# Global warming case frustrating

tonyjeffs
Effect of doubling CO2 - why logarithmic and not linear?

How does a 30% to 50% increase in CO2 predict a 2-5 deg C temperature rise?
Is there a formula?

Why is the effect on low altitude atmosphere temperature effect of increasing CO2 logarithmic?
I would have thought it would be linear.
(This is something I've read on both pro and anti man-made global warming sites, but frustratingly no explanation for the assertion. Here is a graph from an anti site.)

http://www.junkscience.com/Greenhouse/co2greenhouse-X2.png [Broken]

Tony

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Andre
Tony,

The short answer is about 0.7 - 1 degree Celsius per doubling CO2 at thermal radiative equilibrium, excluding feedbacks.

You won't find much reference to that but you can calculate it yourself using http://geosci.uchicago.edu/~archer/cgimodels/radiation.html [Broken]

For instance start off with the default in that page and hit "submit the calculation"

At the right side we see:

Iout, W / m2 = 227.87
Ground T, K = 272.20

Now we double CO2 from 375 to 750 ppmv and submit again, to get

Iout, W / m2 = 225.546
Ground T, K = 272.20

Now we assume that the first value was at steady state thermal radiative equilibrium so with more CO2 less radiation leaves the Earth than comes in and with the second value there is unbalance. Now, we have to raise the earth temperature to get back to the radiative equilibrium radiation out matching radiation in again. Therefore we enter in "Ground T offset, C" the value of 0.769 degrees and we get

Iout, W / m2 = 227.87
Ground T, K = 272.97

Now if you understand the trick you can play with the scenario's and numbers to see that it's always about 0.7-1 degree sensitivity for doubling.

But then there is the positive - negative feedback dispute and the interpretation of the proxies of the past ice ages. My threads here are loaded with that, for instance:

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tonyjeffs
Thanks Andre,
Interesting reply, and thanks for introduction to Modtran, but my question really is:
Why is it 0.7-1 deg C per doubling of CO2.
Why isn't it a linear relationship - What is the basis in physics for the calculation that gives that result?

Thanks

Tony

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Andre
Look in the Modtran graph right below the output data, that should answer both questions. CO2 only radiates only with a certain narrow frequency spectrum. the first ppmv's saturate it rather quickly, then at higher CO2 concentrations the adjacent frequencies only get slowly affected.

Compare it with painting with a little transparant paint. The first layer gives the strongest coloring. Additional layers of paint only deepen the color slightly. This a 100% identical process, only in another frequency band.

tonyjeffs
Thanks again;
I understand the analogy, and can now why this should apply to CO2, and to transparant paint too, which I'd never thought about.
I will work on the Modtran material to understand it properly.

cbacba
Thanks Andre,
Interesting reply, and thanks for introduction to Modtran, but my question really is:
Why is it 0.7-1 deg C per doubling of CO2.
Why isn't it a linear relationship - What is the basis in physics for the calculation that gives that result?

Thanks

Tony

I'm not too sure what that modtran calculation is doing but it looks a bit high. I'm wondering if they've included pressure broadening, etc. with that calculation. It looks quite sophisticated but without the details....

What is happening is the attenuation of the light is an exp(-tau) type of thing. As one gets more molecules in the way, the transmission decreases. For much of the effects like with co2, most happens within a few feet but like any exponential value it never truly reduces to 0 on paper. Basically, that means that roughly, every doubling of co2 gives a diminishing return of about the same increment in absorption. I was thinking that at present a doubling should produce about 3.6W/m^2 additional absorption or around 1.6 W/m^2 since 1750 to now. The diminishing return means that both you've got to double the amount of co2 to add the increment that the previous doubling provided and that this increment has also shrunk a bit, something like 1/1.1 or 1/1.2 from the previous doubling.

As for temperature rise with power absorption, that's a much more complex situation even at the most fundamental rendition of reality. For a straight blackbody, it's denoted by stefan's law Power/m^2 = sigma * T^4 where sigma is boltzman's constant. This is the radiated power at all wavelengths where thermal radiation is occuring (planck's law stuff). Also a factor called emissivity enters in which in reality is a function of wavelength, angle and the nature of the material. It's usually called epsilon and multiplies times sigma... and ranges between 0 and 1 where 1 is a perfect black body. The earth's surface appears to run around 0.98 or 0.99 on average for the wavelengths of most interest in the IR.

When some co2 is added to the atmosphere, it means that some radiation is blocked going from the surface to space. For equilibrium to occur, this means that the incoming solar radiation must be compensated for, in this case by the atmosphere since there's energy radiating from the surface and from the atmosphere. Also, there are substantial amounts of clouds. A cloud cover of 62% was used by Kiehl and Trenberth in their 1999 energy balance paper whose balance chart is in common use. Clouds will substantially block radiating light going out as well as coming down from the sun. That means roughly 62% of the surface radiation is blocked by clouds anyway, leaving only about 38% of surface radiation (after attenuation by the atmosphere) to escape into space. It also basically means that only 38% of the incoming solar radiation reaches the surface.

