In which my shortcomings as a research scientist led to a nice e-mail from the only Nobel Prize winner in my discipline’s history
The Who and the Why
Atmospheric chemist Paul Crutzen died last Thursday at the age of 87. His passing was international news, because he shared the Nobel Prize in Chemistry in 1995 with Drs. Sherwood Rowland and Mario Molina for their pioneering work that explained how stratospheric ozone loss could occur due to human activity. This work paved the way for an understanding of the ozone “hole” discovered over Antarctica in 1985 — a landmark moment in scientific history and our world’s understanding of the fragility of nature in the face of industrialization.
I’m not a Nobel Prize winner, and don’t intend to be one. In fact, I am a member of a entire area of science — the geosciences — that is snubbed, decade after decade, when it comes to Nobel Prizes, no matter the magnitude of the discovery. (Plate tectonics? Nope. Modern chaos theory? Nope. Numerical weather prediction? Nope. The oceanic conveyor belt? Nope. Anthropogenic climate change? Nope. Global climate modeling? Nope.) No Nobel in Physics has ever gone to a geoscientist, no matter how closely related to physics the discovery is. Crutzen was recognized by the more open-minded chemists. And I’m more of a teacher than a researcher, anyway; my place is with the students and always has been.
In the late 1990s, when I was a post-doctoral scientist reaching the conclusion in that last sentence, my fly-on-the-wall-of-history status intersected in a tiny way with Dr. Crutzen’s career. It’s a fun little story with quirks in it that couldn’t be made up, and it also illustrates how and why I do the small-fry research that I do. I may be a small fly and small-fry, but it ended up with a nice e-mail from the only Nobel Prize-winning meteorologist in world history. Read on.
There is a semi-apocryphal quote ascribed to Isaac Asimov that says,
“The most exciting phrase to hear in science, the one that heralds new discoveries, is not ‘Eureka!’ (I found it!) but ‘That’s funny …’”
Sometimes the remark is even more pungent than that.
There’s much truth in this, and this takes us back to Sweden some 50–60 years ago.
Paul Crutzen ended up in meteorology rather by accident. He was a bright Dutch kid during the tough WWII and post-WWII years in the Netherlands. But he was sick during the all-important examinations in high school, didn’t score high enough, and didn’t win a coveted place at the university. So he went to a lower-level technical school and became an engineer to make a living; his family was poor. Then he went into the military.
With zero computer experience, in 1958 Crutzen somehow landed a job as a programmer at the Department of Meteorology at Stockholm Högskola, where he worked on some of the very early numerical weather prediction models (which, again, didn’t garner a Nobel Prize for their creation). Sweden, owing to the leadership of returning native son Carl-Gustaf Rossby (another non-Nobelist), had become the leading nation in numerical weather prediction in the 1950s. Rossby also was one of the early pioneers of a field called atmospheric chemistry, in which the chemicals and the chemical reactions in the air were looked at and not just considered passive tracers.
While doing his job, Crutzen was able to pursue a long-held dream of an academic career that had been deferred by his test scores. While running computer models of the atmosphere, he earned first a Master’s and then a doctorate in meteorology. For his doctoral work, instead of the then-hot topic of acid rain — another atmospheric chemistry subject of importance — he picked a much more esoteric subject: the chemistry of stratospheric ozone. From there his career intersected with world history; you can read his own account of it in Crutzen’s Nobel Lecture. It’s great reading; here’s a taste (emphasis added):
By comparing these with the production of NOx by reaction R13, I realized immediately that we could be faced with a severe global environmental problem. Although the paper in which I proposed the important catalytic role of NOx on ozone destruction had already been published in April, 1970, clearly the participants in the study conference had not taken any note of it, since they concluded “The direct role of CO, CO2, NO, NO2, SO2, and hydrocarbons in altering the heat budget is small. It is also unlikely that their involvement in ozone photochemistry is as significant as water vapour”. I was quite upset by that statement. Somewhere in the margin of this text I wrote “Idiots”.
The Unproductive Post-Doc
My own story intersects with Crutzen’s at the NASA/Goddard Institute for Space Studies on Broadway and 112th Street in Manhattan in 1997. I was an unproductive post-doctoral research fellow at the time. I was not making progress on the second research project I was assigned; the first one — which I’d agreed and turned down good offers to come to New York City to do — was killed off on day one by my advisor because “we turned up the Rayleigh drag in the model and got rid of” an actual atmospheric phenomenon in the simulations.
After a year at a world research center for climate modeling, I’d determined that I was not cut out for that life. It was time to apply for jobs at small universities and teach, as my own mentors had done. Publishing my own work and getting out of the city were the priorities. I still feel kind of bad about that, but I needed a job to support my wife and infant son. And I wasn’t ever going to be like the other post-docs at NASA/GISS, three of whom have gone on to become some of the most-cited researchers in the world in all of climate science (two with h-indices over 100 before age 55, for those who value this kind of bean-counting).
What I did do, though, was read widely in my field and think about science. My idea of science, though, was less lofty than my colleagues’ work. I came at scientific research through the other end of the telescope, from the students’ perspective. I’d seen my professors and advisors grapple with topics of great complexity, but not be able to explain little niggling details. The cutting-edge people say, “Move on, that detail’s not important.”
But students don’t. They get hung up and say, “But why can you assume that? What does that mean? Stop.” I’m with the students on this one; I can clearly remember thinking in graduate school that I’d like to be the one to clear the thickets and make it all understandable instead of rushing to the front lines of the battle and stepping over the “unimportant” details.
That’s how a teacher thinks, not a big-time researcher. You don’t get big research grants for cleaning up the little details. However, my reading of the history of science tells me that in pursuing those little niggling, unfundable details you can find some nuggets of various sizes. And nobody else is doing it, so why not me?
Taylor, My Best Friend From Calculus Class
Meet Taylor. I met him in calculus class in college. He’s kind of misunderstood. Oddly, those who think they know him best, only see one side of him, the dreamy side. Those from the other side of the tracks know a different Taylor, though, who is practical and down-to-earth.
I’m referring to the subject of “Taylor series.” To mathematicians, Taylor series are about error estimates, theorems, and proofs. This is how this subject is taught to undergraduates majoring in mathematics and the sciences.
Taylor series as I have come to know them are based on all that, but they are practically useful in a powerful way in science. The elegance of the higher mathematics is undeniable, but the utility and versatility of Taylor series is lost in the teaching of it.
It’s as if someone came across a Swiss army knife and deemed it a work of art, a sculpture, and put it on a pedestal and put a spotlight on it and gave deep lectures about its beauty. All well and good, but could we please use the damn thing? Show us what all it can do!
Your Cherrypicking Textbooks
Let’s say you have a complicated equation that is nonlinear. That is, the variable in the formula is squared, or cubed, or there’s a trig function somewhere in it (sine curves aren’t straight lines), or there’s exponential growth or decay, etc.
There are tons of examples of these that describe the world around us. But most people probably encounter only a few such examples in their education, the quadratic formula being probably the most commonly encountered. You might remember that it has the square of x in it. That’s not linear, because x is multiplied by itself. If you graph a quadratic function, it is not a straight line, it is a parabola. And that means it’s non-linear. The compound interest formula is another example, where the all-important variable of time is in an exponent. That doesn’t graph like a straight line, either. Talk to the financial planners and they’ll tell you that, over time, it curves and shoots almost straight up, like a rocket. That, too, is nonlinear.
Most math teachers won’t say this in class, but 1) most problems in life are nonlinear, 2) there are very few nonlinear equations describing these situations that are solvable with pencil and paper, so 3) most mathematics and physics textbooks and classes in high school college tend to focus on the small subset of “analytically tractable” (i.e., pencil-and-paper solvable) linear problems. To be uncharitable, this is a kind of “if we ignore it, maybe it’ll go away” way of thinking about nonlinearity.
But some problems are of the “go big or go home” variety. They don’t go away, and you have to tackle them at some degree of sophistication or else you have to quit your profession. My discipline of meteorology is perhaps uniquely challenged by this clash of complexity and practicality. Everyone wants a good weather forecast. But the equations that describe the atmosphere are highly nonlinear. If we turned them into pencil-and-paper-solvable linear equations, they wouldn’t describe the real atmosphere very well. And so we could reduce the problem to linearity, but then we wouldn’t be meteorologists anymore. We’d be solving a hypothetical, non-real-world problem instead of the pressing real-world problem we are tasked to solve. Therefore, the problem has to be tackled on its own terms, or at least met halfway.
This means that meteorology, along with other applied fields such as aerodynamics and engineering, is powerfully interested in simplifications that, where possible, can reduce the nonlinearity of a problem without losing much accuracy.
But how do you linearize a function or a problem systematically? Do you just toss out a term here or there? How do you know what to keep and what to save?
Step right in and announce yourself, Taylor.
Taylor’s Practical Side
If you’ve read this far, you probably have taken calculus in college. OK, forget anything you ever learned about Taylor series for a moment. Just look at the series itself, on the right-hand side of the equals sign, as an approximation to the elegant-but-nonlinear expression on the left-hand side. That’s it. Don’t try to prove it, or worry about the exact size of the error, or use it to demonstrate some other principle. Just think of it as
HAIRY FUNCTION= POLYNOMIAL OF INFINITE TERMS.
That’s what a Taylor series is. You can express any nonlinear function f(x) at a point a by this polynomial:
The first term in the series is usually a constant, sometimes zero. The next term has an x in it; often it’s a linear term. The third term has the square of x in it, and may end up being nonlinear. And so on.
Repeat after me: a gnarly nonlinear function can often be approximated as a linear function by using just the first few terms of the Taylor series for it. You have to do the math and obey certain restrictions, but this is often what it boils down to. For example:
But you don’t have to keep all the terms! To turn a nonlinear function into a linear one, you just throw away all the terms greater than the first power of x. So cos(x) is approximated as 1; sin(x) is approximated as x; and the exponential function is approximated as 1+x. And these are not half-bad approximations for their ranges of validity.
That is the Swiss army knife-ness of Taylor series: they can whittle down a nonlinear function into something linear and tractable, without ignoring it entirely. This meets the problem halfway, and allows you to gain mathematical/physical insight that you couldn’t do with the full nonlinear expression.
This is just sophomore-level calculus. Yet I have been surprised at how many mathematicians I’ve known who didn’t think about Taylor series from my perspective. And I have been amazed at how often in my career I have been able to explain established empirical rules-of-thumb from first principles using Taylor series approximations. And this isn’t necessarily because the details were so pedestrian that the science gods omitted them. Borrowing a simple-tool-not-seen-as-a-tool from one discipline and using it in another discipline, I have been able to derive new results that get some citations and even a compliment or two from better researchers than I am.
Even a Nobel Prize winner. Which brings us at last to the story of my small encounter with Paul Crutzen.
A Sense of Where You Are in the Stratosphere
In the September 23, 1997 issue of the American Geophysical Union peer-reviewed weekly newspaper Eos, there was a tiny quarter-page piece about a very basic question: how do you convert from one vertical coordinate to another in the atmosphere?
It turns out that there’s no one perfect vertical coordinate for the atmosphere. Altitude, or z, is often used, but air doesn’t travel on altitude lines, and altitude lines run smack into mountains. Pressure is arguably better than altitude, but still isn’t perfect. The best choice is something called potential temperature, in meteorology symbolized by the Greek letter theta (either Θ or more often θ). Air that doesn’t irreversibly gain or lose energy will stay at the same numerical value of potential temperature, in units of Kelvins (like any temperature). That’s a great “conservative” property in meteorology and other physical sciences, so it is common to use theta as a vertical coordinate. But then, because altitude is used as the vertical coordinate in other applications, there’s a need to translate back and forth from one to another.
The problem is that the formula for potential temperature is nonlinear, so it’s not obvious how you easily go from z to θ.
In this article, Crutzen and a co-author provide a simple conversion:
z (in km) = θ/25
and they noted that the errors were small for the region of 12–30 km, which is the lower and middle stratosphere, and smallest in the middle of that region.
I read Crutzen’s postage-stamp-sized article in Eos one day in 1997, while not doing my post-doctoral research. I pondered how they could get such a cute little linear relationship between two variables that are nonlinearly related. Just divide by 25? Really? Where does the 25 come from? The article didn’t say. Why does it work? The authors didn’t elaborate. Where does the rule-of-thumb work or not work, outside of the middle stratosphere? Unsure. This was my “that’s funny” pre-Eureka moment, in the Asimovian sense.
