I created some simulations of different CO2 concentrations using the atmospheric radiation model described (briefly) in Part Two and in detail in Visualizing Atmospheric Radiation – Part Five – The Code.
- surface temperature = 300 K
- lapse rate = 6.5 K/km
- 10 layers (of roughly equal pressure change to keep similar number of molecules in each layer)
- top layer, layer 10, centered at 16.9 km or 98 hPa and 7.0 km thick
- temperature of layer 10 = 220K
- Water vapor RH=80% in boundary layer, 40% in free troposphere
- CH4 at 1775 ppmv, N2O at 319 ppbv, no ozone
- Line width resolution = 0.1 cm-1
Here’s the transmissivity of the atmosphere with CO2 at pre-industrial levels of 280 ppm compared with doubled at 560 ppm. Transmissivity just means “what proportion of incident radiation makes it through”:
Figure 1 – Transmissivity of layer 10 from 666.6-667.6 cm-1
Wavenumber 667 cm-1 = 15 μm. (Wavenumber in cm-1 = 10000/wavelength in μm).
So we see that at 280 ppm a miniscule fraction of 666.6 cm-1 makes it through and at 560 ppm that’s gone to zero. But 0.05% down to 0% in just one small part of the spectrum is not much of an issue.
The equivalent graph for the surface layer is just plain 0.00000% making it through.
So it’s pretty clear that CO2 is already saturated and increasing CO2 has no effect.. well.. not in this 1 cm-1 range of the peak absorption of CO2.
Here’s another graph of this high up layer, across a wider wavenumber range:
Figure 2 – Transmissivity of layer 10 from 650-690 cm-1
We can see that there’s quite a difference as a result of doubling CO2.
Let’s take a look at the same graph for the bottom layer of atmosphere in this model. The center of this layer is at 420 m and it is 840 m thick at an average temperature of 297K:
Figure 3 – Transmissivity of bottom layer from 650 – 690 cm-1
Notice that the transmissivity scale has been dropped considerably from that of the top layer and still the transmissivity is zero – in both cases. So going from 280 ppm to 560 ppm has absolutely no effect on this layer.
Before we move on.. how can a layer with a similar number of molecules have such different characteristics?
The difference is the pressure. The line width of a typical CO2 absorption line is 0.07 cm-1 at the surface. At 100 hPa (17km) its line width will be about 0.008 cm-1(see note 1). The peak gets higher but the line gets thinner. Add lots of lines together at the surface and nothing gets through. Add lots of lines together at 100 hPa and quite a bit gets through.
Wait a bit.. if we consider all the layers together don’t we get back to zero transmissity?
Good point. (On a technical note we multiply transmissivities together, so if 10% gets through one layer and 5% through the next layer the total effect is only 0.5% getting through – and anything x 0 = 0).
So that’s clear then. If we take the entire spectrum from 650 – 690 cm-1 (15.4 – 14.5 μm) and look at the surface emitted radiation, nothing gets through to the top of atmosphere (TOA) regardless of whether we are at pre-industrial levels of CO2 or the future potential doubled CO2.
Right?
Let’s take a look at outgoing radiation at the top of the atmosphere:
Figure 4 – TOA Radiation from 650 – 690 cm-1
Why is there any radiation at all? Because the atmosphere emits radiation as well as absorbs radiation. If a gas molecule absorbs at one wavelength, it can also emit at that wavelength. See Part One and Part Two for the basics.
So, even though the total transmissivity through the atmosphere at these wavenumbers is zero, there is emitted radiation at these wavenumbers. And, there appears to be some small change in TOA radiation. If you reduce outgoing terrestrial radiation (called outgoing longwave radiation or OLR) and still absorb the same amount of solar radiation then the climate must warm.
Let’s look at the magnitude of this change – the calculation in the top right is the sum (the integral) of the curve. The change is less than 0.001 W/m²!! I knew it!
Figure 5 - Difference in TOA Radiation for 280 ppm – 560 ppm CO2 from 650 – 690 cm-1
So doubling CO2 wil cause a tiny tiny reduction in outgoing radiation at TOA in this band (all other things being equal). Except it doesn’t look like much - and it’s not.
If you have grasped why figure 5 is like it is even though total atmospheric transmissivity is zero then you have understood some important points.
Now let’s look at the whole picture. Here’s the top layer again, with the same conditions, but over a wider wavenumber range:
Figure 6 - Transmissivity of layer 10 from 650-690 cm-1 – Click to enlarge
It’s worth clicking on the graph to expand the scale.
