One question that has intrigued me for a while – how much of the transmittance (and change in transmittance) from CO2 in the atmosphere is caused by weak lines, and how much is caused by the “far wings” of individual lines.

Take a look back at Part Nine. Here is the calculated change in transmittance from the surface to the tropopause for a doubling of CO2 from pre-industrial levels:

*Figure 1*

We can see that there is significant change outside of the central range of wavenumbers. (For reference, 667 cm^{-1} = 15 μm).

Here is another of the graphics – showing how an **individual** line absorbs across a range of wavenumbers:

*Figure 2*

There are many lines in the HITRAN database for CO2 (over 300,000 lines), many of them weak.

So back to my original question – is it the “wings” = line broadening, of the individual lines, or the weaker lines that have the biggest effect? And how do we quantify it?

### The Curve of Growth

*The Curve of Growth* is about the change in transmittance with an increase in path length (specifically the increase in number of absorbing molecules in the path).

The reason the problem is not so simple is because each line has a **line shape** that absorbs across a range of wavelengths.

We know that transmittance, t=e^{-τ} – this is the Beer-Lambert law. The spreading out of the line means that the reduction in transmittance as the path increases isn’t as strong as predicted by a simple application of the Beer-Lambert law.

Here is a calculation of this for a typical tropospheric condition (note 1) – for **one isolated** **absorption line** with its line center at 1000 cm^{-1}:

*Figure 3*

Each curve represents a **doubling** of the number of absorbing molecules in the path. So, if this was at constant density, each curve represents a doubling of the path length.

What does this graph show?

At the line center, an increase in the path length soon causes saturation. But out in the “wings” of the absorption line the reduction in transmittance changes much more slowly.

With some maths we can show that for very small paths (note 2), the absorptance is **linearly proportional** to the number of absorbers in the path.

And with some more maths we can show that for very large paths (note 2), the absorptance increases as the **square root** of the number of absorbers in the path.

Here is another way to view the “curve of growth”. I haven’t seen it shown this way before:

*Figure 4*

This is the same scenario, but this time the x-axis (bottom axis) has the effective path length, while **each curve** represents a different distance from the line center. The legend is showing the distance from the line center compared with the “line width”. So “10” = 1 cm^{-1} from the line center, while “0” is the line center.

Notice the difference at a mass path x absorption coefficient = 10^{0} (=1) where the line center has a transmittance of 0.05 while at 1 cm^{-1} distance from the line center the line has a transmittance of 0.97.

What happens when we look at the effect of the whole line?

Here is a curve of the **equivalent line width, W** as the path increases. This equivalent line width is just the total effect across enough bandwidth to take into account the effects of the far wings of the line (note 3):

*Figure 5*

The 2nd graph is the important one. This shows the power relationship between W and Su, as Su increases – where Su = absorption coefficient x mass of molecules in the path. *S is the absorption coefficient, and u is the mass*.

For a given line, S is a constant, and so an increase in Su means an increase in the number of molecules in the path.

So if we believe that W = (Su)^{x}, how do we determine x?

We can do it by taking the log of both sides: log(W)=x.log(Su), so by calculating the slope of this log relationship we can see how this value, x, changes.

The 2nd graph shows:

- at very small paths, the line strength is proportional to Su – because x = 1
- at very large paths, the line strength is proportional to √(Su) – because x = 0.5

This is simply backing up by numerical calculation the earlier claim: “*With some maths we can show..*”

So if you want to understand “the curve of growth” you need to understand that at very small optical thickness the “equivalent line” grows in linear proportion to the mass of molecules in the path, and with very large optical thickness the “equivalent line” grows in proportion to the square root.

And probably the easiest way to see it conceptually is to take another look at Figure 3.

### Line Wings in Practice

So back to my original question. Is it the weak lines or the wings of the strong lines that has the effect?

The first step is to calculate the effect of the line wings. I took the MATLAB program developed for Part Nine and made a few changes. The original program had simply applied the equation for line shape out to the edge of the region under consideration. In this version I added a new factor, c_{a}, which “chopped” the line shape. The factor *c _{a}* limits the line shape for each line to the line center, v

_{0}± (c

_{a}* line width).

The **line width** is a parameter in the HITRAN database and is a measure of the shape of the line, approximately the value where the strength has fallen to half the peak value.