By asking these questions and whipping out my Taylor series army knife, I was able to answer these questions. Explaining their cute little rule-of-thumb took more space in Eos, as you can see below. At the core of the explanation, though, was a Taylor series approximation for theta in terms of altitude. Here’s my article:
I derived the nonlinear relationship between potential temperature and altitude (equation 3 in my little paper above); I linearized it using a Taylor series (equation 4), which leads directly to a linear relationship between z and θ that looks a lot like Crutzen’s (equation 5). And then I compare these three relationships in terms of percent error in parts a, b, and d of Figure 1 above. Here’s a figure of just those three panels:
Clearly, the nonlinear relationship θ = 350 e^[(0.045)(z-13)] is the best one, and works with high accuracy from near the bottom of the atmosphere to the top. But it’s not a memorable rule-of-thumb. The linearized Taylor series version of this nonlinear relationship, which is z = θ/27.75+3.00, is pretty accurate down low but is very inaccurate for the upper stratosphere and above. But that, too, is not very memorable. you get from the linearized Taylor series to Crutzen’s rule-of-thumb z = θ/25 by reducing the two terms on the right-hand side to one term, by arbitrarily lopping off the constant term and then compensating for that by changing the denominator to a nice round number, say 25. And that lopping-and-arbitrary-rounding has the totally unexpected and beneficial result of reducing the error throughout the stratosphere. It’s even more accurate than the nonlinear formula in the upper troposphere and lower stratosphere, fortuitously and fortunately. And that’s why the rule-of-thumb is so good… unreasonably good!
So, by using Taylor series analysis and applying it to a first-principles nonlinear formula I derived that relates potential temperature to altitude, I was able to explain a) where Crutzen’s formula came from, b) why it worked, c) where it worked and where it wasn’t useful, and d) how luckily effective it was for a region outside of the area of validity of the linearized Taylor series it formally stems from. Not bad for college calculus.
Crutzen’s Freie* Response
The publication of my short article in Eos, to my astonishment, elicited a response from Dr. Crutzen. I was at this point an assistant professor with a 4 course-per-semester teaching “load” at Valparaiso University in Indiana. One morning during my first semester there, this e-mail showed up in my queue:
Wow. “Great interest” from a Nobel Prize winner just three years before. “(H)appy with the analysis”?! Just a few months after turning away from big-time research and having the feeling that I’d failed, this e-mail made my day.
But there was the oddest of plot twists, at the end of the e-mail. Somehow, Crutzen’s co-author’s name was transmogrified somewhere in the publication process. I don’t know how “Peter Braesicke” became badly approximated as “P.C. Freie.” Neither did Peter; I happened to know him and ask him about it later. (He’s now a Professor for Theoretical Atmospheric Physics at Karlsruhe Institute of Technology in Germany.)
How odd! Jim Marti, the Eos editor, and I rectified the error a bit by changing the reference to their Eos article in the reference list of my article (look closely, above). But this is how you know my story’s real: that’s just too weird, and there is no reason for me to have invented such an irrelevant, bizarre detail.
A Great and Good Scientist
Paul Crutzen did not have to send me an e-mail and compliment my work. I was a fly on the wall compared to him, and to my colleagues and bosses at NASA/GISS I had fled to go into a career in teaching and some research. But I never forgot that he took the time to write. As my students will attest, when I teach about the stratospheric ozone hole, I always tell the story of how he wrote me a nice e-mail when I was a new assistant professor, and I always show his picture. Being a good person should count as much as winning a Nobel Prize, but the rewards system of academia doesn’t skew that way. Being a Nobel Prize winner and a good person deserves special praise.
And while I am just one data point, the quote from the Washington Post article on Crutzen’s passing made me think that I wasn’t the only recipient of his praise. Crutzen mentee and atmospheric chemist John Birks is quoted as saying:
“Paul Crutzen mentored at least hundreds and promoted the careers of thousands of scientists around the world. Besides being a brilliant scientist, he [is] one of the most caring and generous people I have ever known.”
Too many scientists think of their legacy in terms of their publications, their grants and citation statistics, the number of doctoral recipients they pumped out, and the number of awards and exclusive society memberships they racked up. Instead, aspire to a career and life that will inspire a person you traded one e-mail with almost a quarter-century ago to tell thousands of students, and more readers (if, dear reader, you’ve made it this far), that you were not just a great scientist — you were a good person, too.
Thank you, Dr. Crutzen, for everything.