Shouldn't $CO_2$ forcing depend on the surface temperature?

When I read about radiative forcing due to CO2 it says:

"Radiative forcing is a measure of the influence a factor has in altering the balance of incoming and outgoing energy in the Earth-atmosphere system and is an index of the importance of the factor as a potential climate change mechanism. In this report radiative forcing values are for changes relative to preindustrial conditions defined at 1750 and are expressed in Watts per square meter (W/m2)."

Then it goes on to give an equation of the forcing as a function of only the concentration and some constant:

$$\Delta F = 5.35 \times \ln {C \over C_0}~\mathrm{W}~\mathrm{m}^{-2} \,$$

And reading about the greenhouse effect, wikipedia says:

Greenhouse gases absorb and emit some of the outgoing energy radiated from Earth's surface, causing that heat to be retained in the lower atmosphere.[38]

If the CO2 is reflecting back IR radiation that was emitted from the surface (the amount of which obviously depends on the temperature), shouldn't the forcing depend on surface temperature? Ie, it should be higher at the equator than near the poles?

To use an extreme example, I would expect the CO2 forcing for a planet at 0 K to also be 0 W/m^2, since there would be no IR emitted by the surface for it to absorb.

If that is correct, are there any papers about greenhouse gas forcings as a function of temperature/latitude? I do not mean due to latitude-dependent concentration of CO2, just due to the temperature of the surface.

EDIT:

I did some more research and found some data that shows that IR emissions for earth by latitude ranges from 94-276 W/m^2. So if the forcing is assumed constant, even just for our planet (not considering the rest of the solar system or exoplanets), it is equivalent to assuming that a given concentration of CO2 radiates back the same amount of energy over a ~3-fold difference in input energy.

Edit 2:

I asked this in a comment and think it may be a more direct question than the original:

If a 0.2% CO2 atmosphere receives 300 W/m^2 of IR from the surface, will it radiate back towards the surface the same amount as a 0.2% CO2 atmosphere that receives 100 W/m^2 of IR from the surface?

Edit 3:

I found a derivation for that 5.35 value. Apparently it is: $$F = f_a*f*F_{g►a}*(1 - T_d) = 0.6*0.4256*390*0.05371 = 5.35$$

where:

• $$F$$ is the $$CO_2$$ greenhouse flux density in $$W/m^2$$ ($$F_{a►g}$$)

• $$f_a$$ is the fraction of flux returned downward to the Earth, absorbed or re-emitted by $$CO_2$$

• $$f$$ is the Planck Blackbody Fraction (The fraction of Earth’s heat radiation in the spectral interval over which $$CO_2$$ absorption is significant)

• $$F_{g►a}$$ is the total flux density emitted by the Earth’s surface ($$\sigma T^4$$)

• $$T_d$$ is the diffuse transmittance

• $$(1-T_d)$$ is the fractional absorption

The value of interest would then be: $$F_{g►a} = \sigma T^4 = 390\,Wm^{-2}$$

We know the Stefan-Boltzman constant is $$\sigma = 5.67 x 10^{-8}\,Wm^{-2}K^{-4}$$. So to get a value of $$390\,Wm^{-2}$$ they must have used $$T = 288\,K$$, which makes sense since that is a widely cited value for the average temperature of the earth.

The temperature on earth has been found to vary between $$183.95-367.05\,K$$, which corresponds to values of $$F_{g►a}$$ between $$65-1029\,Wm^{-2}$$. The forcing will then range between:

$$\Delta F = 0.89 \times \ln {C \over C_0}~\mathrm{W}~\mathrm{m}^{-2} \,$$

$$\Delta F = 14.11 \times \ln {C \over C_0}~\mathrm{W}~\mathrm{m}^{-2} \,$$

So that is about a factor of 16 difference in forcing due to different surface temperatures. Correct?

Edit 4:

I should also note that the forcing due to the average temperature is not equal to the average forcing of multiple temperatures: $$\sigma\frac{1}{n}\sum_{i=0}^n T_i^4 \neq \sigma(\frac{1}{n}\sum_{i=0}^n T_i)^4$$

You can easily prove it to yourself by plugging in two temperatures 298 and 278 K that average to 288 K. The $$LHS \approx 393$$ and $$RHS \approx 390$$. Can anyone cite something that tells us which calculation is used in GCMs?

