. For a discussion of such models, to which I'll be returning in another post about water vapor's greenhouse contribution, see Ramanathan and Coakley, 1978. As of that era, one did specify a critical lapse rate. This isn't the lapse rate that the model had to have, rather, it was a limit. If the limit were violated, something has to happen. That something is to conserve energy by mixing the layers that violated the limit. And Energy Balance Models, as I expected, don't even mention lapse rate. See North, 1975 for a discussion of energy balance models.

Either the models are too simple to know about lapse rates (0 dimensional, Energy Balance), or they compute the lapse rate (Radiative Convective, 4 dimensional). Either way, the lapse rate is not assumed before hand. It's an interesting after the fact diagnostic for the Radiative Convective or 4d models, or impossible to speak to.


One thing to do is find some better sources for you to read. I taught an introductory (freshman level) physical geology class with Lutgens and Tarbuck, and liked the text there. They have a text at that level for meteorology, but I haven't read it myself. It should be good, though. John M. Wallace and Peter V. Hobbs, Atmospheric Science: An Introductory Survey is an excellent book. In half the chapters, comfort with multivariate calculus is assumed. But the other half are descriptive/physical rather than quantitative/mathematical so should be approachable already. A second edition is now out, I used the first. Does anyone have suggestions for a good freshman level introduction to meteorology/climate?

Elegant Gas Compression Technology

Both a matter of some fundamental science and a particularly elegant technology. The technology is the Australian Aboriginal firestarter. What makes it elegant is that it is simple, easy to use, makes a good demonstration of a physical principle, and is extremely obvious -- in retrospect. The way it works is that you take an airtight tube with a snugly fitting piston in it. Put the tube, with the piston at the top end, over some dry tinder. Then slam down the piston, while holding the tube hard against the ground.

The principle is that as you compress a gas, while keeping it insulated from the surroundings, it heats up. Compress it enough, and you get to the ignition point of your tinder. As far as I've seen, after doing a little looking when I saw the description of the Australian Aboriginal firestarter, they are the first people by some thousands of years to make use of the technology. Brilliant! elegant!

The next technology to make use of the principle, as far as I know, is the Diesel engine, late 1800s.

For our climate concerns, we don't deal with such extreme or rapid compressions. But the principle holds: If we take a blob of air, insulated from the surroundings and increase the pressure on it (because we're pulling it from lower pressure part of the atmosphere to higher), it warms up. The converse is also true -- if we decrease the pressure (by moving to a higher (lower pressure) part of the atmoshere), then the gas cools off.

This is another part of dealing with potential temperatures -- we'll get rigorous about by just how much the air warms or cools. But that's another note.


If you have other examples of technologies or cultures using air compression heating between the firestarter and Diesel engine, please do mention them here or by email to me at plutarchspam at aim dot net. (It's a valid address, the 'spam' in it is part of the name. You could also use the bobg at radix dot net that is in my profile, but I get so much unfiltered spam there that chances are good I'll miss your note.)

The Atmosphere in the Vertical

I'd started with a note on potential temperature, the vertical structure of the atmosphere, gas laws, and a few other things. After much typing, I realized that I'd done much typing and hence you'd have a lot of reading to do. So I'll divide things up a bit and hope that the result is more manageable and understandable.

In the introducing the atmosphere note, I mentioned that in the lower atmosphere (troposphere) and upper atmosphere (mesosphere), temperatures decreased with height. This is true, but if you've heard about 'hot air rises', I hope it gave you some discomfort. We'll ultimately solve the problem by understanding potential temperature. But first, let's think some more about the vertical.

The troposphere is where we live, and where almost all of what we think of as 'weather' happens. It is between something like 10 and 20 km thick. Above this, is the stratosphere. It is where the ozone layer is, and temperatures are constant or increasing with height. It runs from the top of the troposphere, that 10-20 km elevation, to the base of the mesosphere (something like 50 km). The mesosphere runs from the something like 50 km to something like 80 km. Finally, above the 80 km, is the thermosphere, by which time the atmosphere is so thin that different processes take over.

The 'something like' values there should make you uncomfortable. Even more so, that 10-20 km thickness of the troposphere. Why should it vary by so much? Partly, what is happening is that in hotter parts of the atmosphere (the tropics), pressure drops more slowly with height than in cold parts (the polar regions). If we use a thickness that relates to mass (pressure) rather than temperature, we find that the troposphere is between 700 (polar) and 900 (tropics) mb thick. Still thicker in the tropics, but no longer a factor of 2, so this is progress.

Average atmospheric pressure at sea level is 1013.25 mb. This is a pressure of a column of air over you standing at that point. So the tropospheric thicknesses say that the troposphere is 70-90% of the mass of the atmosphere over a given point. If we used elevations instead, then the troposphere is only 10-20% of your vertical -- the remainder being the very thin gases of the stratosphere, and even thinner gases in the mesosphere.

The stratosphere, then, is from the 100-300 mb level (above 70-90% of the mass of the atmosphere) to about the 1 mb level. In other words, it and the troposphere jointly account for 99.9% of the mass of the atmosphere. The mesosphere runs from about 1 mb to about 0.01 mb. Add this in, and we've got 99.999 % of the atmosphere.