Here’s the transmissivity difference between the two cases for this top layer:
Figure 7 – Difference in transmissivity of layer 10 for 280 ppm – 560 ppm - Click to enlarge
And, as before, let’s compare with the bottom layer – and with the same scale of transmissivity as figure 6:
Figure 8 - Transmissivity of layer 10 from 650-690 cm-1 - Click to enlarge
Let’s see the difference, on the same transmissivity scale as figure 7:
Figure 9 – Difference in transmissivity of layer 10 for 280 ppm – 560 ppm - Click to enlarge
Why is figure 9 so different from figure 7? There are two reasons:
- first, as before, the CO2 lines are narrower high up in the atmosphere meaning that there are more gaps in the spectrum
- second, water vapor absorption is very high around the 550 cm-1 region
The mixing ratio of water vapor molecules in the surface layer is more than 100 times greater than the top layer we are considering (in this scenario). So water vapor absorption is considerable near the surface, and very small in the top layer. (I might do a simulation with zero water vapor to see what the water vapor impact is around 800 cm-1 - as the water vapor continuum may be playing a role here).
As before, let’s consider total atmospheric transmissivity. From the graphs above of top and bottom layer of the atmosphere we might expect some wavenumber regions to be close to zero and others possibly not.
That’s what we find:
Figure 10 - Transmissivity of total atmosphere from 650-690 cm-1 - Click to enlarge
And the difference between 280 ppm – 560 ppm:
Figure 11 – Difference in transmissivity of total atmosphere from 650-690 cm-1- Click to enlarge
Now let’s look at the TOA spectrum.
As before when we considered a much narrower bandwidth, the value of TOA is not zero, and there is also a significant difference between the cases at 280 ppm and 560 ppm, even where the total atmospheric transmissivity is zero (compare with fig. 11):
Figure 12 – TOA radiation from 650-690 cm-1 - Click to enlarge
The reason should be clear – the atmosphere emits radiation. The radiation that escapes to space is NOT surface radiation attentuated by the transmissivity of the whole atmosphere. It is a sum of radiation from the surface and from all different levels in the atmosphere, all very dependent on wavelength. In the wavelengths where the atmosphere absorbs strongly the emission to space is from higher levels.
Take a look at Part Three – Average Height of Emission and Part Four – Water Vapor for more insight.
Here is the difference in TOA radiation for 280 ppm – 560 ppm. The total flux difference between the two cases in this wavelength region = 4.3 W/m².
Figure 12 – Difference in TOA radiation for 280 ppm – 560 ppm - Click to enlarge
Conclusion
Simple considerations of transmissivity of radiation in the most absorbing wavelengths of CO2 have led many people in the blog world to conclude that increases in CO2 will have no impact on outgoing radiation and that CO2 is “already saturated”.
Others have stated that water vapor totally overwhelms the effect of CO2.
We can see that these both misunderstand the actual, more complex, situation.
The data for the model comes from the HITRAN database (reference below) compiled over decades by spectroscopy professionals. The formula for absorption is the Beer-Lambert law. The formula for emission is the Planck emission law, modified by the emissivity at the wavelength in question. The formula for line width changes under atmospheric conditions have been known for 50 years or more and published in hundreds of papers.
I created this model from scratch using these equations. The equations can be seen in the Matlab model in Part Five.
The results look very similar to those published by atmospheric physicists who have run more detailed models under more exacting conditions – for example, the graph shown in CO2 – An Insignificant Trace Gas? – Part Eight – Saturation from Radiative forcing by well-mixed greenhouse gases: Estimates from climate models in the IPCC AR4, W.D. Collins et al, Journal of Geophysical Research (2006).
Hopefully, the data presented here helps to verify the approximate magnitude of the net change in absorbed radiation due to CO2 doubling (see note 2).
Much more important – the aim to to help the reader see more clearly how radiation interacts with the atmosphere under different conditions.
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 Six – Technical on Line Shapes - absorption lines get thineer as we move up through the atmosphere..
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
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)
Notes
Note 1: The formula for line half-width in the troposphere is basically bl0.(p/p0).(T0/T)0.5
where bl0 is the value measured at p0 & T0 (usually 1013 hPa and 296K) and p, T describe the pressure and temperature where we want to know the new line half-width. For in depth material on this see Part Six – Technical on Line Shapes
Note 2: This is often known as radiative forcing. That term has some more restrictive conditions around it – primarily it is the case before any response from the lower atmosphere (i.e., the surface and atmospheric temperature are held constant) but after allowing for the stratosphere to return to radiative equilibrium, which takes a few months.
