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Why is the greenhouse effect logarithmic?

Steve McIntyre at climateaudit.org is trying to locate the provenance ;-) of the logarithmic formula for the greenhouse effect. Instead of joining him, let me post my explanation why I personally think that the idealized greenhouse warming is a logarithmic function of the concentration under semi-realistic idealized assumptions. This posting is a technical supplement for

Dynamics of greenhouse effect
Sublinear CO2 climate sensitivity
Let us first notice that as Wikipedia explains, Svante Arrhenius published a paper in April 1896 that attempted to quantify the warming caused by "carbonic acid" (the term was used even for CO2 itself at that time). And he abused the effect to incorrectly explain the alternation of ice ages and interglacials (the reasons currently believed to be relevant are the Milankovitch cycles i.e. fluctuations of the geometry of Earth's orbit and axis and perhaps some waves inside the Sun).

Because this blog likes to offer you the original sources, here it is:
Svante Arrhenius: On the influence of carbonic acid in the air upon the temperature of the ground (PDF, 1896)
I admit that I am impressed by this paper. It is as technical and as detailed as the fourth IPCC report except that it was written by 1 man instead of 2500 people and it was written 111 years earlier. Much like the IPCC, Arrhenius obtained something comparable to 5 °C for the climate sensitivity and just like the IPCC, his numerical result was wrong. The only difference from the IPCC is that climatologists already agree that Arrhenius' results were wrong.

A footnote on page 238 contains logarithms but it doesn't talk about the CO2 concentration. Instead, you must go to the top of page 267 which says:
... Thus if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression. ...
This rule - one that Arrhenius only deduced "experimentally" on page 266, without a derivation - says that the warming in Celsius degrees is proportional to the logarithm of the ratio of the initial and final concentrations:
Delta T = alpha log(C/C0)
That's why it is slowing down as the concentration increases, much like the effect of 10th painting of your bedroom. And that's also one of the reasons why our worries should be diminishing even if the CO2 production stays constant. The logarithmic formula guarantees that even though we will probably produce substantially (twice or thrice) more CO2 in the 21st century than we did in the 20th century, it will contribute - via the greenhouse effect - roughly the same amount to the warming.

Clearly, the law is not completely universal. It breaks down for very small C because the logarithm would otherwise go to -infinity. For small C, the correct relationship becomes asymptotically linear: each molecule of CO2 causes pretty much the same warming if you only have a couple of them. Phenomenologically, the linear increase for small C and the logarithmic law for the large values of C is often interpolated by a function that is also quadratic in the middle but it is just one of possible conventions for curve-fitting. Another popular function is
Temperature = Temperature0 + ln(1 + 1.2 x + 0.005 x2 + 0.0000014 x3)
where "x" is the CO2 concentration in ppmv. This formula works pretty well up to 1,000 ppmv.

However, the asymptotic logarithmic behavior for large C is more than a convention. It can be derived as a result of an idealized calculation that is relatively realistic - a kind of calculation that theoretical physicists, especially condensed-matter physicists, should like. One reason for the logarithm could be found if we were looking how new spectral lines and their "wings" become relevant for the absorption. The old lines eventually get saturated but the total greenhouse warming never quite stops because new spectral lines emerge: it just slows down. However, below we focus on a different effect related to the lapse rate that, I believe, is dominant for the dependence of the greenhouse effect on the concentration.

The setup

If the sensitivity with respect to the CO2 doubling is 1 °C, the warming obtained from each multiplication of the CO2 concentrations by a factor of "e=2.718..." should be around 1 °C / ln(2) = 1.44 °C. That's the quantity that we would like to derive here.

Consider the Earth with CO2 only. The density of CO2 decreases exponentially with the height, being proportional to the Maxwell-Boltzmann factor exp(-height/height_0) where height_0 is something over 5 kilometers, I don’t know exactly, for CO2. The precise number doesn't matter for the qualitative result.

This exponential decrease is a standard result of college thermodynamics, coming from the maximization of entropy of a gas given a conserved energy. Fellow readers can remind you about the derivation of the result from statistical physics if you need it.

Lifting the tropopause

Now, if you increase the total concentration of CO2 e-times, the level where the concentration is equal to a reference value, say C_r, increases exactly by height_0 in the direction up. I conveniently choose C_r to be a representative for the concentration above which the whole atmosphere may be considered transparent for the infrared radiation we consider, with some accuracy. The height where this concentration is C_r may be referred to as the tropopause, the boundary between the troposphere and the stratosphere above it. It is somewhat fuzzy but I can choose a convention about the percentage how transparent it should be, and then the tropopause will be a well-defined sharp shell. For example, define the tropopause as the plane such that the whole atmosphere above it only absorbs 10% of the black body radiation corresponding to the temperature of Earth.