Edit 5:

At least in the GISS model E v2.1.1 GCM it appears the same forcing based on an average temperature of 288 K is applied everywhere. From model/RAD_DRV.f:

#ifdef HEALY_LM_DIAGS
FCO2=FULGAS(2)*CO2R
FN2O=FULGAS(6)*N2OR
FCH4=FULGAS(7)*CH4R
c
c      write(6,*) 'RJH: GHG: CONC=',
c     * FCO2,FN2O,FCH4
ghg_totforc=5.35d0*log(FCO2/CO2I)
*  +0.036d0*(sqrt(FCH4)-sqrt(CH4I))
*  -(Fe(FCH4,N2OI)-Fe(CH4I,N2OI))
*  + 0.12d0*(sqrt(FN2O)-sqrt(N2OI))
*  -(Fe(CH4I,FN2O)-Fe(CH4I,N2OI))
c      write(6,*) 'RJH: GHG: FORC=',ghg_totforc
#endif

Notice the 5.35d0*log(FCO2/CO2I) at line 2156. I don't know fortran though, so perhaps this code is doing something else.

Edit 6:

Looking further at GISS model E2.1.1, it appears I have answered the title question. Below is a extract of what they do to calculate the impact of CO2 on the downward solar flux within a given spectral interval for a single atmospheric layer at a given solar zenith angle:

C ULGAS  = Array of gas concentrations by layer (CO2 is column 2)
C N      = Atmospheric layer
C TLN    = Temperature at the middle of layer N
C PLN    = Pressure at the middle of layer N
C ULN    = Concentration of CO2 in layer N
C TAU    = Spectral Optical Thickness
C K      = Spectral Interval (eg, K = 1 corresponds to 3000-3400 nm)
C DKSO   = Fraction of solar flux in interval K (eg, .01 for K = 1)
C S0COSZ = Solar constant (defined as 1366 W/m^2) times cosine of the solar zenith angle
C DKS0X  = Solar flux in spectral interval K at zenith angle Z
C NL     = Top atmospheric layer
C SRDFLB = Vector of total solar downward flux at bottom edge of each layer

C Line 5586:
DKS0X=DKS0(K)*S0COSZ

C Line 5605:
C     Select parameterized k-distribution gas absorption by H2O, O2, CO2

C Line 5678:
CASE (9)
C--------K=4-------CO2       DS0=.002
ULN=ULGAS(N,2)
TERMA=(50.73-.03155*TLN-PLN*(.5543+.00091*TLN))*(1.-.1004*PLN)
TERMB=(1.+.006468*ULN)*(1.+PLN*(49.51+.8285*PLN))
TAU  =(TERMA/TERMB)*ULN
IF(PLN < 175.0) TAU=(.00018*PLN+0.00001)*ULN

CASE (10)
C--------K=3-------CO2       DS0=.003
ULN=ULGAS(N,2)
TERMA=(1.+.01319*TLN)*(PLN*(.008001*ULN+.4589E-03)-.8396*ULN)
TERMB=ULN*(PLN+295.7+1.967*ULN)+.15126*PLN
TAU  =(TERMA/TERMB)*ULN

CASE (11)
C--------K=2-------CO2       DS0=.003
ULN=ULGAS(N,2)
TERMA=(1.+.02257*TLN)*(PLN*(.002295*ULN-.5489E-04)-.7571*ULN)
TERMB=ULN*(PLN+803.9+2.477*ULN)-.09899*PLN
TAU  =(TERMA/TERMB)*ULN

C Line 5823:
XANX   = EXP(-TAU*SECZ)
DX     = 0

C Line 5844:
SRDFLB(NL)=SRDFLB(NL)+DKS0X*(XANX+DX)

By seeing how they use TLN (Temperature of layer N) to determine TAU (optical depth), we see the answer is that the forcing due to CO2 is a function of the temperature of that atmospheric layer (which would be a function of the surface temperature). It is also a function of the pressure, and concentration of the gas in that layer.

I isn't clear to me what all those magic numbers are in the calculation for TAU, nor what they are doing with XANX = EXP(-TAU*SECZ), but definitely see that the forcing is a function of the temperature.

The "simplified" calculation for the CO2 forcing appears to be worse than useless... it is not even the correct average. So I don't know what that was doing in their code, hopefully it was not used to tune the model.

• Calculation of average temperature used in GCM's: The GCM's should be dividing the atmosphere up into "cells", where the size of the cell is limited by the computing power of the supercomputer that the simulation is run on. Currently, these cells are several miles square, and I don't know whether or not they include all atmosphere in that area or if they divide the cell up into chunks of fixed altitude. This means that there is no "average" temperature used in the climate models, or at least not the kind of average that you are anticipating – David White Oct 15 '19 at 1:41
• @DavidWhite Basically, I suspect now that the 5.35*ln(C/C0) calculation is probably irrelevant. It is some kind of "example of the math" for people to look at who won't think very hard about it. Now, I want to see exactly what is input to calculate the effect of CO2 (+ etc) in these GCMs. – Livid Oct 15 '19 at 1:52
• @DavidWhite However, I do not know what that 5.35 value is doing in the GCM code either... – Livid Oct 15 '19 at 1:53
• Livid, I suspect that the equation is a REALLY averaged value over the whole earth, which obviously greatly over-simplifies what is happening in the real world. – David White Oct 15 '19 at 2:36
• @DavidWhite Yes, but look at my edit 4. It is also the wrong average, even for that. – Livid Oct 15 '19 at 3:18

