The Azimuth Project
Blog - the log forcing (part three)

Third part of a blog article in progress, see also Blog - the log forcing and Blog - the log forcing (part two)

Are we still talking about climate sensitivity?

Before we talk about radiation physics, let us look more closely again at the climate sensitivity. In a textbook that dates back from 1982 I found:

There is little doubt that in the absence of other climatic perturbations an increase in atmospheric carbon dioxide concentrations will give rise to globally averaged warming of the lower atmosphere. The degree of warming for a given increase in the carbon dioxide concentration is, however, difficult to predict and will depend on the complex interactions of physical processes in the [earth system] which control climate.

This sounds very reasonable, it’s difficult to predict the degree of warming for a given increase in the carbon dioxide concentration. So let’s give it a name first, and call it climate sensitivity. Actually, the link between carbon dioxide concentrations and temperature changes consists of two major steps:

(1) changes in carbon dioxide influence the radiation budget of the Earth directly (this is the greenhouse effect, i.e. parts two and three).

(2) subsequently, there are feedback processes within the earth system triggered by the change in radiative forcing.

To some extent, the second step also impacts the first step, because we will see that the radiative forcing of carbon dioxide depends on the temperatures in the atmosphere. As I mentioned, in this blog series, we will purely concentrate on the greenhouse effect! For the second step we would need some good understanding of the Earth system, and for the first step it suffices to know some atmospheric physics and chemistry.

But I’ve mentioned degree of warming, so it would be good, just in case this number makes it to a question in quizzes, that we have an idea how many Kelvin can we expect this degree to be? In Week 301 the host of this blog wrote below the summary of some report:

It’s worth noting that they get their best fit if each doubling of carbon dioxide concentration causes a 3.1 ± 0.3°C increase in land temperature. This is consistent with the 2007 IPCC report’s estimate of a 3 ± 1.5°C warming for land plus oceans when carbon dioxide doubles. This quantity is called climate sensitivity, and determining it is very important.

Practically, that’s an important question, since a typical consumer of the Western world has a major hobby in burning carbon (perhaps it’s even more a job than a hobby…). Unfortunately, to determine this number exactly is way beyond the goal of this blog series. Let us be satisfied if we can start to understand how raising carbon dioxide levels influences the Earth’s temperature. Or, perhaps, let’s check this claim of 3 degrees first.

Verifying numbers

And where to look better than in the IPCC report? In questions like these googling doesn’t really help because sometimes the world wide web resembles somewhat Borges’ Library: for every statement one can somehow also find a refutation – and quite likely also another similar statement but with spelling errors. For example in climate science it’s safer to look at reports written by experts. So let’s go to the IPCC homepage, download their most recent report (still AR4) and start to leaf through.

On page lookup we find:

Carbon dioxide is the most important anthropogenic greenhouse gas (see Figure SPM.2). The global atmospheric concentration of carbon dioxide has increased from a pre-industrial value of about 280 ppm to 379 ppm3 in 2005

The combined radiative forcing due to increases in carbon dioxide, methane, and nitrous oxide is +2.30 [+2.07 to +2.53] W m–

The equilibrium climate sensitivity is a measure of the climate system response to sustained radiative forcing. It is not a projection but is defined as the global average surface warming following a doubling of carbon dioxide concentrations. It is likely to be in the range 2°C 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.

Explain difference radiative forcing and climate sensitivity.

So that turns out to be:

ΔTΔCC 1, \frac{\Delta T}{\Delta C} \propto C^{-1},


ΔTΔlog(C). \Delta T \propto \Delta \log(C).

Number of about 2 degrees Celsius

This is the famous log forcing of the earth’s temperature in response to changes in carbon dioxide concentration. Actually there are two immediate questions here. First, the logarithm. Where does it come from? And how does it arise: is it exact or only approximate? And it simple to understand why it’s there? Second: the factor of 2 degrees (we’re conservative and disrespect the best estimate). Can we get that by some simple calculations?

One question at a time suffices, and in this little blog series I will concentrate on the mechanisms behind the logarithmic relation. As I wrote earlier, the factor of two degrees follows when feedbacks are taking into account, and that would lead us too far. To explain the logarithmic relation, we’ll start simple, and then gradually add complexity:

(a) we’ll take a closer look at the greenhouse effect, but purely qualitatively.

(b) we will derive the increase in radiative forcing due to increases in carbon dioxide concentration.