In terms of building models of the atmosphere, and for observing it, the pressure levels make many things easier. If we took meters (or feet) in the vertical, and spaced evenly, then we waste 80-90% of the levels on parts of the atmosphere that don't have the weather we're interested in. If we take pressure (the mb -- millibars), then we only 'waste' 10-30%, a big improvement. For observing, it is easier to build a pressure gauge than to sense the elevation above the ground. (Often, even elsewhere, pressure is used instead of elevation, such as for aviation). Radiosondes, then, report in terms of the pressure they're experiencing at the time of an observation.

If we were being strictly correct, I should be replacing references to millibars (mb) with hectopascals (hPa). The latter (well, Pascal) is the official scientific unit for pressure. 1 mb = 1 hPa, though, so we can just swap them. In other units, 1 mm mercury is a bit more than 1 mb, there being 760 mm Hg in a standard atmosphere to the 1013.25 mb. Inches of mercury ... well, my great grandmother's barometer uses that (only) but it's not even close to what's used in science and hasn't been for a long time. Then there are pounds per square inch (14.7 being standard atmosphere), and a host of others. I'm lazy about the mb versus hPa label because a) it's what I learned first b) it's in more common use in the US than hPa and c) most importantly, the conversion factor is 1. Still, if you're a younger reader, get used to the hPa and practice translating to it in your mind rather than perpetuate the erroneous mb.

Introducing the Atmosphere

A while back I introduced the ocean. a bit. Time for the atmosphere, which turns out to be much simpler than the ocean, or maybe that just is a matter of being more subtle.

In the ocean, we have names for areas bounded (usually) by land. This doesn't work for the atmosphere, since it covers the whole earth. It does turn out, however, that the circulation itself (as for the Southern Ocean) helps divide the atmosphere. For a first approximation, little air crosses the equator. (Some does, to be sure.) So we can divide the atmosphere to northern and southern hemisphere circulations. Even though the two hemispheres are greatly different in their amounts and distribution of land and ocean, it turns out that their major circulation is quite similar.

In both hemispheres, air rises from the equator, move towards higher latitude at elevation (5-15 km, 3-9 miles up) and then sinks to the surface around 30 degrees latitude (north or south). The surface circulation is from that sinking latitude back towards the equator. This is the Hadley cell. There's a similar cell between about 60 degrees latitude and the pole, called the Ferrel cell. In between the two is not so much a cell as a storm zone.

A couple of questions I haven't seen firmly answered (I do have leads on a couple of papers, but haven't read them yet) are: Why are the circulations so similar between the hemispheres? Why are there 3 zones of circulation instead of 2, 4, 5, 75, ...?

In the vertical ... how high is up? :-) There's actually a way of approaching this by way of a different question:

What is the atmosphere made of? If we sample air all around the world, and through all levels of the atmosphere, we discover a few striking things. First, only 3 gases account for almost the entire atmosphere -- Nitrogen (N2), Oxygen (O2), and Argon (Ar). They are, in order, 78%, 21%, and 1% of the atmosphere in terms of counting molecules, and 76%, 23%, and 1% by mass. Note that these figures have been rounded, but also note that they do add to 100%. A further feature is that this proportion is constant throughout the atmosphere -- until you get to about 100 km (about 60 miles) up. Above that point heavier gases separate from lighter ones.

So that's our answer to 'how high is up?' -- about 100 km.

But back to the matter of what the atmosphere is made of. We've all heard about water (H2O), Carbon dioxide (CO2), and Ozone (O3) and how important they are. They are indeed important. But, added up through the atmosphere, they account for only about 0.24% of the molecules. Considering all gases, only about 1 in 400 molecules is not one of the big 3 of Nitrogen, Oxygen, or Argon.

Yet it is these other, rare, molecules which account for almost all the interesting processes in the atmosphere! This includes rain, hurricanes, the fact that the earth is not frozen, protecting surface life from solar ultraviolet, constructing the stratosphere, ...

Ok, the stratosphere brings us back to the vertical structure of the atmosphere. Starting from the surface, we have the troposphere. Here's where almost all the weather happens, and temperatures generally decrease with height. (So it's usually cooler in the mountains than the flat land around them.) In the stratosphere, temperatures are constant or increasing with height. Above this is the mesosphere, where temperatures go back to decreasing with height. The top of the mesosphere is that 100 km (or so) boundary. Above this, things get very different, and usually are studied by people who aren't meteorologists or climatologists. (Aeronomers, chemists, physicists instead.)

The stratosphere being a warm place, compared to the air above or below it, is because of the ozone absorbing ultraviolet. Without the ozone (and the chemistry that maintains that molecule), there's nothing to be absorbing energy to warm the layer. At most, ozone is about 12 parts per million of the molecules in that layer (0.0012 % !)

Since my first cut is always to take a look at the largest scale things, the most common, and such, we're already done. 3 molecules make up almost all of the atmosphere (N2, O2, Ar), there are 3 circulation zones between equator and pole in each hemisphere, there are 3 layers in the vertical, and the atmosphere goes up about 100 km.

To understand most of what's interesting, we have to get much more subtle. Always a fun and interesting thing to find that understanding some very small part of a big system will let us understand a lot about that large system.