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.
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.
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
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.
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.
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)
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.