(c) we will look at the temperature rise due to the increased radiative forcing (i.e. in the absence of other feedbacks). For this, we need to consider all wavelengths.

But, in order to take a closer look at the greenhouse effect, we’ll need to summarize some relevant physics. Some of that has appeared in earlier blog posts, so I will be brief, and give references.

Almost inevitably, we will have to satisfy ourselves with less rigor, since too much rigor might detract us from understanding the mechanisms. All concepts and arguments are only approximate, but I claim that the derivations are robust and that the results are stable: as we refine our assumptions, we won’t drift too far away from the approximate answer. And before a critical mind may comment that you can’t trust unknown internet sources, we’ll try to patch the current lack of rigor in a later post – with a date due before the current baktun is finished.

Anyway, let’s stop contemplating and finally move on! We’ll start off without any rigor at all. Let us first summarize the mechanisms behind the log forcing.


Recapitulation of previous two posts - what’s going to come here

Summary of important points of MathEnvPt3

but it’s still possible for the Earth to have an average emissivity near 1 at the wavelengths of infrared light and near 0.7 at the wavelengths of visible light.

The tendency of a substance to absorb light at some frequency equals its tendency to emit light at that frequency.

I’d like to add one point

principle is good to find the relation between epsilon and alpha (that is, they are equal) BUT it does not mean that at some wavelength, the same radiation (per wavelength) is emitted as absorbed.

Take thought example of monochromatic laser light.

But, this is only part of the , if hit by monochromatic source, energy absorbed, till reaches temperature T, emission is a factor

(Actually there are some small factors because of the diffuse radiation, but let’s absorb that in some redefinitions. B νB_\nu is the Planck function.)

Schwarzschild’s equation made dimensionless

Now it’s time to analyze the Schwarzschild equation. How shall we do this? Perhaps a good way would be to put it in nondimensional form first, so we get some appreciation for the important variables, and we can analyze the remaining equation more simply. If we look at the equation again

(1)dI νdz=α ν(B ν(T(z))I ν), \frac{d I_\nu}{d z} = \alpha_\nu \left( B_\nu (T(z)) - I_\nu\right)\,,

we see that we have two quantities of dimension irradiation (more detailed name!)namely I and B, one of dimension length (height above the earth’s surface z), and one of dimension inverse length (absorption coefficient alpha).

At the top of the atmosphere there is practically only outgoing longwave radiation (the incoming longwave is tiny compared to the incoming visible) and we will divide all terms by I(TOA). Does this make sense?

z a bit different. Yes, we could divide this by the height of the atmosphere, but that would not be clever. z is actually a variable here, and we can simplify the equation if we think a little bit. Mention not all z the same.

To make height dimensionless we will need the absorption coefficient, so let’s take a closer look at that first.

The absorption coefficient

In fact, the functional dependence of alpha is very important for our problem.

proportional to pressure and density of active absorbers

  • α ν\alpha_\nu includes density of active absorbers, here CO 2\mathrm{CO}_2, cross-section; split again in two terms, then in accordance with textbooks and MathEnvPt3

since pressure and density similar scale heights and approximately exponential, the absorption coefficient is an exponential too, with half the scale height

Optical depth

Now that we understand the response of alpha to changes is carbon dioxide concentration, we can proceed with making zz dimensionless. To discuss the Schwarzschild equation it will be use to make a change of variables from zz to the so-called optical depth χ ν\chi_\nu. It is a dimensionless variable that expresses the attenuation of the radiation. The optical depth is related to the height by:

(2)dχ ν=α nudz d \chi_\nu = - \alpha_nu \, d z

Through alpha chi scales with Optical depth obviously proportional to CO2 concentration

Better physical explanation of optical depth needed.

At the surface z=0z=0 I νsI_{\nu s}, if larger than blackbody radiation

(3)χ nu(z)Cexp(z/H χ) \chi_nu (z) \propto C \exp\left( - z / H_\chi\right)

(If the optical depth at the surface is large, the earth behaves more like a black body, and the effective emission height lies high in the atmosphere. If the optical depth at the surface is small for a certain wavelength, the emission height lies close to the ground and for those wavelengths one can almost see the earth’s surface.)

