This is a technical thread – not really “visualization” at all, just part of the series – looking for comment and verification.

Interested 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 subject in this article.

I’ve been looking at the absorption line shapes in the upper troposphere/lower stratosphere.

Lorenz line shapes (collisional broadening) dominate in the lower atmosphere, Doppler broadening dominates in the upper atmosphere – and in some middle ground there is a “convolution” which is a Voigt line shape. This is bit of a pig because instead of being a nice well-formed function (like Lorenz or Doppler), each point on each line shape is calculated by an integral from -∞ to ∞. That’s one absorption line requiring many many integrals of a complex function between -∞ to ∞.

Here are a few pages explaining the topic from the excellent (very technical) *Radiation & Climate*, Vardavas & Taylor (2007).

The key point is the value of the mixing parameter, a, that determines whether the line needs to be modeled as a Voigt profile at all.

Note especially the last paragraph. **If a > 1 for all lines of interest then we can use the simple Lorenz formula** (pressure broadening).

So after playing around comparing line shapes for a while I realized that in the region of the atmosphere that we have been getting to in this series so far, perhaps the Voigt line shape wouldn’t be needed.

So let’s look at the likely values of a=b_{l0}/γ_{d}.

My calculation, using eqns 4.5 & 4.11 and using v_{0} instead of ω_{0}:

a = b_{l0}.(p/p_{0}).(T_{0}/T)^{nair}.(c/v_{0}).(m/2RT)^{1/2}

Also note that the formula for the Lorenz broadening (eqn 4.5) is an approximation, and the more accurate formula has an exponent which is measured (and can be retrieved from the HITRAN database) and here called nair.

nair varies from 0.41-0.78 for the CO2 lines between 200-2500 cm^{-1}, and, just for interest, the measured line width at 1013 hPa and 296K (here called b_{l0}) ranges from 0.055 – 0.095 cm^{-1}.

Let’s re-arrange this formula to group things a bit better:

a = C . b_{l0}/v_{0} . pT_{0}^{nair}/T^{nair+0.5}.

where the constant, C = c/p_{0} . (m/2R)^{1/2} = 152

p_{0} = 1013 hPa and T_{0} = 296K are the pressure and temperature at which the HITRAN measurements are made. Note that the formula requires SI units so p_{0}=101300.

Just a note that the Matlab program seen in earlier articles in this series and described in Visualizing Atmospheric Radiation – Part Five – The Code uses **gama** for the line half width, **nair** for nair and **v** for v_{0}.

So now we can plug some values for the lower stratosphere into the equation, take the whole HITRAN CO2 database and plot line strength, S vs a.

The reason for plotting line strength was expecting that there would be some values of a <1 and therefore requiring Voigt treatment and so wondering if they were so weak they could be ignored. Always want to skip the extra mile if possible..

So here is such a graph, with all 53,757 lines (in the range of interest) at 50 hPa and 210K:

*Figure 1*

The minimum value of the mixing parameter, a = 1.4. We are in the clear.

Comments please – mistakes in the formula re-arrangement?

Here is the MATLAB code for plotting the above graph:

% Voigt/Lorenz determination – value of mixing parameter, a,for CO2

% ref Vardavas & Taylor 2007, p91

% uses HITRAN database to determine where in atmosphere and line database

% a<1. When a>1 line profile is Lorenzian

p0=1.013e5; % std pressure for the HITRANS database in Pa

T0=296; % std temperature for HITRANS database in K

c=3e8; % speed of light

mco2=44.01e-3; % mass of mole of CO2

R=8.3143; % gas constant

C=c/p0*(mco2/2/R)^.5;

% for now pick a temperature and pressure

p=5e3; % 50mbar

T=210; % temperature in K

% read in HITRAN for CO2 for the range considered in RTE program

[ v S iso gama nair ] = Hitran_read_0_2(2, 200, 2500, 1);

a=C.*gama.*p.*T0.^nair./(v.*T.^(nair+0.5));

semilogx(S,a)

xlabel(‘Line Strength’)

ylabel(‘Mixing parameter’)

title(['p= ' num2str(p/100) ' hPa, T= ' num2str(T) ' K, Min a= ' num2str(min(a))]);

Here’s a histogram of values:

*Figure 2*

I was also interested in the breakdown of the above results in relevant bands. It’s clear that the mixing parameter will be lower when v is highest, and the 2500 cm^{-1} region is of less concern anyway as the atmospheric radiation is comparatively very low at this point (4 μm).

