Climatic Distortions Due to Diminutive Denominators

**Guest essay by Thomas P. Sheahen**

We all learned in elementary school that “you can’t divide by zero.” But what happens when you divide by a number very close to zero, a small fraction? The quotient shoots way up to a very large value.

Pick any number. If you divide 27 by 1, you get 27. If you divide 27 by 1/10, you get 270. Divide 27 by 1/1000 and you get 27,000. And so on. Any such division exercise blows up to a huge result as the denominator gets closer and closer to zero.

There are several indices being cited these days that get people’s attention, because of the big numbers displayed. But the reality is that those big numbers come entirely from having very small denominators when calculating a ratio. Three prominent examples of this mathematical artifact are: the feedback effect in global warming models; the “Global Warming Potential”; and the “Happy Planet Index.” Each of these is afflicted by the enormous distortion that results when a denominator is small.

**A. The “Happy Planet Index” is the easiest to explain: ** It is used to compare different countries, and is formed by the combination of

(a x b x c) / d.

In this equation,

a = well-being “How satisfied the residents of each country feel with life overall” (based on a Gallup poll)

b = Life expectancy

c = Inequalities of outcomes. (“the inequalities between people within a country in terms of how long they live, and how happy they feel, based on the distribution in each country’s life expectancy and well-being data”)

d = Ecological Footprint (“the average impact that each resident of a country places on the environment, based on data prepared by the Global Footprint Network.”)

How do the assorted countries come out? Using this index, Costa Rica with a score of 44.7 is number 1; Mexico with a score of 40.7 is number 2; Bangladesh with a score of 38.4 is number 8; Venezuela with a score of 33.6 is 29; and the USA with a score of 20.7 is number 108 — out of 140 countries considered.

Beyond such obvious questions as “Why are so many people from Mexico coming to the USA while almost none are going the other way?”, it is instructive to look at the role of the denominator (factor d) in arriving at those numerical index values.

Any country with a very low level of economic activity will have a low value of “Ecological Footprint.” Uninhabited jungle or barren desert score very low in that category. With a very small number for factor (d), it doesn’t make a whole lot of difference what the numbers for (a), (b) and (c) are — the tiny denominator guarantees that the quotient will be large. Hence the large index reported for some truly squalid places.

The underlying reason that the “Happy Planet Index” is so misleading is because it includes division by a number that for some countries gets pretty close to zero.

**B. The second example of this effect is the parameter “Global Warming Potential,” which is used to compare the relative strength of assorted greenhouse gases. **

The misuse of numbers here has led to all sorts of dreadful predictions about the need to do away with very minor trace gases like methane (CH_{4}), N_{2}O and others.

“Global Warming Potential” was first introduced in IPCC second assessment report, and later formalized by the IPCC in its *Fourth Assessment Report *of 2007 (AR-4). It is described in section 2.10.2 of the text by *Working Group 1*. To grasp what it means, it is first necessary to understand how molecules absorb and re-emit radiation.

Every gas absorbs radiation in certain spectral bands, and the more of a gas is present, the more it absorbs. Nitrogen (N_{2}), 77% of the atmosphere, absorbs in the near-UV part of the spectrum, but not in the visible or infrared range. Water vapor (H_{2}O) is a sufficiently strong absorber in the infrared that it causes the *greenhouse effect* and warms the Earth by over 30 C, making our planet much more habitable. In places where little water vapor is present, there is less absorption, less greenhouse effect, and it soon gets cold (think of nighttime in the desert).

Once a molecule absorbs a photon, it gains energy and goes into an excited state; until that energy is lost (via re-radiation or collisions), that molecule won’t absorb another photon. A consequence of this is that the total absorption by any gas gradually *saturates* as the amount of that gas increases. A tiny amount of a gas absorbs very effectively, but if the amount is doubled, the total absorption will be less than twice as much as at first; and similarly if doubled again and again. We say the absorption has *logarithmic *dependence on the concentration of the particular gas. The curve of how total absorption falls off varies according to the *exponential* function, exp (-X/A), where X is the amount of a gas present [typically expressed in parts per million, *ppm*], and A is a constant related to the physics of the molecule. Each gas will have a different value, denoted B, C, D, etc. Getting these numbers within __+__ 15% is considered pretty good.

There is so much water vapor in the atmosphere (variable, above 10,000 ppm, or 1% in concentration) that its absorption is completely saturated, so there’s not much to discuss. By contrast, the gas CO_{2} is a steady value of about 400 ppm, and its absorption is about 98% saturated. That coincides with the coefficient A being roughly equivalent to 100 ppm.

This excursion into the physics of absorption pays off when we look at the mathematics that goes into calculating the “Global Warming Potential” (GWP) of a trace gas. GWP is defined in terms of the *ratio* of the slopes of the absorption curves for two gases: specifically, the slope for the gas of interest divided by the slope for carbon dioxide. The slope of any curve is the first derivative of that curve. Economists speak of the “marginal” change in a function. For a change of 1 ppm in the concentration, what is the change in the *radiative efficiency*?

At this point, it is crucial to observe that every other gas is compared to CO_{2} to determine its GWP value. In other words, whatever GWP value is determined for CO_{2}, that value is re-set equal to 1, so that the calculation of GWP for a gas produces a number *compared to CO _{2}*. The slope of the absorption curve for CO

_{2}becomes

*the denominator*of the calculation to find the GWP of every other gas.

Now let’s calculate that denominator: When the absorption function is exp (-X/A), it is a mathematical fact that the first derivative = [-1/A][exp(-X/A)]. In the case of CO_{2} concentration being 400 ppm, when A = 100 ppm, that slope is [-1/100][exp (-4)] = – 0.000183. That is one mighty flat curve, with an extremely gentle slope that is slightly negative.

Next, examine the gas that’s to be compared with CO_{2}, and calculate the numerator:

It bears mentioning that the calculation of GWP also contains a factor related to the atmospheric lifetime of each gas; that is discussed in the appendix. Here we’ll concentrate on the change in absorption due to a small change in concentration. The slope of the absorption curve will be comparatively steep, because that molecule is at low concentration, able to catch all the photons that come its way.

To be numerically specific, consider methane (CH_{4}), with an atmospheric concentration of about Y = 1.7 ppm; or N_{2}O, at concentration Z = 0.3 ppm. Perhaps their numerical coefficients are B ~ 50 or C ~ 150; they won’t be terribly far from the value of A for CO_{2}. Taking the first derivative gives [-1/B][exp{-Y/B)]. Look at this closely: with Y or Z so close to zero, the exponential factor will be approximately 1, so the derivative is just 1/B (or 1/C, etc.). Maybe that number is 1/50 or 1/150 – but it won’t be as small as 0.000183, the CO_{2} slope that appears in the denominator.

In fact, the denominator (the slope of the CO_{2} curve as it nears saturation) is guaranteed to be a factor of about [exp (-4)] smaller than the numerator — for the very simple reason that there is ~ 400 times as much CO_{2} present, and its job of absorbing photons is nearly all done.

When a normal-sized numerator is divided by a tiny denominator, the quotient blows up. The GWP for assorted gases come out to very large numbers, like 25 for CH_{4 }and 300 for N_{2}O. The atmospheric-lifetime factor swings some of these numbers around still further: some of the hydrofluorocarbons (trade name = *Freon*) have gigantic GWPs: HFC-134a, used in most auto air conditioners, winds up with GWP above 1,300. The IPCC suggests an error bracket of __+__ 35% on these estimates. However, the reality is that every one of the GWPs calculated is enormously inflated due to division by the extremely small denominator associated with the slope of the CO_{2} absorption curve.

The calculation of GWP is not so much a warning about other gases, but rather an indictment of CO_{2}, which (at 400 ppm) would not change its absorption perceptibly if CO_{2 }concentration increased or decreased by 1 ppm.

**C. The third example comes from the estimates of the “feedback effect” in computational models of global warming. **

The term “Climate Sensitivity” expresses how much the temperature will rise if the greenhouse gas CO_{2} doubles in concentration. A relevant parameter in the calculation is “radiative forcing,” which can be treated either *with* or *without* feedback effects associated with water vapor in the atmosphere. Setting aside a lot of details, the “no feedback” case involves a factor l that characterizes the strength of the warming effect of CO_{2}. But *with* feedback, that factor changes to [ l /(1 – bl)], where b is the sum of assorted feedback terms, such as reflection of radiation from clouds and other physical mechanisms; each of those is assigned a numerical quantity. The value of l tends to be around 0.3. The collected sum of the feedback terms is widely variable and hotly debated, but in the computational models used by the IPCC in prior years, the value of b tended to be about b = 2.8.

Notice that as là 1/3 and b à 3, the denominator à zero. For the particular case of l = 0.3 and b = 2.8, the denominator is 0.16 and the “feedback factor” becomes 6.25. It was that small denominator and consequent exaggerated feedback factor that increased the estimate of “Climate Sensitivity” from under 1 ^{o}C in the no-feedback case to alarmingly large estimates of temperature change. Some newspapers spoke of “11 ^{o}F increases in global temperatures.” Nobody paid attention to the numerical details.

In more recent years, the study of various positive and negative contributions to feedback improved, and the value of the sum b dropped to about 1, reducing the feedback factor to about 1.4. The value of the “Climate Sensitivity” estimated 30 years ago in the “Charney Report” was 3 ^{o}C __+__ 1.5 ^{o}C. Today, the IPCC gingerly speaks of projected Climate Sensitivity being “near the lower end of the range.” That sobering revision can be traced to the change from a tiny denominator to a normal denominator.

The Take-Home lesson in all of this is to beware of tiny denominators.

*Any numerical factor that is cranked out is increasingly meaningless as the denominator shrinks.*

When some parameter (such as “Climate Sensitivity” or “Global Warming Potential” or “Happy Planet Index”) has built into it a small denominator, don’t believe it. Such parameters have no meaning or purpose other than generating alarm and headlines.

**APPENDIX**

Duration-Time Factor in “Global Warming Potential”

The “Global Warming Potential” (GWP) is not merely the ratio of the slopes of two similar curves at different selected points. There is also a factor that strives to account for the length of time a particular gas stays in the atmosphere. The idea is to integrate over a long interval of time and thus capture the *total* energy absorbed by a gas during its residence time in the atmosphere.

For methane, there is general agreement that the mean lifetime is 12 years. Unfortunately, for carbon dioxide there is no agreement at all; estimates range across 5 years to 200 years. It’s debatable whether to count the time for a CO_{2} molecule to *enter* a tree, or the lifetime of that tree. The IPCC does not settle this issue by choosing a single number for the CO_{2} lifetime; rather, in a footnote to table 2-14 of AR-4, it presents a formula for the *response function* to a pulse of CO_{2}. That extremely obscure formula has 7 free parameters to fit data, as it tries to accommodate 3 different plausible lifetimes. This makes it nearly impossible for outsiders to discern what numerical value goes into the denominator of every GWP calculation. A plausible guess for the average lifetime of CO_{2} is 55 years, but 100 years is not *im*plausible.

When it is too difficult to carry out an actual integral over data that is very uncertain and widely variable, the next best thing is to select one number for the lifetime and multiply by it. We thus obtain the simple form

GWP = ____Slope (gas) x lifetime (gas) __

Slope (CO_{2}) x lifetime (CO_{2})

The guesswork involved in that will probably afflict both numerator and denominator is roughly the same way.

If we take 55 years as the lifetime of CO_{2}, 12 years as the lifetime of CH4; and use 0.000183 as the slope in the denominator, and 1/B = 1/50 as the slope in the numerator, the numbers work out to GWP = 23.85 – a number close to 22, 24 and 25, each of which has sometimes been stated as the GWP of methane.

The same procedure can be used for N_{2}O, for the *Freon*s, for any other gas. The guesswork about lifetimes definitely enlarges the error brackets, but the really enormous factor that drives the calculated GWP numbers up is the extremely flat slope of the CO_{2} absorption curve (0.000183 in the example above). That is entirely due to the fact that the absorption by CO_{2} is very near saturation.

Superforest,Climate Change

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