The original program was run for two values of atmospheric CO2: 280ppm (pre-industrial); and 560ppm (doubling of CO2) – with 15 layers and a calculation every 0.01 cm^{-1} across the band of 500 cm^{-1} – 850 cm^{-1}. These are called the “Standard” results. *And for these, and all following simulations, we are only considering the main isotopologue of CO2, which accounts for over 98% of atmospheric CO2*.

Then the revised program was run for a number of values of c_{a}: 5000, 1000, 100 & 10.

Given that the typical line width is 0.05 – 0.1 cm^{-1} this means that each line is considered between two extremes: across 250-500 cm^{-1} down to only 0.5 – 1 cm^{-1}.

As expected, with c_{a} = 5000, the difference between that and the Standard is almost nil. And as c_{a} is reduced the differences increase:

*Figure 6*

By the time ca = 100, the differences are noticeable, and at ca = 10, the differences in some parts of the band are huge. The above result (figure 6) is dominated by the ca=10 result so here is ca=100 separately:

*Figure 7*

So this demonstrates that **even when** the line shape is “limited” to 100x the line width, there is still a noticeable effect in the transmittance calculation for the troposphere.

For completeness, here is the same comparison at figure 6, but for 560ppm:

*Figure 8*

Across this band, the standard case at 280ppm: t = 0.4980, and at 560ppm, t = 0.437.

Here is how the mean transmittance changes as we crop the line shape. Cropping the line shape means that we make the atmosphere **more transparent**.

*Figure 9*

We can see from figure 9 that the change in transmittance from 280 – 560 ppm is of a similar magnitude at each artificial “cropping” of the line shape.

### Weak Lines

So let’s take a look at the effect of weak lines. For this case I used the same MATLAB program developed for Part Nine but with a user-defined selection of lines. For example, the top 10% of lines by strength.

Here is the transmittance change @ 280ppm through the troposphere, for all lines (standard) less the transmittance of the strongest 10% of lines:

*Figure 10*

And here is the graph of all lines – the strongest 1%:

*Figure 11*

And here is the mean transmittance for both 280ppm and 560ppm as only the strongest lines are considered:

*Figure 12*

It’s clear that the weakest 90% of lines have virtually no effect on the transmittance of the atmosphere. There are a lot of very weak lines in the HITRAN database.

The calculated transmittance for 100% of the lines at 280ppm = 0.4974 and for the top 10% = 0.4994 – meaning that the top 10% of lines account for 99.6% of the transmittance.

At 560ppm the top 10% of lines account for 99.3%.

### Conclusion

Some of this analysis is of curiosity value only.

However, it is very useful to understand the “curve of growth” – and to realize how absorptance increases as the mass in the path increases.

And it’s at least interesting to see how the “far wings” of the individual lines have such an effect on the transmittance through the atmosphere. Even “cropping” the effect at 100x the line width has a significant effect on the atmospheric transmittance.

And for the question posed at the beginning, both the weak lines and the far wings of individual lines have an effect on the total atmospheric transmittance.

Many people have appreciated the massive absorption at the peak of the CO2 band (around 15 μm). But as we have seen in earlier parts of this series (and as shown in Figure 1), it is towards the “edges” of the band where the largest changes take place as CO2 concentration increases.

*Remember as well* that total transmittance is not really a complete picture of **radiative transfer** in the atmosphere. The atmosphere also emits radiation, and so the temperature profile of the atmosphere is just as important for seeing the whole picture.

*Other articles in the series:*

*Part One – a bit of a re-introduction to the subject.*

*Part Two – introducing a simple model, with molecules pH2O and pCO2 to demonstrate some basic effects in the atmosphere. This part – absorption only.*

*Part Three – the simple model extended to emission and absorption, showing what a difference an emitting atmosphere makes. Also very easy to see that the “IPCC logarithmic graph” is not at odds with the Beer-Lambert law.*

*Part Four – the effect of changing lapse rates (atmospheric temperature profile) and of overlapping the pH2O and pCO2 bands. Why surface radiation is not a mirror image of top of atmosphere radiation.*

*Part Five – a bit of a wrap up so far as well as an explanation of how the stratospheric temperature profile can affect “saturation”*

*Part Six – The Equations – the equations of radiative transfer including the plane parallel assumption and it’s nothing to do with blackbodies*

*Part Seven – changing the shape of the pCO2 band to see how it affects “saturation” – the wings of the band pick up the slack, in a manner of speaking*

*Part Eight – interesting actual absorption values of CO2 in the atmosphere from Grant Petty’s book*

*Part Nine – calculations of CO2 transmittance vs wavelength in the atmosphere using the 300,000 absorption lines from the HITRAN database*