*Figure 3*

It does appear that the lines getting close to the Voigt threshold are those in the higher wavenumber (lower wavelength region, 4-5μm) that is anyway of less impact on the radiative balance in the atmosphere.

**Related Articles**

Part One – some background and basics

Part Two – some early results from a model with absorption and emission from basic physics and the HITRAN database

Part Three – Average Height of Emission – the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions

Part Four – Water Vapor – results of surface (downward) radiation and upward radiation at TOA as water vapor is changed

Part Five – The Code – code can be downloaded, includes some notes on each release

Part Seven – CO2 increases – changes to TOA in flux and spectrum as CO2 concentration is increased

Part Eight – CO2 Under Pressure – how the line width reduces (as we go up through the atmosphere) and what impact that has on CO2 increases

Part Nine – Reaching Equilibrium – when we start from some arbitrary point, how the climate model brings us back to equilibrium (for that case), and how the energy moves through the system

Part Ten – “Back Radiation” – calculations and expectations for surface radiation as CO2 is increased

Part Eleven – Stratospheric Cooling – why the stratosphere is expected to cool as CO2 increases

Part Twelve – Heating Rates – heating rate (‘C/day) for various levels in the atmosphere – especially useful for comparisons with other models.

### References

*Radiation and Climate*, I.M. Vardavas & F.W. Taylor, *Oxford Science Publications; International Series of Monographs on Physics 138* (2007)

The data used to create these graphs comes from the HITRAN database:

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

The HITRAN 2004 molecular spectroscopic database, by L.S. Rothman et al., Journal of Quantitative Spectroscopy & Radiative Transfer (2005)

on January 12, 2013 at 7:53 am |scienceofdoomHere’s the minimum value of a as a function of pressure and temperature. The blue is >1, the red is <1. Probably can do shading if I looked how.

on January 12, 2013 at 8:56 am |scienceofdoomHistogram of CO2 line widths:

on January 12, 2013 at 9:11 am |scienceofdoomI played around with minimum line strength (S) and instead of minimum value of a, when the 1% percentile or 10% percentile of a is below 1.

The values of minimum pressure for a>1, for temperatures from 200-250K don’t really change that much.

When considering all line strengths or line strengths >10

^{25}or the top 90% or the top 50%, the minimum pressure where the Voigt profile should be considered is still is around 20-40 hPa. For reference, the pressure of the tropopause is around 200 hPa.The minimum pressure used in the model so far is 50 hPa.

on January 12, 2013 at 10:03 am |Pekka PiriläSoD,

There’s no point of even starting to worry about Doppler broadening until you use in your calculations a resolution bandwidth that’s comparable or smaller than the Doppler broadening effect. As the Doppler broadening half-width is never more than 0.002 1/cm you are always far from that point with your standard parameters. It may be worthwhile to study the impact of resolution bandwidth on the results, but worrying about Doppler broadening is futile unless it’s concluded that very narrow bands are needed in some calculations.

Furthermore the case cannot be such that neglecting Doppler broadening would be reason for not observing need for narrower bands as its effect would go in the opposite direction.

The resolution that you apply is rather low in comparison with the line structure at low pressure. Thus it could influence the results significantly but on that point the easiest way of finding out is to experiment, i.e. to do some calculations with very narrow bands overnight and compare the results with those calculated with a standard choice putting particular emphasis on such details that are most likely to depend on the resolution. Results specific for the stratosphere are probably those most strongly affected.

You have done some tests along these lines, and I had a higher resolution run taking 2.5 hours last night. Some spectral details change clearly but overall results very little.

on January 12, 2013 at 12:33 pm |scienceofdoomPekka,

Your point on the Doppler profile and my linewidth resolution is correct.

This is my concern – that too coarse a wavenumber resolution misses the impact of the linewidth shrinking that takes place at lower pressure.

Of course, if I don’t implement the Voigt profile when I should, then a high resolution model will have more gaps and higher peaks than it should.

But forgetting about the Voigt profile, if I have too coarse a resolution then the (collision broadening) gaps at lower pressure might be missed. You could say that it will all “average out in the wash”, but the peaks are “saturated” already so too many peaks adds nothing, but missing the gaps might have some significant impact. (I’m not certain that this will be the outcome – it’s a work in progress)

As you say, the important bit is to run the model with varying linewidths to see how things change.

