### Recap

In Part Five we finally got around to seeing our first calculations by looking at two important papers which used “numerical methods” – 1-dimensional models – to calculate the first order effect from CO2. And to separate out the respective contribution of water vapor and CO2.

Both papers were interesting in their own way.

The 1978 *Ramanathan and Coakley* paper because it is the *often cited* paper as the first serious calculation. And it’s good to see the historical perspective as many think scientists have been looking around for an explanation of rising temperatures and “hit on” CO2. Instead, the radiative effect of CO2, other trace gases and water vapor has been known for a very long time. But although the physics was “straightforward”, solving the equations was more challenging.

The 1997 *Kiehl and Trenberth* paper was discussed because they separate out water vapor from CO2 explicitly. They do this by running the numerical calculations with and without various gases and seeing the effects. We saw that water vapor contributed around 60% with CO2 around 26%.

I thought the comparison of CO2 and water vapor was useful to see because it’s common to find people nodding to the idea that longwave from the earth is absorbed and re-emitted back down (the “greenhouse” effect) – but then saying something like:

Of course, water vapor is 95%-98% of the whole effect, so even doubling CO2 won’t really make much difference

The question to ask is – how did they work it out? Using the complete radiative transfer equations in a 1-d numerical model with the spectral absorption of each and every gas?

Of course, everyone’s entitled to their opinion.. it’s just not necessarily science.

### The “Standardized Approach”

In the calculations of the “greenhouse” effect for CO2, different scientists approached the subject slightly differently. Clear skies and cloudy skies, for example. Different atmospheric profiles. Some feedback from the stratosphere (higher up in the atmosphere), or not. Some feedback from water vapor, or not. Different band models (see Part Four). And also different comparison points of CO2 concentrations.

As the subject of the exact impact of CO2 – prior to any feedbacks – became of more and more concern, a lot of effort went into standardizing the measurement/simulation conditions.

One of the driving forces behind this was the fact that many different GCMs (Global Climate Models) produced different results and it was not known how much of this was due to variations in the “first order forcing” of CO2. (“First order forcing” means the effect before any feedbacks are taken into account). So different models had to be compared and, of course, this required some basis of comparison.

There was also the question about how good **band models** were in action compared with **line by line** (LBL) calculations. LBL calculations require a huge computational effort because the minutiae of every absorption line from every gas has to be included. Like this small subset of the CO2 absorption lines:

Band models are much simpler, and therefore widely used in GCMs. Band models are “paramaterizations”, where a more complex effect is turned into a simpler equation that is easier to solve.

### Averaging

Does one calculation of CO2 radiative forcing from an “average atmosphere” gives us the real result for the whole planet?

Asking the question another way, if we calculate the CO2 radiative forcings from all the points around the globe and average the radiative forcing do we get the same result as one calculation for the “average atmosphere”.

This subject was studied in a 1998 paper: *Greenhouse gas radiative forcing: Effects of average and inhomogeneities in trace gas distribution*, by Freckleton et al. They ran the same calculations with 1 profile (the “standard atmosphere”), 3 profiles (one tropical plus a northern and southern extra-tropical “standard atmosphere”), and then by resolving the globe into ever finer sections.

The results were averaged (except the single calculation of course) and plotted out. It was clear from this research that using the average of 3 profiles – tropical, northern and southern extra-tropics – was sufficient and gave only 0.1% error compared with averaging the calculation at 2.5% resolution in latitude.

### The Standard Result

The standard definition of radiative forcing is:

The change in net (down minus up) irradiance (solar plus longwave; in W/m

^{2}) at the tropopause after allowing for stratospheric temperatures to readjust to radiative equilibrium, but with surface and tropospheric temperatures and state held fixed at the unperturbed values.

What does it mean? The extra incoming energy flow at the top of atmosphere (TOA) without feedbacks from the surface or the troposphere (lower part of the atmosphere). The stratospheric adjustment is minor and happens almost immediately (there are no oceans to heat up or ice to melt in the stratosphere unlike at the earth’s surface). *Later note added – “almost immediately” in the context of the response of the surface, but the timescale is the order of 2-3 months.*

The common **CO2 doubling** scenario, from pre-industrial, is:

**278ppm -> 556 ppm**

And the comparison to the present day, of course, depends on when the measurement occurs but most commonly uses the 278ppm value as a comparison.

**IPCC AR4 (2007) pre-industrial to the present day (2005), 1.7 W/m**^{2}

**IPCC AR4 (2007) doubling CO2, 3.7 W/m**^{2}

Just for interest.. Myhre at al (1998) calculated the effects of CO2 – and 12 other trace gases – from the current increases in those gases (to 1995). They calculated separate results for clear sky and cloudy sky. Clear sky results are useful in comparisons between models as clouds add complexity and there are more assumptions to untangle.

They also ran the calculations using the very computationally expensive Line by Line (LBL) absorption, and compared with a Narrow Band Model (NBM) and Broad Band Model (BBM).

CO2 current (1995) compared to pre-industrial, clear sky – 1.76W/m^{2}, cloudy sky 1.37W/m^{2}

(The NBM and BBM were within a few percent of the LBL calculations).