This Matlab model does not have a proper treatment of the stratosphere, and stratospheric radiation change does have an impact on the tropospheric radiation balance. The intent of this model is not to redo what atmospheric physicists have done, but to provide a reasonably realistic tropospheric model which allows us to look behind the scenes.
The model has the basic mistake, that the greenhouse effect is not considered. The assumptions used for 280 ppm can be made – but doubling the concentration to 560 ppm has an impact on the greenhouse effect. The surface temperature can be assumed with 303 K, the temperature of the layer 10 is then 211 K and the average pressure of layer 10 is reduced (for example 97 Pa), because the rises of height of tropopause.
These changes result from the higher disability rating of radiation propagation through more CO2 while maintaining the total radiation of all wavelengths. The cumulative curve is beginning with a higher performance by level 0 and larger slope at the upper end.
I think the calculation with the transmission is counterproductive, because where is strongly absorbed, is also emitted strong. A calculation with intensities that are much higher than the intensity of thermal emission, says nothing: The absorption onto the Beer-Lambert law neglected the thermal emission – only the radiative transfer equation is considered for the thermal emission, only by neglecting of the thermal emission in the radiative transfer equation is the Beer-Lambert law correct, ie the intensity change by the use of Beer-Lambert law requires intensities that are far above the actual intensities and therefore unrealistic.
Sincerely
The results are deliberately shown with the same surface and atmospheric conditions.
As I stated in the conclusion to Part Four – Water Vapor (but for slightly different reasons):
So the intent here is to show how the same atmospheric conditions but different concentrations of CO2 cause a change in outgoing radiation.
For people familiar with the basic physics this is, of course, very obvious and not so interesting.
For people getting to grips with the physics of radiative transfer this is essential knowledge. What wavelengths cause what changes and so on.
Comparing with fig 12, we can see the fine difference in atmospheric radiation at TOA when the simulation (for 280 vs 560 ppm) is run at Δv = 0.01 cm-1 vs Δv = 0.1 cm-1. The view zooms in on one wavelength region of 580 – 590 cm-1.
Click for expanded view
In terms of total upward flux at TOA, the calculated value is the same whether the atmosphere is evaluated at 0.1 or 0.01 cm-1.
SoD,
What’s the change in downward flux to the surface. That would answer almost directly some of the recent questions presented in other threads as far as this model is considered representative for the whole atmosphere.
Pekka,
Do you mean the DLR for the difference of Δv of 0.1 cm-1 vs 0.01 cm-1? If so, the answer is also 0.0 W/m2.
Or the change as a result of introducing the diffusivity factor to the continuum?
I plan to do a separate article on the downward flux.
I meant the effect of doubling CO2.
I did my own calculation with slightly different input (and a little modified model with 10 azimuthal segments). That gave an decrease of 5.05 W/m^2 in the flux up at TOA, 3.24 W/m^2 in flux down at surface and thus a net rate of 1.81 W/m^2 was left to warm the atmosphere.
This simple clear atmosphere model gives a high value for the forcing, but what I had in mind was the point that about 64% of the warming goes initially to the surface and 36% stays in the atmosphere.
Due to the smaller heat capacity the atmosphere warms at that point much faster but that will soon affect convection, and at that stage the energy the atmosphere takes in further warming is very little. What’s initially 64% is soon almost 100%.
Pekka,
That sentence reads flux down at the surface decreases. That must be an increase or the net wouldn’t be smaller. That’s a rather large increase in flux down for doubling CO2. That’s what I would expect to see for very low humidity, like the sub-Arctic winter atmosphere.
DeWitt,
You’re right. The flux down to surface increases. Words “.. and an increase ..” were missing from my previous message.
In this calculation the boundary layer humidity is 80% and the top of the boundary layer at 920 hPa / 400 m. Above that altitude the humidity is 40%. I haven’t paid much attention to the parameters. Thus the values are perhaps the most appropriate ones.
However well the model is made to calculate the radiative transfer, it’s still based on one clear sky profile only and therefore not quantitatively right for global averages of the real Earth system.
Pekka,
That profile would give very low total column precipitable water for a surface temperature that high. Looking at the different profiles, only the US 1976 standard atmosphere likely has total column water that low. At 300 K, you get ~+2.5 W/m² increase in DLR for doubling CO2 and a reduction of 4 W/m² at the tropopause looking down.
It makes me suspect that there might be a problem with the water vapor pressure calculation.
DeWitt,
I haven’t studied any part of the code related to water vapor, just used the code as SoD has written it. The moisture levels are also the same he has been using.