I’m running the model now with varying CO2 concentrations, each time with a run of 1 cm

^{-1}and then of 0.1 cm^{-1}. I am saving the whole dataset each run so later I can compare the spectra as well as TOA flux and DLR.on January 12, 2013 at 1:10 pmPekka PiriläSoD,

As I write below the resolution is after all not that critical as long as the set of frequencies is statistically representative enough. The analysis is done for a set of exact frequencies rather than for a set of bands. This means that the set of frequencies must represent closely enough the full continuous set of frequencies. Any single peak may be misrepresented as long as the errors done for individual peaks compensate each other and the sample is not biased.

There are a few very strong peaks that may be totally missed by a coarse set of frequencies but that’s not important, if we are not looking layers or distances so small that the absorption is not saturated even at the nearby frequencies that get included.

On the other extreme the few narrow minimums of absorptivity cannot dominate the outcome. Even if the valleys are very narrow, there are many of them and statics performs its role in picking to the calculation an essentially right number of such frequencies.

Basically the set of frequencies provides an essentially random sample of absorptivities. Reducing the frequency step increases the sample size and improves the accuracy through that.

on January 12, 2013 at 10:43 am |Pekka PiriläI was wondering why the approach you use in handling the spectra works so well in spite of the fact that the resolution cannot express well the lineshapes. The collisional broadening leads to peak half-widths at half-maximum that are less than 0.1 1/cm. How can a resolution of 1.0 1/cm be successful with that?

Then I realized that the point is that the approach does not integrate over each resolution band but uses the value at the center of the line. Thus the full spectrum is not really divided in bands but rather the values at selected grid points are used as representative sample values picked from the accurately calculated spectrum.

This realization means that my previous message is not fully correct. The results could be influenced by Doppler broadening even when the resolution is much worse. You are right that collisional broadening dominates. In case of CO2 all significant lines have an air-broadening half-width near or above 0.06 1/cm at surface pressure and temperature. Water and to lesser extent methane have some much more stable excitations that lead to narrower lines in air. In general collisional broadening dominates very clearly even in the upper half of the stratosphere. Ozone and N2O are closer to CO2 in this respect.

on January 12, 2013 at 3:18 pm |DeWitt PayneSoD,

I remember looking into this in reference to someone else writing his own LBL program some years ago. When the FWHM of the line is less than the resolution of the program, calculating the optical density for each frequency band becomes, more interesting shall we say. What happens is that you overestimate the optical density more and more as the line narrows. It’s not even Doppler/Voigt/Lorentzian. This leads to overestimation of the absorption and emission in the stratosphere. As I remember, it was about an order of magnitude too high.

As Pekka points out above, using your approach will result in a spectrum that looks correct. In fact it should look like a spectrum with much higher resolution than 1 cm-1. But if you try to integrate to calculate total emission or absorption, you’ll probably get the wrong answer at high altitude.

on January 12, 2013 at 3:44 pm |Pekka PiriläDeWitt,

What I’m trying to explain is that the approach used by SoD does actually lead to the right total emission and absorption. Individual values may be highly unrepresentative of frequencies close to that point but the errors cancel in the sum.

This approach calculates correctly each of the values. It may be missing strong structure between the points, but the values represent basically an unbiased sample.

When the structure that influences the final outcome is strong, it’s important to have a large sample as a small fraction of the points may have a sizable influence and a fair number of values must be included from that small fraction. Therefore getting good accuracy requires more points, but a smaller number does not imply bias, only great uncertainties.

on January 12, 2013 at 4:06 pmPekka PiriläTo explain what happens a bit more i show too graphs that present optical depth of a layer between 17 km and 24 km in the stratosphere near the 15 µm absorption peak.

This graph is calculated with 0.05 1/cm steps

and

This graph is calculated with 1.0 1/cm steps

Comparing those two graphs you may see that the values agree at every common data point but also that the data points of the coarse graph represent badly the average behavior in their neighborhood. That effect is visible also in the other graph as the highest value does not correspond to the highest value of the complete spectrum. The highest maxima are close to the peaks on both side of 667.5 but those narrow peaks fall between the frequencies calculated. As the peaks are so narrow that’s enough to leave the 668 value as the highest. There the line falls almost exactly at the data point.

