Doubling CO2 and basic physics

From Clive Best:

I have been struggling to understand exactly how increasing CO2 levels leads to global warming using  basic physics, and the story is complex. The trapping of certain bands of infrared radiation emitted from the Earth by greenhouse gases is well known. The effective cross-section of CO2 of absorption for CO2 in its rotational bands is also well known (HITRAN). Using the current concentration of CO2 in the atmosphere it is also clear that essentially all the radiation emitted by the surface in these bands is already absorbed by CO2 at pre-industrial levels. In fact spectra from space show that the main CO2 bands are saturated in the lower atmosphere with minimal emission from the high atmosphere. The absorption of radiation follows a logarithmic law with distance assuming a uniform concentration of CO2 in air. All that happens if you double the concentration of CO2 in air is that the absorption length is halved. So more radiant energy is absorbed and reflected back to earth at  lower levels of the atmosphere than before. However the total energy balance would appear at first sight to be almost unchanged. In fact just such an experiment was performed by Herr. Koch and led Angstrom to  dismiss theories of man induced warming already back in the early 20th century (Realclimate: what Angstrom didn’t know).

There are 3 main CO2 bands of IR absorption at wavelengths 1388, 667, 2349 cm-1 (HITRAN) and these are already saturated at current levels of CO2 in the atmosphere. Infra Red measurements from space show that the atmosphere is  opaque at these wavelengths. However at high levels in the atmosphere this is not no longer true and this is the only argument for the enhanced greenhouse warming by CO2 warming which makes sense to me (see below).

Spectral distribution of incoming radiation from the sun and outgoing Infrared radiation from the Earth to space. Taken from Atmosphere, Weather and Climate Barry & Chorley.

There is a very interesting paper here : which describes the basic physics. The absorption length for the existing concentration of CO2 is around 25 meters i.e. the distance to reduce the intensity by 1/e. All agree that  direct IR radiation in the main CO2 bands is absorbed well below 1 km above the earth. Increasing levels of CO2 merely cause the absorption length to move closer to the surface. Doubling the amount of CO2 does not double the amount of global warming. Any increase could be at most logarithmic, and  this is also generally agreed by all sides.

Lab experiments using tubes filled with CO2 to represent the atmosphere show that the IR transmission is essentially saturated and doubling the amount changes the transmission very slightly . So what is wrong and how are these arguments refuted by the majority of climate scientists? There are two main arguments  why global warming is important despite apparent saturation when man adds CO2 to the atmosphere (see  Realclimate ). The first of these is rather convincing.

1. IR scatters repeatably upward through layers of the atmosphere until at between 5-9 km  the air is so thin that the the atmosphere becomes transparent allowing CO2 emissions here to radiate out into space. At these levels there is little water vapour and CO2 dominates the energy loss. As CO2 concentrations increase so this level shifts to  higher levels in the atmosphere since a critical density must be reached for the radiation to escape.  These levels are colder (until we reach the troposphere) and IR loss is proportional to T**4 (Stefan Boltzman’s law). This means that slightly LESS energy is radiated to space than before and since the total energy must balance, the Earth warms up to radiate more heat to compensate. Don’t forget that there are windows in the IR spectrum with no absorption other than water vapour allows extra energy loss through evaporation and IR emission from clouds. Clouds also increase albedo. The vertical concentration of H2O and clouds are both temperature dependent. The temperature profile of the atmosphere is called the (adiabatic) lapse rate and is approximately -7 degrees per km falling to -4 degrees per km in the tropics. This is valid up to the Tropopause after which temperature rises again in the stratosphere. So greenhouse warming depends on falling temperature with height, and the anthropic enhanced greenhouse effect due to CO2 emissions is equal to the reduction of IR emissions to space in CO2 bands from the upper troposphere.

2. It is not completely true that the CO2 absorption bands are saturated as the fine structure is quite complex and in the side bands there is still energy left for the atmosphere to absorb. This is a much smaller effect than the first point. The extra absorption caused by an increase of a factor 4 is just a few percent as shown in the figure. In fact already 95% of the radiant energy is absorbed by CO2 at pre-industrial levels leaving a maximum of just 5% for any increase you like. It is often estimated  that the CO2 green house warming component of  the planet is about  3 degrees C so this extra absorption and emission to the surface  would appear to only add just 0.15 degrees of warming.

