Science Briefs

A Sense of Where You Are in the Stratosphere

Where are you right now? You might answer, "New York City" or "Topeka" or even "Oslo" or "Nairobi." Scientists like to quantify things, and so they describe spatial locations in terms of coordinates: one number apiece for where you are in the west-east, north-south, and the up-down directions. Most of us are familiar with longitude and latitude as numerical measures of west-east and north-south locations, respectively. Here I discuss different types of vertical coordinates used for the atmosphere.

There is no one perfect way to measure up-down in the atmosphere. You can do it using a geometric distance, such as miles or kilometers, but because the atmosphere is concentrated near Earth's surface, a small change in vertical distance equals a huge change in atmospheric temperature, pressure, and so forth. So although altitude is an easy idea for humans to grasp, it's not a natural vertical yardstick for the atmosphere.

An alternative is to use atmospheric pressure as a vertical coordinate, ranging from sea-level pressure [roughly 30 inches of mercury = 1000 millibars or hecto-Pascals (hPa)] at the bottom to zero at the top of the atmosphere. An even better, but less familiar, vertical coordinate is "potential temperature." a combination of temperature and pressure, abbreviated as the Greek letter θ (theta). Its value gets larger the higher you go in the atmosphere, sort of like altitude. Winds in the atmosphere blow along surfaces of constant potential temperature most of the time, and so it's an extremely good choice for a vertical coordinate.

There's just one problem — some researchers like to study the atmosphere in terms of geometric height, others in terms of pressure, and still others in terms of potential temperature. This can lead to confusion when an atmospheric scientist is reading a graph in someone else's research paper, because the scale on the vertical side of the graph may be in a unit he or she's not familiar with! It's kind of like converting miles into kilometers, only even more puzzling. This leads to Tower-of-Babel communication gaps between different atmospheric scientists and different research groups, an unfortunate and all-too-common situation.

This confusion struck Nobel Prize winner Paul Crutzen and his collaborator P.C. Freie as a problem worth addressing. In the Sep. 23, 1997 issue of the American Geophysical Union publication Eos, Crutzen and Freie published a rule-of-thumb for converting between altitude and potential temperature in the stratosphere. Using their scheme, scientists can quickly convert from potential temperature to altitude in kilometers, just by dividing by 25. However, the authors didn't describe where their rule came from, why it worked, or if it worked outside of the low-to-mid stratosphere (from 200 hPa, or 12 km in altitude, to 10 hPa, or 30 km).

That's where I came in. Using some concepts straight out of college-level calculus, I derived the rule-of-thumb Crutzen and Freie published but didn't elaborate upon. In the course of obtaining their formula, I found that their formula depends on a clever arithmetic trick, without which their rule-of-thumb would fail outside of a very small region of the atmosphere.

Figure 1: See caption
Fig. 1: Latitude-pressure cross sections of percent error in the estimation of altitude z from potential temperature θ averaged along latitude lines for the region 80°S-80°N, 500-0.0001 hPa (6-105 km) in a typical April, using the following conversion methods from left to right: a) my nonlinear formula θ = 350 e0.045(z-13); b) the linearized Taylor series expansion 0.0635 θ - 9.22 = z that Crutzen and Freie started with; and c) Crutzen and Freie's rule-of-thumb θ / 25 = z. (All z are in km.)

Best of all, though, I discovered a slightly more complicated, but much more general formula for converting between altitude and potential temperature. Figure 1 shows the percent error from my formula (left), the approximation to it that Crutzen and Freie started with (middle), and the rule-of-thumb Crutzen and Freie obtained using their clever little trick (right). The bigger the percent error, the worse the approximation. It's pretty obvious from Figure 1 that my formula is the best of the bunch in a huge swath of the atmosphere, from about jet-stream level (300 hPa, or about 10 km in altitude) all the way up into the thermosphere (0.0001 hPa, or about 105 km in altitude). Armed with nothing more than a calculus book, I found a much-improved way to help scientists know where things are in the stratosphere, and above!

Is this little discovery going to earn me a Nobel Prize? Hardly. But it is the kind of day-to-day work that makes up a large portion of a scientist's professional life. It's my philosophy that you can't do the really important work if you're confused about the fundamentals. This is one case where knowing the fundamentals of mathematics and meteorology helped me explain what a Nobel winner was thinking — but didn't tell anyone — about his own research.


Crutzen, P.J., and P.C. Freie 1997. Converting potential temperature to altitude in the stratosphere. Eos 78, 410.

Knox, J.A. 1997. On converting potential temperature to altitude in the middle atmosphere. Eos, submitted.