There is also an albedo effect which reflects about 1/3 of the incoming solar radiation and most of this is caused by clouds as the earth is really good at absorbing incoming energy and clouds are fairly good at reflecting it.

What this boils down to is that the extra xx W/m^2 of increased atmospheric absorption is really only going to apply to what radiates from the ground to space. If it radiates to clouds - well - it's stuck in the atmosphere anyway. Ultimately, most of the energy is radiated from the atmosphere anyway as it's rather opaque overall in the IR. This xx W/m^2 effect only counts for the 38% of the radiated energy that isn't going to be stopped by clouds anyway. That's more like 0.38*xx .

One can do an energy balance (0 dimension model) for what happens with this stuff. One does need to have good information on molecular absorption - such as the hitran database. Care should be taken to make sure proper resolution and significant bandwidth is used as tiny amounts of power per unit wavelength add up to quite a bit over huge amounts of bandwidth. I did one going from 200nm out to 65535 nm and there is still a few W/m^2 of energy missing between stefan's law and a summation of the results from planck's law. Also, incoming solar radiation is less than 50% in the visible and the IR contribution is about the same as visible to within a few percent.

What I found doing the energy balance was that a change of 1W/m^2 increased absorption affected the surface temperature by 0.117 deg K and the atmospheric temperature mean by about 0.130 deg K. This translates to a doubling of CO2 (3.6W/m^2) to around 0.4 deg K rise.

Another thing I did was to take the total warming over an earth with no atmosphere and divided by the total energy absorption value for the whole atmosphere, around 230W/m^2. This forms a chord on the function of warming versus 'forcing'. The tangent to this function is the real instantaneous 'sensitivity'. The official IPCC definition is based on a chord from a co2 level to double that level. Since this 'function' actually includes all existing feedback mechanism at work and since the function is a rising value that has diminishing slope, the chord is actually the tangent at a point between 0 ghg effect and now, and it is steeper than the tangent at today's point. Nominally, the increase in temperature total is 33K and about 230W/m^2 of total ghg absorption in the atmosphere. This comes out to be about 0.14 K/W/m^2 which should be the upper limit of sensitivity that occurred at lower levels of GHGs on the curve.

The unknown feedbacks, especially the mythical positive feedback, are contained in this number for their contributions to date. Even though the feedbacks are in this number and our nonfeedback containing value of 0.117 is quite reasonable wrt this value, one must realize that the feedbacks are going to be a net negative and the actual effect of the increased absorption is going to reduce this sensitivity to be even less than 0.117 K/W/m^2.

What's more, these results indicate that of the supposed 0.7 deg K rise in temperature only has a co2 + methane + other manmade GHGs (total about 2.5W/m^2) of around 0.3 deg K , leaving 0.4 deg K to be accounted for by all those factors not understood and / or ignored. This even raises the question of if 0.4 K is unaccounted for, could this be responsible for added co2 releases assumed to be caused by man.

Canuck
the first ppmv's saturate it rather quickly, then at higher CO2 concentrations the adjacent frequencies only get slowly affected.

I've heard this before, but am admittedly a tad confused with it. The concentration of a gas affects the absorption band of the gas? Can you explain that a bit further?

In my linear brain, a gas' absorption band would be separate from the concentration of the gas.

sneez

cbacba
I've heard this before, but am admittedly a tad confused with it. The concentration of a gas affects the absorption band of the gas? Can you explain that a bit further?

In my linear brain, a gas' absorption band would be separate from the concentration of the gas.

The absorption band is broadened as more and more minor absorption lines become significant due to the increase of that molecule in the gas. Some are so strong they totally absorb the energy in a few feet or even a few inches.

absorption or transmission is affected by the optical depth. This is an exp(-tau) function. As more gas molecules of a type are in the way, the liklihood that one will absorb some more energy goes up. Some absorption lines are extremely likely and some are much less likely. The likely ones grab up the energy rather quickly, the less likely ones are all that's left to absorb the energy.

Essentially, the result is that every time you double the number of a molecule type in the column, you will get a certain change in total absorption, but it took lots more of the molecule to have the same additional effect as previously added molecules had. For instance, take the co2 doubling from 1750 to whenever it doubles. That's an addition of molecules to increase the co2 from 280 ppm to 560ppm. This should add some absoprtion to the total of about 3.6W/m^2. Once that level is reached, for co2 to to absorb another 3.6W/m^2, the amount will have to double again - to 1120ppm. Just adding the same number of molecules used to go from 280ppm to 560ppm (280ppm worth)is not nearly enough to gain that extra 3.6W/m^2.

At present in the atmosphere, there is some amount of energy radiated from the surface and some radiated from the sky. As these ghg molecules increase the atomspheric absorption, then more energy must be radiated by the atmosphere as it won't be coming from the surface so that the heat flow will balance out. It's a rather small amount, unrelated to the alarmist industry propaganda. It doesn't block the energy going out, but rather causes shifts in how that energy can get out and will cause some minor changes to temperature levels. My guess is less than 0.4 deg K for a doubling of co2. That also means that all ghg increases since 1750 amount to less than half the claimed warming.