Going through the full calculation every point calculated is correct (up to the other approximations made in the analysis). Having a rather dense set of correct values we can calculate also correct integrals. The accuracy of the outcome is helped by the fact that transmissivity has far less structure than optical depth, because any optical depth far larger than one gives essentially zero transmissivity.

on January 12, 2013 at 4:47 pmDeWitt PaynePekka,

It’s been years, so I didn’t remember the details. The problem was at higher altitudes. What comes to mind is that, for example, at 40 km looking up using MODTRAN tropical atmosphere, you see ~2 W/m² while the wrongly calculated value was more like 20 W/m².

After looking up some of the correspondence, I don’t think there’s a problem at pressures above 50hPa. For some reason, the person in question was trying to calculate emission/absorption above 100 km and finding very large values.

on January 12, 2013 at 5:02 pmPekka PiriläDeWitt,

Going that far up the lines get very narrow and the calculation must use a narrow frequency spacing to get sufficient statistical accuracy.

Another possibility is that the approach is not the one used in SoD’s code. Some alternative approaches may lead to strong biases, while they might avoid better the random uncertainties of SoD’s method.

on January 12, 2013 at 7:16 pmDeWitt PaynePekka,

I believe that may have been the case. I think there may have been something odd about calculating optical density from concentration, or something like that. Whatever it was, it didn’t go completely off the rails until the line width was narrower than the program resolution, which, as I remember, was a lot finer than what SoD is using. It’s definitely not worth the time searching the archives to find out.

LBLRTM uses the Voigt profile at all altitudes, but then it’s compiled, not interpreted, and I don’t know how efficient the code is.

on January 12, 2013 at 3:46 pm |DeWitt PayneI found the spreadsheet. Now all I have to do is remember what it all means.

Here’s a normalized plot of Doppler vs. Lorentz lines.

As I remember, the key concept is equivalent line width. A narrow line can be very intense and still have a small effective absorptivity/emissivity/cm-1 at low resolution.

on January 12, 2013 at 4:29 pmPekka PiriläOne consequence of the fat tails of the Lorentz distribution is that the convolution of Lorentz and Doppler shapes can be calculated using a almost tailless finite range approximation for the Doppler broadening as long as collisional broadening is not completely negligible. That helps in calculation of the convolution, if that’s ever needed.

on January 12, 2013 at 7:31 pm |scienceofdoomI did 10 runs with CO2 concentrations of 280, 360, 400, 450, 560ppm each with δv=0.1 and 1 cm

^{-1}. These were with a surface temperature of 300K and water vapor BLH=80%, FTH=40%. I took O3 out for speed but CH4 and N2O are still included at current concentrations.This set took 4 hours. I’m doing another set with surface temperature at 288K for comparison. Then I will see if δv=0.01 cm

^{-1}can run on 2 of these sets.The first 10 show that the TOA flux is not significantly affected by δv, but at the same time, in comparison with the changes from CO2 they could be considered significant. From 280ppm to 560 ppm TOA flux reduces from 273 to 266 W/m

^{-1}. This is not radiative forcing as per IPCC definition of course but there is no stratospheric adjustment. My recollection is that stratospheric adjustment reduces the impact of changes in CO2, must find that reference.I had a quick look at the absorptivity changes in a few layers vs wavelength and it’s pretty interesting. I’ll write up those results in the next post.

on January 13, 2013 at 6:02 pm |DeWitt PayneSoD,

If you don’t allow the stratosphere to cool, you see too much upward radiation at the TOA. That is, the apparent forcing reduces with altitude in the stratosphere. Also, there is more radiation downward at the tropopause. That affects the radiative balance in the upper troposphere. Whether the effect translates all the way to the surface isn’t clear, at least to me. As I remember, the stratosphere cools because the lapse rate decreases. The temperature at the tropopause doesn’t change.

on January 13, 2013 at 6:26 pmPekka PiriläIn the defining case of radiative forcing radiation from below to the stratosphere decreases and the emissivity of the stratosphere increases. As the temperature is defined by the local radiative balance at every altitude of the stratosphere both factors contribute significantly to the cooling.

In the upper stratosphere the increase in the emissivity is the more important factor when the main source of heat is absorption of solar UV and that doesn’t change as much with more CO2.

on January 14, 2013 at 9:31 amscienceofdoomDeWitt,

That’s my working understanding as well.

It’s probably not that hard to introduce at least a parameterized stratosphere. I would like to be able to finally explain in layman’s terms why the stratosphere cools with more CO2.