Unsaturated spectra (note LOG scale)

Therefore the main physics arguement supporting enhanced global warming caused by increasing levels of CO2 is the in height and thereby lower temperature of the effective radiating level of the atmosphere to space. The first comment to make is that we never  hear this crucial explanation in  the popular descriptions of the greenhouse effect. We just hear that more CO2 absorbs more heat radiated from the earth and radiates it back to the surface thereby heating us up just  like a thicker blanket does in bed (blankets actually work by cutting down convection losses). However the real explanation above concerns just the outermost layers of the atmosphere. The enhanced greenhouse effect depends on  a decreasing temperature gradient with height so that as the effective radiating level for IR  by CO2 rises so the energy loss falls and the earth must rise in temperature to compensate. Energy balance demands that there is a perfect match between incoming solar energy and outgoing IR energy. So lets look at this in more detail.

The effective temperature for all IR emissions is around -20 degrees C radiating at a height h of 5km. Now suppose that we double CO2 concentrations and the effective last radiating level rises to say 6 km. The temperature would now be about -27 degrees C as the decrease is almost linear in the troposphere. However the effective surface area is now also greater by an amount 8*PI*R (where R is radius of the Earth = 6350) making 1.6*10**5 sq km larger than at 5 km height.  The IR radiation falls off with temperature as T**4 which gives us a reduction of about 18% relative to before.

However if we just look at the main CO2 emission band and use the measurements from space (taken from Houghton’s book Global Warming – A complete Briefing). It would appear that the effective temperature of CO2 band alone  is -53 degrees C which is almost at the tropopause ( -60 degrees). So with all else remaining the same (water vapour, methane etc.) the drop in energy loss is just 3% in that single band.

CO2 causes the  lower atmosphere to be opaque at the main absorption bands. The mean free path is only  about 25 meters, so at these wavelengths the lower atmosphere is already like a thick fog where IR radiation is scattered in all directions. As we rise up in the atmosphere so the density falls exponentially and only at heights of 8-9 kms  does the atmosphere then become transparent in the main CO2 bands allowing energy loss direct to space. Doubling concentration rises that level nearer to the tropopause  which radiates at a lower temperature. The estimate given for the Earth’s warmng in Houghton’s book is 1.2 degrees for each doubling – so 2.4 degrees would be the heating if  CO2 concentrations were to increase by a factor 4. These figures are based on a radiation reduction of 4 watts/meter**2 caused by this effect of the effective radiation level rising to a colder level. In his book Prof. Houghton says this can easily be proved, but I have not  understood where this figure comes from nor how it has been calculated.

Feedback Effects

The IPCC predictions of future warming are based on model assumptions of positive feedback effects which are supposed to result from the initial warming caused by CO2 emissions. The main feedback effect is that of increased evaporation of the oceans leading to an increased greenhouse effect of water vapour (already 80 – 90% of greenhouse effect). However it is known that cloud cover in general leads to a net cooling effect on the Earth by blocking incident solar radiation. An increase in cloud cover of 10% would be enough to cancel out global warming effcets of increased CO2 ( Barry & Chorley).

Radiative Forcing Update: I have now found this reference to the equations used to derive the 4 watts/sq m radiative forcing by doubling the amount of CO2 in the atmosphere.

rf = f * ln([CO2]/[CO2]prein)/ln(2)    in watts/m ** 2

It would appear that in order to derive the factor f the IPCC assume that all of the 0.6 degrees warming apparently seen since the industrial revoluton is due to CO2 and thereby derive the constant

AF = 5.35 ln(C/Co)

Then we get simply  5.35*ln(2)  = 3.7 watts/sq m for the radiative forcing of  doubling CO2 !

If it is really true that this formula has been derived only by assuming that all “observed” temperature rise since 1750 is caused only by CO2 increases, then I fear this is a circular argument !  Many skeptics argue that the recent rises in temperature is dominated by a natural recovery from the little Ice Age. In order to be convinced that CO2 is the primary cause of recent warming then I would prefer  that this  formula could be derived from basic physical arguments  rather than introducing a fudge factor preset to prove a theory.

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