Also, the actual lines as calculated or measured for something like Hitran have no width to them. Pressure broadening and doppler broadening are the terms used to describe the width for molecules in practical situations. For instance, molecules bouncing around due to their temperature will have relative velocities that are non zero and will cause a doppler spreading of the frequency or wavelength that the molecule is going to absorb. Some will be moving towards and some away from the direction of incoming radiation, spreading out the width of the absorption line. This isn't the same thing as absorption bands spreading out due to more material in the column as the bands are combinations of many individual lines of varying strength.

sneez
The absorption band is broadened as more and more minor absorption lines become significant due to the increase of that molecule in the gas. Some are so strong they totally absorb the energy in a few feet or even a few inches.

Even though my work is not with co2 i would like to supplement. The lines at the wings of absorbtion bands for co2 may become also saturated due to higher number of molecules but speaking about broadening may be hurting our understanding.

What you describe is that more absorbers make will make weak lines at the wings become more important. It will not broaden the lines. The actuall broadening is pressure/doppler mainly in the atmosphere.

That should cover this question in that it shows that absorbtion bands are affected by gas concentration.
In my linear brain, a gas' absorption band would be separate from the concentration of the gas

What will happen when we double co2 that the radiation coming from the surface will be absorbed closer to the surface, but the co2 bands are saturated bellow 10km already.

...absorption bands spreading out due to more material in the column...
This is effectivelly pressure broadening, as I understand it, due to more frequent collisions. Band refers to vibrational-rotational transitions. The line strength is lab measured, available for example in HITRAN.

cbacba
You're right about the 'wings' widening out for the band areas and the former 'wings' saturating. However, I'm recalling that there is a partial pressure factor of the same molecule that effects the actual line broadening approximation that should be used with Hitran. I'd have to go check it to make sure but I think it's there.

As for below 10km - I think you should probably use 10m although it's a somewhat subjective comment. There is substantial amounts of blocking going on over very short distances. It also seems that 5km is more like the halfway point through the atmospheric mass. It only takes about 8km at 1 atm pressure to provide the same amount of molecules as the real atmospheric column does up through 120km (99.999...%).

Hitran appears to be a composite of measured and calculated lines, some both. It's the 1960s physics equivalent of those log and trig tables created by otherwise unemployeed mathematicians in the 1930s great depression era that you can stumble across in the back of an old library.

Unfortunately, there is still possibly some confusion between band and line. A band is a wide section of spectrum generally composed of numerous lines. A band broadens as lower quality lines become more prominant due to added materials - the 'wings' spreading out. The line itself cannot just be a true monochromatic value but has some width as well, doppler/pressure broadening. I believe doppler is essentially the temperature bouncing molecules around. the pressure broadening is the pressure of the overall gas (and I believe the partial pressure of the same type of molecule being involved differently than the gas) causing additional spreading of the line's width - which remains a fatter line - not a band.

cbacba
Andre,

I came across an astounding revelation this weekend. The stefan's law equation indicates a T^4 relationship for power vs. temperature - nothing new. However, when dealing with it, one tends to ignore epsilon, the emissivity. The atmosphere has an emissivity as does the earth's surface. Emissivity is a very complex function of wavelength and is not a simple constant as is often assumed in simple engineering applications.

Emissivity also has the criteria of reciprocity with regard to incoming absorbed radiation and outgoing emitted radiation. Good absorbers are good radiators, bad radiators are bad absorbers. It is also linear in the stefan's law equation.

When a green house gas is increased in concentration, the absorption in the atmosphere increases. However, this is a change in emissivity and it means that the emissivity of the atmosphere increases and the radiation for a fixed temperature increases as well.

While this is a function of wavelength and the temperature of the atmosphere is not going to be the temperature of the surface, the IR emissions of both have tremendous overlap and are quite wide and so the change in actual emissivity for the different temperature spectra is quite similar, virtually equal. If they were perfectly equal, the change in emissivity due to say - a doubling of co2 concentration - would actually become a total wash as the extra energy absorbed would be equal to the extra (or new) energy radiated at the same original temperature - meaning that no additional temperature change (T^4) would be needed to balance the equations of power in / power out.

Running the numbers on my KISS project, I discovered that the actual results were almost zero and were negative. Differences in emissivity between the incoming and radiating spectra could evidently be positive or negative, depending upon where the changes in emissivity occur and whether it affects incoming or outgoing more. The key though is it is extremely insensitive to change.

Other types of of power absorption changes - like albedo change, cloud cover, solar insolation do not change the emissivity of the atmosphere and will cause the full effect of a variation in temperature (the T^4 factor which means that the sqrt(sqrt(new power/(sigma*epsilon))) gives the new temperature).

Again, calculating the epsilon (emissivity) for the atmosphere for an old and new value using atmospheric absorption from the Hitran database results in a negative result when the new emissivity is applied. That means the temperature drops because the radiation output of the atmosphere becomes more efficient with the new increased emissivity.

That brings to question, what does cause the earth to be warmer if ghgs don't have much of anything to do with it. It would appear that cloud cover might be the actual reason - that combined with albedo - as well as being the dominant reasons for variations.