My implementation differs in two ways:
- I read the HITRAN data from text files downloaded from SpectralCalc rather than HITRAN-files directly.
- I have added the possibility of separating segments of different azimuthal angles. In many runs I use ten segments as the program runs fast enough even with those (the most time consuming step need not be modified for that). It turns out that this added complexity makes little difference for the results. (The up and down fluxes change by +0.4 and -0.5 W/m^2, when 10 segments are taken into use rather than one with diffusion factor 1.66).
When I use more closely his parameter values the changes from doubling CO2 are -4.2 W/m^2 up and +2.8 W/m^2 down.
Pekka,
Can I see the code for the different azimuthal angles?
Did you do this rather than use the Zhao & Shi method, because it’s potentially more accurate?
DeWitt,
The code for water vapor is:
————————————-
function [ es ] = satvaph2o( T )
% Saturation vapor pressure in Pa at temperature T in K
% Using wiki formula
% T can be a vector
Tc=T-273.15; % temperature in ‘C
es=610.94.*exp(17.625.*Tc./(Tc+243.04));
end
—————————————-
In the main function this is called like this:
for i=1:numz-1
% now calc mixing ratio in molecules for water vapor at prescribed RH
% = RH * es / p
% currently satvaph2o() gives es=610.94.*exp(17.625.*Tc./(Tc+243.04)); where Tc in ‘C
if i==1
mixh2o(i)=BLH*satvaph2o(Tinit(i))/p(i);
else
mixh2o(i)=FTH*satvaph2o(Tinit(i))/p(i);
end
end
————————————–
And the number of molecules is worked out via:
na=(Na*rho(i)/mair)/1e6; % number of air molecules per cm^3
….
if mol(m)==1 % if water vapor, special case
mix(m)=mixh2o(i); % precalculated for each layer using prescribed RH
….
nummol=isoprop(m,iso(im(j)))*mix(m)*na;
….
At the end the function returns the vector mixh2o() so the mixing ratio by volume can be seen for each layer.
SoD,
I just sent it to you (before I read this message).
Here are the mixing ratio values (%) for a 10 layer model with a surface temperature = 288K, BLH = 80% and FTH = 40%.
Height= 0.4 km, Pressure= 966.1588 hPa, Mixing Ratio= 1.1785%
Height= 1.26 km, Pressure= 870.9433 hPa, Mixing Ratio= 0.44898%
Height= 2.21 km, Pressure= 774.2521 hPa, Mixing Ratio= 0.3269%
Height= 3.27 km, Pressure= 677.3866 hPa, Mixing Ratio= 0.22389%
Height= 4.45 km, Pressure= 580.6494 hPa, Mixing Ratio= 0.14243%
Height= 5.81 km, Pressure= 483.9863 hPa, Mixing Ratio= 0.080612%
Height= 7.41 km, Pressure= 387.3887 hPa, Mixing Ratio= 0.03837%
Height= 9.36 km, Pressure= 290.8015 hPa, Mixing Ratio= 0.01374%
Height= 11.95 km, Pressure= 194.1372 hPa, Mixing Ratio= 0.0031486%
Height= 16.2 km, Pressure= 97.5971 hPa, Mixing Ratio= 0.006263%
SoD,
You asked also for the preference of this over Zhao and Shi.
When I realized that the extra computational work is not too heavy, this approach seems clearly preferable. Using this approach the whole diffusion approximation can be dropped as the path-lengths are real path-lengths, not parametrizations. The need of using some “fudge factor” like 1.66 is not there any more.
The Zhao and Shi approach is likely to give results that depend a little on the selected layer thickness while this other method is approximate only in the same sense as using a finite number of layers is approximate. (Using a discrete set of frequencies is on still better basis as that does not introduce bias, but only effectively random errors.)
From practical point it’s nice to see that the simple diffusion factor of 1.66 works so well, while it’s a little disappointing that the theoretically better method is so little better in practice. Differences are so small that they can be ignored for most purposes. There are certainly again some detail results where the difference is larger.
One example is the spectrum of radiation that escapes directly from surface to space. That’s a quantity for which I get quite different results from some early calculations and from my present version. I’m not quite certain on the reason. There may be an error in either one calculation. That’s anyway just one of those “curiosity values” as it’s not possible to tell in reality what’s the point of emission of each photon. The diffusion factor is certainly important for that, but in this case we know by definition that the photons must traverse the whole atmosphere. Thus the angular dependence of the optical thickness is also known exactly and can be used in a separate calculation as I have done.