There are lots of other papers looking at the subject. All reach similar conclusions, which is no surprise for such a well-studied subject.

### Where does the IPCC Logarithmic Function come from?

The 3rd assessment report (TAR) and the 4th assessment report (AR4) have an expression showing a relationship between CO2 increases and “radiative forcing” as described above:

**ΔF = 5.35 ln (C/C**_{0}**)**

where:

C_{0} = pre-industrial level of CO2 (278ppm)

C = level of CO2 we want to know about

ΔF = radiative forcing at the top of atmosphere.

(And for non-mathematicians, **ln** is the “natural logarithm”).

This isn’t a derived expression which comes from simplifying down the radiative transfer equations in one fell swoop!

Instead, it comes from running lots of values of CO2 through the standard 1d model we have discussed, and plotting the numbers on a graph:

From *New estimates of radiative forcing due to well mixed greenhouse gases*, Myhre et al, *Geophysical Research Letters* (1998).

The graph reasonably closely approximates to the equation above. It’s very useful because it enables people to do a quick calculation.

E.g. CO2 = 380ppm, ΔF = 1.7W/m^{2}

CO2 = 556ppm, ΔF = 3.7 W/m^{2}

Easy.

### Benefit of Using “Radiative Forcing” at TOA (top of atmosphere)

**First of all**, we can use this number to calculate a very basic temperature increase at the surface. Prior to any feedbacks –** or can we?** [*added note, James McC kindly pointed out that my calculation of temperature is wrong and so maybe it is too simplistic to use this method when there is an absorbing and re-transmitting atmosphere in the way. I abused this approach myself rather than following any standard work. All errors are mine in this bit – we’ll let it stand for interest. See James McC’s comments in About this Blog)*

In Part One of this series, in the maths section at the end (to spare the non-mathematically inclined), we looked at the Stefan-Boltzmann equation, which shows the energy radiated from any “body” at a given temperature (in K):

Total energy per unit area per unit time, j = εσT^{4}

where ε= emissivity (how close to a “blackbody”: 0-1), σ=5.67×10^{-8} and T = absolute temperature (in K).

The handy thing about this equation is that when the earth’s climate is in overall equilibrium, the energy radiated out will match the incoming energy. See The Earth’s Energy Budget – Part Two and also Part One might be of interest.

We can use the equations to do a very simple calculation of what ΔF = 3.7W/m^{2} (doubling CO2) means in terms of temperature increase. It’s a rough and ready approach. It’s not quite right, but let’s see what it churns out.

Take the solar incoming absorbed energy of 239W/m^{2} (see The Earth’s Energy Budget – Part One) and comparing the old (only solar) – and new (solar + radiative forcing for doubling CO2 values), we get:

T_{new}^{4}/T_{old}^{4} = (239 + 3.7)/239

where T_{new} = the temperature we want to determine, T_{old} = 15°C or 288K

We get T_{new} = 289.1K or a 1.1°C increase.

Well, the full mathematical treatment calculates a 1.2°C increase – prior to any feedbacks – so it’s reasonably close.

*[End of dodgy calculation that when recalculated is not close at all. More comments when I have them].*

**Secondly**, we can compare different effects by comparing their radiative forcing. For example, we could compare a different “greenhouse” gas. Or we could compare changes in the sun’s solar radiation (don’t forget to compare “apples with oranges” as explained in The Earth’s Energy Budget – Part One). Or albedo changes which increase the amount of reflected solar radiation.

What’s important to understand is that the annualized globalized TOA W/m^{2} forcing for different phenomena will have subtly different impacts on the climate system, but the numbers can be used as a “broad-brush” comparison.

### Conclusion

We can have a lot of confidence that the calculations of the radiative forcing of CO2 are correct. The subject is well-understood and many physicists have studied the subject over many decades. (The often cited “skeptics” such as Lindzen, Spencer, Christy all believe these numbers as well). Calculation of the “radiative forcing” of CO2 does not have to rely on general circulation models (GCMs), instead it uses well-understood “radiative transfer equations” in a “simple” 1-dimensional numerical analysis.

There’s no doubt that CO2 has a significant effect on the earth’s climate – 1.7W/m^{2} at top of atmosphere, compared with pre-industrial levels of CO2.

What conclusion can we draw about the cause of the 20th century rise in temperature from this series? None so far! How much will temperature rise in the future if CO2 keeps increasing? We can’t yet say from this series.

The first step in a scientific investigation is to isolate different effects. We can now see the effect of CO2 in isolation and that is very valuable.

Although there will be one more post specifically about “saturation” – this is the wrap up.

Something to ponder about CO2 and its radiative forcing.

If the sun had provided an equivalent increase in radiation over the 20th century to a current value of 1.7W/m

^{2}, would we think that it was the cause of the temperature rises measured over that period?

*Update – CO2 – An Insignificant Trace Gas? Part Eight – Saturation is now published*

### References

*Greenhouse gas radiative forcing: Effects of average and inhomogeneities in trace gas distribution*, Freckleton at al, *Q.J.R. Meteorological Society* (1998)

*New estimates of radiative forcing due to well mixed greenhouse gases*, Myhre et al, *Geophysical Research Letters* (1998)