# Earth Cubed

## log(CO2) and Scary Graphs

After reading Aurther’s blog post “New Congressional Budget Office Report on Climate Change” I got curious as to what the temperature response to CO2 would look like if it is actually logarithmic.

I descovered (what I should have realized from the start) that the curve is concave up but will converge to a linear curve for large T

It is commonly believed that the response of the earth to greenhouse gasses is logarithmic. I heard people suggest on other forums that this view was obtained empirically though climate models and perhaps more specifically radiative transfer models.

For analytic justification one would derive an expression for how the spectral width of an absorption peak grows with CO2 concentration (as I’ve done here). However, I am not convinced that this is sufficient justification as there will always be some radiative transfer at a given wave length even if the majority of it is absorbed over a very short distance. This is because the temperature gradient produced by the lapse rate produces an outward radiative flux that would exceed radiative feedback. I discuss this in more detail in my post:
Tropospheric Feedback

The logarithmic response is important because it is a type of saturation, in other words the more CO2 that is added to the atmosphere the less effective the next unit of CO2 will be in contributing to the warming. What I learned from reading Arther’s blog is that the current CO2 levels have not yet overwhelmed the natural levels of CO2.  This can be seen in the following graph:

More specificity from about 1000-1800 the CO2 concentration in the atmosphere stayed around 280 ppm. The following graph is more useful for measuring the current growth in CO2 concentration:

This graph is surprisingly very linear. If the growth in CO2 is truely exponential then it should be possible to estimate in from the slope on this graph which is given as 1.4203 PPM per year. For an exponential function:

$y=y_o+Aexp(\lambda (t-t_o))$
The derivative is:
$y'=A\lambda exp( \lambda (t-t_o))$
And the second derivative is:
$y''=A\lambda^2 exp( \lambda (t-t_o))$
The second derivative was taken because two equations are needed to find both $A$ $\lambda$ can be found.

The site also where I obtained the above figures gives a quadratic fit which can be used to estimate the first and second derivatives:

$44690.5-46.1486x+0.0119942x^2$

Therefore at year 2007 the first derivative is given by:

$-46.1486+2*0.0119942*2007=1.9961$

and the second derivative is 0.0119942

Giving:

$A exp( \lambda (t-t_o)) \lambda =1.9961$
$A exp( \lambda (t-t_o)) \lambda^2 =0.0119942$

Dividing the second equation by the first:

$\lambda=0.0119942/1.9961=0.006008817$

From this the doubling time can be obtained as follows:

$ln(2)/\lambda=ln(2)/0.0060=115.5245$ years.

$y_o$ is taken to be the base level of CO2. That is:

$y_0=280$

A third equation can now be found as:

$384-280=Aexp(0.006(1997-t_0))$

giving:

$A=\frac{384-280}{exp(0.006(1997-t_0))}$

If t_o is taken to be 1800 this gives:

$A=31.8931$

which suggests the CO2 growth rate has decreased over the last 200 years.

The CO2 is estimated to follow this function:

$CO_2(t)=280+31.8931 exp(0.006 (t-1800))$

The question now is how does this growth rate in CO2 effect the temperature. There are several estimates for the sensitivity of the climate to changes in CO2. Lucia’s one box model “lumpy” suggest a sensitivity of:

1.7 degrees Celsius per CO2 doubling.
The IPCC estimates the lower bound for sesitivity to be:
1.5 degrees Celsius per CO2 doubling (see CO2 Climate Sensitivity)
Isaac M. Held suggests a climate sensitivity of about 2.8C/CO2 doubling. See:

http://www.gfdl.gov/isaac-held-homepage

#### Selected recent papers on climate sensitivity:

Here is what wikipedia has to say:

In Intergovernmental Panel on Climate Change (IPCC) reports, equilibrium climate sensitivity refers to the equilibrium change in global mean near-surface air temperature that would result from a sustained doubling of the atmospheric (equivalent) CO2 concentration. This value is estimated, by the IPCC Fourth Assessment Report (AR4) as likely to be in the range 2 to 4.5°C with a best estimate of about 3°C, and is very unlikely to be less than 1.5°C. Values substantially higher than 4.5°C cannot be excluded, but agreement of models with observations is not as good for those values. This is a slight change from the IPCC Third Assessment Report (TAR), which said it was “likely to be in the range of 1.5 to 4.5°C” [1]. AR3 defined climate sensitivity alternatively in systematic units, equilibrium climate sensitivity refers to the equilibrium change in surface air temperature following a unit change in radiative forcing and is expressed in units of °C/(W/m2) or equivalently K/(W/m2). In practice, the evaluation of the equilibrium climate sensitivity from models requires very long simulations with coupled global climate models, or it may be deduced from observations. Therefore the 2007 AR4 renamed the alternative climate sensitivity to climate sensitivity parameter adding a new definition of effective climate sensitivity which is “a measure of the strengths of the climate feedbacks at a particular time and may vary with forcing history and climate state”.

The logarithmic law of CO2 forcing is given as:

$\Delta T=k*log_2(CO_2(t_2)/CO_2(t_1))=\frac{k}{log_n(2)}log_n((CO_2(t_2)/CO_2(t_1))$

where $k$ is the CO2 sensitivity for doubling CO2

I plotted this function for several values of the doubling sensitivity k

The labels on the right hand side of the plot are the climate sensitivities for each curve. This is actually a considerably smaller response then one would expect given the doubling time is around 100 years. However, while this is a nearly sufficient time for the exponential part of the curve to double,  the CO2 only increases by a factor of 1.2 since at 1997 the CO2 concentration is 384 ppm and in 2100 the CO2 concentration is projected by this fit to be  473ppm. This reduces the expected response by a factor of:

$log(473/384)/log(2)=0.3$

Notice that if the sensitivity ranges given by the IPCC are used then with this fit to CO2 emission growth is around 0.5-1.2 degrees which is hardly the doomsday scenario shown in the following graph which was posted on Aurther’s blog.

As a final note, the MATLAB code I used to produce the above graph is:

clear all
CO2= @(t)280+31.8931*exp(0.006*(t-1800))
CO2_0=CO2(1997)
t=linspace(1997,2100)
K=[1.5 4 7 11 15 21 25 29 33]
CO2s=CO2(t);
DT=@(k)(k/log(2))*log(CO2s/CO2_0)
for i=1:length(K)
DTs=DT(K(i))
plot(t,DTs)
AXIS([1997 2100 0 10])
gtext(num2str(K(i)))
hold on;
end
xlabel('Year')
ylabel('Temperature Change in Degrees Celcius')
hold off,