This is more of an observation than an answer. The term "radiative forcing" is a linear, one-dimensional way of describing the effect of CO2 in the atmosphere. There are MANY effects going on that influence global mean temperature, and I can only guess at the most important, as this is an active area of research:

1) The total incoming solar radiation adds to the energy balance of the earth;

2) The total outgoing IR radiation subtracts from the energy balance of the earth;

3) A heat balance is of the form "Incoming solar ebergy - outgoing IR energy = energy accumulation", where for the global warming problem, the accumulation is positive

4) A positive energy accumulation causes phase changes (melting ice) and a temperature increase

5) The outgoing IR radiation is proportional to the fourth power of temperature, so temperature rises slightly until outgoing energy equals incoming energy

These effects are definitely NOT linear. At somewhat low atmospheric concentrations (e.g., 40-50 ppm), CO2 is already absorbing as much outgoing IR radiation as it can, so at the current atmospheric CO2 concentration, the IR absorption band is saturated. The absorption band can be slightly broadened by higher concentrations, but the effect is logarithmic (as your equation indicates) relative to atmospheric CO2 concentration, so the more you add, the less effect it has.

Competing effects: water vapor is the most important greenhouse gas on earth. It's also a fact that water vapor absorbs IR in more than one IR band, and one of those IR bands overlaps with the CO2 IR absorption band. In addition, there are several other atmospheric gases which have their own IR absorption bands.

Regarding water vapor, it is an excellent heat transfer medium, so there is some amount of convective heat transfer from the surface of the earth to high altitudes, where the water vapor condenses and radiates IR to outer space. This condensation obviously causes clouds, which affect the albedo (reflectance) of the earth. Low clouds reflect incoming solar radiation back to outer space, and high clouds reflect outgoing IR back towards the earth. It's also obvious that at high latitudes, clouds produce snow, which also increases the albedo of the earth.

On top of these effects, there are interactions between ocean currents and air temperatures, and ocean CO2 concentrations and air CO2 concentrations. The point of all this typing? It is DEFINITELY very short-sighted to worry only about "CO2 forcing" when discussing global warming and climate change, as the physics behind this issue is VERY multi-variable, non-linear, and mathematically chaotic. A brief reading of one Wikipedia article is definitely not sufficient to draw reasonable conclusions regarding atmospheric CO2 concentrations and their effect on global mean temperature and climate.

• The question is the same for water vapor, etc... I would expect the forcing should be dependent on the temperature of the surface. Go back to my surface at 0 K example... then the forcing should be 0 W/m^2 right? – Livid Oct 12 '19 at 2:41
• "CO2 is already absorbing as much outgoing IR radiation as it can, so at the current atmospheric CO2 concentration, the IR absorption band is saturated." Then we should deduce that adding additional CO2 to the atmosphere should not contribute to additional greenhouse effect. – Livid Oct 12 '19 at 2:45
• It will contribute a small amount as the radiation absorption band broadens, but the more CO2 you add, the less effect you get for the next ppm of addition. – David White Oct 12 '19 at 2:49
• There are no planets at 0 K. The microwave background radiation is supposedly 2.3 K, or thereabouts, so no planet can get colder than that, and everything that has a temperature radiates IR at some frequency. – David White Oct 12 '19 at 2:51
• If a 0.2% CO2 atmosphere receives 300 W/m^2 of IR, will it radiate back towards the surface the same amount as a 0.2% CO2 atmosphere that receives 100 W/m^2 of IR from the surface? – Livid Oct 12 '19 at 2:53

Radiative forcing is the change in the net, downward minus upward, radiative flux (expressed in Watts per square metre; W m-2) at the tropopause or top of atmosphere due to a change in an external driver of climate change, such as, for example, a change in the concentration of carbon dioxide (CO2) or the output of the Sun. WGIII
...
For the purposes of the WG1 AR5 report, radiative forcing is further defined as the change relative to the year 1750 and, unless otherwise noted, refers to a global and annual average value.

https://www.ipcc-data.org/guidelines/pages/glossary/glossary_r.html

The phrase "otherwise noted" suggests that there may also be a concept of local radiative forcing, as you describe.

• Thanks, can you find where someone considers the effect of the forcing changing due to local surface temperature though? – Livid Oct 13 '19 at 5:31