Schwarzschild’s equation (again)

(4)dI νdχ ν(B ν(T(z(χ ν)))I ν) -\frac{d I_\nu}{d \chi_\nu} \propto \left( B_\nu (T(z(\chi_\nu))) - I_\nu\right)

Then the Schwarzschild equation becomes:

(5)dI νdχ ν=B ν(T)I ν - \frac{d I_\nu}{d \chi_\nu} = B_\nu(T) - I_\nu

where T=T(z(\chi_\nu))

Minus sign because monotonically decreasing when z is going up

We don’t want to solve the Schwarzschild equation here (that is, I don’t want to, if you want to, go ahead!) so we have to make a leap of trust here, the effective emission height lies around the scale height of the optical depth, inside troposphere.

How to understand this better physically?

(Goldilocks wavelength, where changing concentration matters most for the irradiation, are those which have optical depth in troposphere - hard too argue, but is it necessary for temperature sensititvity?)

Which optical depths matter most for irradiance? Irradiance most sensititve to, where is this located wrt z, absolute value of effective z is important too? Or rather z effective from optical depth moderate value, and then for relevant lambda this is in troposphere, but this is harder to make precise without formulas…

Effective emission height

Radiation balance

The outgoing radiation at the effective emission height z *z^* depends on the temperature at that height:

(6)T(z *)=T 0+ΓH χlog(χ ν(z *)C) T(z^*) = T_0 + \Gamma H_\chi \log\left( \frac{\chi_\nu(z^*)}{C}\right)

Where the the carbon dioxide concentration is interpret as a parameter.

(Ok, we see where the concentration of carbon dioxide appears in the equation, and we have a logarithm too, but there are two temperatures we could take as a variable, the question is which.)

Keeping T 0T_0 constant, we get:

(7)ΔT(z *)Δlog(C^)=ΓH χ \frac{\Delta T(z^*)}{\Delta \log(\hat{C})} = -\Gamma H_\chi

Well, this is certainly a reduction of the temperature, not an increase. Did we just denounce the greenhouse effect?! Unfortunately not: it’s indeed a reduction, but we’re looking at the temperature level higher in troposphere at the effective emission height. The temperature does drop there, and there is less radiation going out (through Planck’s law).

(write out better)

Wait, so maybe we can suppose the following: if less radiation is going out, then it means that we’re out of radiation equilibrium, and the average temperature at the earth’s surface will rise. When z *z^* moves upward with rising carbon dioxide concentration, the radiation balance can be restored if the temperature at the new z *z^* rises till it becomes equal to the temperature at the original effective emission height lower in the troposphere. Keeping keeping T(z *)T(z^*) constant under changes of C^\hat{C}, we obtain:

(8)ΔT 0Δlog(C^)=ΓH χ \frac{\Delta T_0}{\Delta \log(\hat{C})} = \Gamma H_\chi

Voila, there we have our logarithmic response of the temperature at the surface. What remains is to evaluate the term ΓH χ\Gamma H_\chi. That’s about 1K/100m×5km1 K/100 m \times 5 km, so we get 50 K. This is pretty big, and we should start to worry, not just about the environment, but rather about our derivation in the last paragraph!

(This is not really correct. T_0 should be the same for all wavelengths.)

Luckily, at this moment we remember that our derivation was only valid for one wavelength, and that the number of 50 K is the temperature response at the wavelengths for which the irradiation is most sensitive to CO2 doubling. Our derivation is correct, but it leads to a brightness temperature, which is only defined for a given wavelength. The real temperature would only rise 50 K if for every wavelength we would have such a strong response. In reality, at most wavelengths the irradiation hardly changes at all, with no effect on the brightness temperature. If the temperature would really rise that much, there would be much more outgoing radiation than incoming radiation.

We rather expect a temperature rise somewhere between 0K and 50 K. How much? Well, unfortunately I don’t know any quick method to derive that… That will have to wait for a more rigorous blog series. The main message from this series was to explain the mechanisms behind the logarithmic response, I didn’t promise we could predict the proportionality factor. But wait, maybe you now say that you agree that the temperature rise will be between 0k and 50K, but how do we know that the logarithmic response is still valid? Well, if every single wavelength responds logarithmically (those that approximately don’t respond, just have a zero factor in front of the logarithm), we would be tempted to assume that the response, integrated over all wavelengths, would remain logarithmically too, not? Remember, I didn’t promise rigor. At this point it’s time to realize that the post is already long enough. See you next time when we’ll consider all wavelengths. Goodbye!