There’s certainly a risk that my code has errors, but the results are, in general, in good agreement with expectations. In particular it gives exactly the same results as your code, when the 1/cos(theta) factor is replaced by 1.66 for all angles.
As I expected, your atmosphere is really dry. Here’s a comparison of water vapor mixing ratios with pressure for your atmosphere and the US 1976 standard atmosphere with humidity corrected to a surface temperature of 300 K. If I did my sums correctly, the total water vapor content in atm cm for your atmosphere is 2597 compared to 3855 for the US 1976 standard atmosphere relative humidity profile at the same surface temperature.
SoD,
I went here: http://www.humidity-calculator.com/index.php
I plugged in the numbers for the boundary layer, 80% RH, 297.4 K and 96616 Pa, the calculated water vapor mixing ratio was 1.6076% w/w or 2.5847% v/v. Something’s wrong somewhere.
SoD,
I somehow missed that the surface temperature in your table was 288 K. The web site calculated volumetric mixing ratio for 285.4 K and 966.16mbar is 1.19945% compared to your 1.1785%. That’s not far off. So your atmosphere is wetter in the boundary layer and dryer at higher altitude with ~2 cm precipitable water compared to ~1.4cm for the US 1976 atmosphere.
DeWitt,
Quite high on my list of things to do with this model is to run it with some US standard atmospheres.
Do you have the data (atmospheric temp & water vapor mixing ratio or RH) available in a convenient table, at the tropospheric resolutions I’d need (at least 10 layers in the troposphere)?
I haven’t looked very hard but when I did look some time ago I found the data up to 100km but not at all good resolution for the troposphere.
SoD,
MODTRAN will give you 1 km resolution up to 25 km and then 5 km to 50 km. Just click on the ‘view the whole output file’ link at the bottom of the right hand pane after you submit the calculation for a particular atmosphere. All the spectral data are there too. The spectral data, however, are only for the selected observation height and direction. Also, the atmosphere data are for the specified surface temperature. If you change the surface temperature, you really need to change the atmosphere pressure profile too which will change the number densities of the various molecules. MODTRAN only changes the temperature up to 13km when you change the surface temperature offset. It probably doesn’t make much difference for small changes in temperature.
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scienceofdoom on January 14, 2013 at 9:08 am
“The results are deliberately shown with the same surface and atmospheric conditions.”
This allowed for calculations. But then you have to point out that the energy balance is violated in any height. This violation of the energy balance is stored in the atmosphere – and leads to temperature changes at the new temperatures lead back to the energy balance.
Pekka Pirilä on January 14, 2013 at 8:20 pm
“I have added The Possibility of separating segments of different azimuthal angles.”
The integration over all angles is not spread as Beer-Lambert law, but the Exponential integral function.
Sincerely
The exponential integral is obtained from a calculation of transmission trough a single layer of given thickness. It cannot be applied exactly to the succession of several layers because it cannot describe properly the probability of photons passing trough several layers. The approach of handling the azimuthal angle as an extra parameter does this as well. Therefore this approach is fundamentally correct while the use of exponential integrals remains an approximation in the multilayer case.
Doing the calculation of transmission trough one layer having the azimuthal angle as one variable leads exactly to the exponential integral when integrated over azimuthal angles. The numerical method based on a few discrete values for the azimuthal angle is naturally an approximation but it’s accuracy can be improved by increasing the number of these discrete values. It’s easy to find out empirically, how many values are needed for accurate enough results. That number seems to be rather small.
Pekka,
If I remember correctly, Miskolczi used either seven or nine azimuthal angles in his LBL program. As far as I know, there’s nothing wrong with his LBL program. It’s the rest of his theory that leaves something to be desired.
DeWitt,
Miskolczi has 9 azimuthal angles. I have also got the impression that his radiative calculation is correct, or at least I haven’t found anything wrong in it. Concerning his 2010 paper it’s a bit strange that he gets so far essentially correctly and then just misses the opportunity of reaching reasonable results on GHE by doing one more very simple step correctly.
[...] But the effect of increasing CO2 on the TOA radiation balance is completely different. High surface humidities have little or no effect on this TOA balance. And there, doubling CO2 has a significant impact (all other things being equal) as shown in figure 12 of Part Seven – CO2 increases. [...]
[...] Visualizing Atmospheric Radiation – Part Five – The Code Visualizing Atmospheric Radiation – Part Seven – CO2 increases [...]
[...] Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased [...]
[...] Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased [...]
[...] Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased [...]
[...] Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased [...]