*Part Ten – spectral measurements of radiation from the surface looking up, and from 20km up looking down, in a variety of locations, along with explanations of the characteristics*

*Part Eleven – Heating Rates – the heating and cooling effect of different “greenhouse” gases at different heights in the atmosphere*

*Part Twelve – The Curve of Growth – how absorptance increases as path length (or mass of molecules in the path) increases, and how much effect is from the “far wings” of the individual CO2 lines compared with the weaker CO2 lines*

*And Also -*

*Theory and Experiment – Atmospheric Radiation – real values of total flux and spectra compared with the theory.*

References

*The HITRAN 2008 molecular spectroscopic database*, by L.S. Rothman et al, *Journal of Quantitative Spectroscopy & Radiative Transfer* (2009)

### Notes

**Note 1** – As you can see in Part Nine, the line shape differs as the pressure reduces.

**Note 2** – Technically speaking, for “very small paths” we are really considering the case where optical thickness, τ <<1 (very much less than 1). And for “very large paths” we are considering the case where optical thickness, τ>>1 (very much greater than 1).

**Note 3** – For a *line strength* = W, if we want to calculate absorptance, a = 1 – t, where t= transmittance, across any given band, Δv, the calculation is very simple:

a = 1-t = W / Δv

on April 30, 2011 at 1:11 pm |DeWitt PayneSince the far wings do have an effect on the absorptivity, that raises the question of how good is the Voigt (or Lorentz) profile in the far wings? According to Petty (p264-5, Second Edition):

on April 30, 2011 at 4:57 pm |james kennedyExcellent post. You have saved me a lot of

time. I was probably not going to get around

to making all the data on CO2 absorption

more comprehensible and relevant as you have done.

thanks.

This is worth some serious study.

on May 1, 2011 at 10:43 am |maitI’m slightly confused about this to be honest – shouldn’t the important value to be looking at be how long the distance in atmosphere is where the transmittance becomes zero?

I don’t see how the transmittance values above zero can have a very big effect on the overall picture (from what I’ve understood the atmosphere is quite optically dense near the ground).

Considering the height of the troposphere, even changing the transmittance from 0.9 to 0.1 wouldn’t make much of a difference.

on May 1, 2011 at 10:55 am |scienceofdoommait:

This is the transmittance from the surface to the tropopause (calculated at one surface temperature and lapse rate).

Transmittance is a very wavelength dependent parameter.

And no, the important value isn’t where the transmittance becomes zero.

Transmittance itself doesn’t give the full picture, but if you want to see where transmittance has the most effect for more radiatively-active gases (=”greenhouse” gases) it is around a transmittance of 0.5.

Changing the transmittance from 0.9 to 0.1 through the troposphere would make a huge difference.

on May 1, 2011 at 12:45 pm |MaitI’m not sure I understand what you mean by transmittance here (my knowledge of terminology is quite terrible unfortunately), but I meant by transmittance the part of *surface* radiation reaching directly to the location we are observing (tropopause in this case I understand).

In this case the radiation absorbed by the atmosphere would be quite spread out for both 0.9 and 0.1 transmittance values (the amount of radiation absorbed by the first 100 meters would be considerably lower than for example a wavelength where transmittance at 400 meters would be 0 already).

on May 1, 2011 at 5:27 pm |DeWitt PayneMait,

For a non-reflecting atmosphere, absorptivity = emissivity = 1-transmittance (t). Optical density, τ = -ln(t). Looking down from the top of the atmosphere, optical density increases as you go towards the surface. The effective altitude of emission is where the optical density = 1 or a transmittance of 0.37. Obviously, this only applies if the optical density at the surface is >> 1.

on May 15, 2011 at 5:41 am |NikFromNYCI love the graphics but my interest in math waxes and wanes. These days I’m into graphic arts. Enjoy’

on May 15, 2011 at 8:24 am |scienceofdoomNikFromNYC:

You have a comment that I found in the moderation queue. It doesn’t seem to have any relevance to this article – more a graphic collage of a wide-ranging field.

A picture paints a 1000 words, but only if it is the right picture.. Your graphic talents will be well employed if you can something relevant to the article in question. Or other articles.

on January 12, 2013 at 6:55 am |Visualizing Atmospheric Radiation – Part Six – Technical on Line Shapes « The Science of Doom[…] people can read a little about line shapes in Atmospheric Radiation and the “Greenhouse” Effect Part Twelve – The Curve of Growth – and of course there’s no problems with questions, but I haven’t aimed to explain the […]