Explaining in layman’s terms requires “first being sure I understand it”.

I am currently not sure.

on January 14, 2013 at 4:38 pmDeWitt PayneSoD,

The way I look at it is that you get a reverse greenhouse effect in the stratosphere because the absorptivity of incident radiation for the SW band is higher than for the LW. If you increase CO2, the absorptivity in the LW goes up and the reverse greenhouse effect gets smaller as a result.

Ignoring the importance of SW and LW relative absorptivities is, IMO, a major flaw in using only lapse rate and emission altitude to explain how the greenhouse effect works.

on January 14, 2013 at 9:10 am |scienceofdoomI posted in a comment in Part Five and realized that it should go with the above comment, so reposting here..

—-

I reran the set of results for changing CO2 from 280ppm, 360, 400, 450, 560 ppm at Ts=288K and 300K with the diffusivity factor (1.66) applied to the continuum.

[Pekka, I note your comments of January 13, 2013 at 6:03 pm, need a little time to review and digest].

The results were run at Δv = 1 cm

^{-1}& Δv = 0.1 cm^{-1}. The differences between the two cases are generally small, somewhere around 0.2 – 0.5 W/m^{2}for TOA and 0.1 – 0.3 W/m^{2}for DLR.Then I reran two cases with Δv = 0.01 cm

^{-1}– Ts=300K & CO2 = 280 ppm & 560 ppm.The runs took 4.5 hours each. That’s 230,000 individual calculations of optical thickness for each of 4 molecules (water vapor, CO2, NO2, CH4) at each of 10 levels. About 10 million calculations in total for optical thickness / transmissivity / emissivity.

The change in TOA and DLR from the Δv = 0.1 cm

^{-1}results in both cases was 0.0 W/m^{2}.on January 13, 2013 at 4:59 am |Visualizing Atmospheric Radiation – Part Seven – CO2 increases « The Science of Doom[...] « Visualizing Atmospheric Radiation – Part Six – Technical on Line Shapes [...]

on January 19, 2013 at 3:40 am |Eli RabettThe series and the conversation is really top class. The only thing that I would add for readers less sophisticated than the author and those who have commented is that collisional broadening is best understood as a small change in the molecular potential energy surface as the collision partner approaches and recedes. Averaged over each collision and all collision this has the effect of broadening the energy levels.

Increasing the number of collisions (pressure) and the average energy of a collision (temperature) affects the effective broadening. Energy transfer, obviously occurs on the collisional potential energy surface which is what determines the collisional lifetime.

on January 19, 2013 at 7:32 am |Pekka PiriläEli,

I don’t fully agree. Whenever we see the Lorentzian lineshape, we have a situation where the lifetime of the excitation determines fully the lineshape. That means that the only properties of collisions that matter are their frequency and the likelihood that a collision leads to a state transition. Other details of the collision affect the form of far tail, i.e. its deviation from pure Lorentzian.

Temperature affects the linewidth trough its influence on density and average speed of the molecules. At constant pressure the density is inversely proportional to density. The speed is proportional to the square root of the temperature. The time between collisions is inversely proportional to both density and speed. These factors lead to the proportionality of the linewidth to

pressure/temperature^0.5. The exponent of the temperature is not exactly 0.5. I haven’t checked what literature tells about that. Perhaps it’s due to the influence of collision energy on the probability of a transition in a collision, which is often close to one but may be less.on January 19, 2013 at 10:49 am |Pekka PiriläI checked literature and that confirmed my speculation. The temperature dependence can, indeed, be determined through quantum mechanical calculation of the transition probability. Here is one publication that discusses that point

http://faculty.uml.edu/robert_gamache/papers/CO2_preprint.pdf

In the previous comment I discussed the time between collisions and likelihood of a transition in a collision. That kind of division cannot really be made as telling whether a collision has occurred when that has not led to a transitions cannot be done unambiguously. There’s no precise dividing line between the cases where two molecules pass each other so far that their interaction cannot be considered a collision and a clear collision. The idea should, however, be clear. In some cases the is a sharp change from very weak interaction to a collision that leads to a transition with almost 100% certainty while in other cases the change goes more smoothly and leads to stronger temperature dependence.

One observation from that paper is that the power law dependence of the linewidth on temperature is not exact. For CO2 it’s a good approximation while it’s worse for water vapor.

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