The fun is that the behavior around the tropopause is pretty much universal, regardless of its height. The other assumption I need to use is a pretty much constant lapse rate - the decrease of the temperature with the height above the Earth. This is another law I need to assume, with all disclaimers about its inaccuracy etc. The lapse rate law holds because it is a form of the adiabatic law.

So if the multiplication of the total CO2 volume by "e" lifted the tropopause by height_0, the temperature at the tropopause dropped additively by the "lapse_rate times height_0". Because the lapse rate is about -5 °C per kilometer, you will get approximately 25 °C decrease of the tropopause temperature from multiplying CO2 by "e".

A linear decrease of the temperature means that the radiation that is emitted by the tropopause decreases by a linear term, too.

Now, I must impose the overall equilibrium of incoming and outgoing energy in order to balance the Earth's energy budget. So if the tropopause radiation dropped by a certain amount E and the incoming solar radiation is unchanged, the radiation directly from the Earth surface must increase by E to compensate the drop from the tropopause, which means that the surface temperature must increase by a linear piece.

Putting the arguments together

So if you combine all these things, you see that a geometric increase of the total CO2 volume - and I could have divided the e-folding into several smaller fixed percentage increases - means a linear increase of the surface temperature. This conclusion is valid assuming that various linear relationships mentioned above hold.

So the lapse rate should be pretty well-defined i.e. constant between the old and new tropopause; the change of the percentage of the energy emitted by the surface vs tropopause should be much smaller than 100%; the predicted change of the temperature should be much smaller than the absolute temperature of the surface, and several other limiting assumptions that you might realize should be satisfied, too. Then the linearizations mentioned above are legitimate.

Don't forget that the logarithm of a power is still proportional to the logarithm so the logarithmic shape for high enough concentrations is probably more robust than you might a priori think.

With the assumptions listed above, and they are kind of - although not perfectly - satisfied for the doubling from 280 to 560 ppm of CO2 as one can check (the temperature change comparable to 1 °C is much smaller than the 300K absolute temperature, the percentages change from 92:8 to 95:5 or something like that is relatively small), the Arrhenius' law is a law. It is all about the Maxwell-Boltzmann distribution, the lapse rate, and the black body law. A geometric/exponential increase of the concentration moves the physical phenomena linearly in altitude and makes standardized linear contributions to various terms.

A rough numerical calculation

Let us try to end up with the 1 °C sensitivity. First of all, as we have already suggested, fundamental physicists respect "e" and not "2" as the right base of exponentials and logarithms so the goal will be to show that multiplying CO2 volume by "e" will warm up Earth by a certain amount comparable to 1 °C / ln(2) = 1.44 °C. Let’s see how close to 1.44 °C for this e-normalized climate sensitivity we can get.

With the e-multiplication of CO2 (between 1800 and 2150 or so, assuming fossil fuels to go on), the tropopause shifts by height_0 = 5 km, the temperature at the tropopause drops by 25 °C. If the tropopause and the surface were emitting 50% of the radiation each, then the surface would have to warm up by 25 °C: with this change, the decrease of the thermal radiation by the cooler troposphere would be compensated by the increase of the thermal radiation from a warmer surface. However, 25 °C would indeed be a pretty high, catastrophic e-sensitivity. Fortunately, the surface emits a vast majority of the radiation, so a small increase of the surface temperature is enough to compensate the small cooling at the tropopause.

Assuming the average percentage composition of the radiation from surface vs tropopause to be 94:6, you see that the Earth is 17 times more important than the tropopause for the energy budget. So you need to change the Earth surface temperature by 25 °C / 17 in the opposite direction to compensate them which is 1.47 °C. A pretty good agreement. OK, I cheated a bit by saying that the effective distribution was 94:6 but what is important is the framework of the calculation and the qualitative logarithmic form of the result. You may try to put better numbers into it if you want to improve it.

You should also think how you could properly incorporate heat convection and some basic influence of different forms of water in the atmosphere.

Climate models: gullibility vs cynicism

Finally, I would like to write a few sentences about "what is known". Although the derivation above is a caricature primarily designed to understand some qualitative features of the greenhouse effect and make some order-of-magnitude estimates, I am convinced that the contemporary climate models should be able to get the right results for the flow of radiation and its absorption and emission by CO2 at different altitudes (unless all of their creators are doing something really silly). For example, the phenomenological formulae written above were constructed to agree with the climate models.

In my opinion, doubts about the climate models only start to be legitimate once we include clouds, precipitation, turbulence of both the atmosphere and the ocean, and other "non-uniform" and "time-dependent" features of the climate.

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snail feedback (1) :

reader Rami Niemi said...

How about this: