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Black body limits: climate sensitivity parameter can't possibly be high, a proof

Originally posted on March 29th. Put on top as the most discussed recent article.

Let me show you another simple way to see that the climate sensitivity can't really be close to the IPCC's mean value (and certainly not close to the upper end of their interval).

A black body minimizes the reflection which is why it maximizes thermal emission at the same wavelength (heat it to 6000 K first). The emissivity may only be reduced - by a light coating or by an atmosphere with a greenhouse gas - but it can't be increased.

The reason I will be able to show that the IPCC figure contradicts the basic laws of physics is that we will look how the relevant theories behave at somewhat more extreme temperatures which are nevertheless physically possible - and not too extreme. We will see that the IPCC estimates require a behavior that contradicts some universal laws of Nature - namely the fact that the emissivity can't exceed one.

First, we must begin with some definitions. Let us distinguish two concepts, the climate sensitivity and the climate sensitivity parameter. The climate sensitivity is the expected increase of near-surface air temperature from a CO2 doubling. It will be claimed to be near 1.1 °C.

The climate sensitivity parameter, λ (pronounce "lambda"), is more general and isn't necessarily connected with CO2. It is the change of the near-surface air temperature, ΔT, that you obtain from a unit change of the radiative forcing, RF - which is the net irradiance measured at the tropopause (the boundary between troposphere and stratosphere, i.e. the upper boundary of the atmosphere where "weather occurs"):

ΔT = λ . RF
The idea is that if you add some extra incoming energy through the troposphere (e.g. increased solar energy), or if you remove some outgoing energy, the Earth will find a new, warmer equilibrium. And the ratio between the warming that you get after the system gets stabilized and the forcing - in W/m^2 - is the climate sensitivity parameter.

The terminology, "climate sensitivity parameter", is borrowed from the 2007 IPCC report and I hope you will forgive me this linguistic choice. I will not use the actual scientific content because it is unusable, as we will see.

Now, let's think how the Earth would behave if you tested its behavior in much cooler temperatures - between 0 K (-273.15 °C) and the current global mean temperature, 288 K (15 °C). The climatologists rarely test their ideas in these more extreme conditions - which prevents them from seeing that their assumptions are physically stretched if not impossible.

But physicists like to be sure that their theories don't predict inconsistent phenomena. So I will assume that the dear reader believes me that the minimal temperature of "normal objects" that can be achieved is close to 0 K or -273.15 °C. That's the temperature at which the thermal radiation goes away.

We will ask a fundamental question: change the total incoming energy per unit area that crosses the tropopause, with the photons that get scattered and reflected not being counted, from 0 W/m^2 to the current value of 235 W/m^2. Alternatively, this figure may be viewed as the total outgoing thermal radiation of the Earth. The total average energy coming from the Sun is 342 W/m^2 but 107 W/m^2 is scattered and/or reflected so only 235 W/m^2 is captured and "transformed" by the Earth: these 235 W/m^2 are re-emitted as the thermal infrared radiation.

See Earth's energy budget.

We want to know how the near-surface global mean temperature is being changed as a function of the energy flow. Everyone should know what the behavior of the black body is: the total emitted energy is proportional to the fourth power of the absolute temperature (in Kelvin degrees).
RF = c T4
OK? How can you find out the "slope" of the curve at the real temperature around 288 K? Well, just differentiate the equation above:
d RF / d T = 4c T3 = 4 RF / T
You may invert the ratio to find that
d T / d RF = T / (4 RF) =
= 288 K / (4 . 235 W/m2) =
= 0.3 °C / (W/m2)
Note that the temperature differences of 1 K and 1 °C are the same. So 0.3 - with the right units - is the black body estimate for the climate sensitivity parameter. The IPCC mean value is something like 0.8, almost thrice as large.

The feedback-free value of the radiative forcing from a doubled CO2 is 3.7 W/m^2 and it is calculable. If you multiply it by 0.3, you get 1.1 °C. That's almost exactly equal to 1.2 °C, the no-feedback climate sensitivity!

In other words, the calculable climate models without the feedbacks almost exactly agree about the climate sensitivity with the assumption that the radiation-temperature relationship obeys a power law with the Stefan-Boltzmann exponent (four). On the other hand, the tripled IPCC value of the climate sensitivity parameter obviously gives you a tripled climate sensitivity, too: that's where those 3 °C come from.

Now, we're gonna check how several graphs look like. This is the graph of the fourth power.

The x-axis goes from 0 K to 288 K while the y-axis goes from 0 to 235 W/m^2. Note that the left portion of the graph is extremely flat: the growth of the fourth power starts "almost suddenly". For example, at one-half of the current temperature, namely at 144 K which is -129 °C, the energy flow is just 1/16 of the current value, less than 15 W/m^2.

The radiation emitted by cool object is really small.

If you want to get to the IPCC value of the climate sensitivity parameter which is about 3 times higher than the black body value, you will have to change the slope of the curve at the right upper corner. The simplest function that can do the job is another power law: but the exponent has to be 1.3 instead of 4.

The black-body curve is the lower boundary of the khaki region; the curve matching the IPCC slope is the upper boundary which is nearly flat. Can you spot the difference?

We must realize that when we discuss a small interval of temperatures around 15 °C, it's easy to inflate various numbers by a factor of three and pretend that everything is fine. However, if you try to use the model for a broader interval of temperatures or energy flows, you will easily see that the IPCC figure is unphysical.

First, let us see that the particular simple power law above - with the exponent being 1.3 - is absurd. For example, when the temperature is those 144 K i.e. -129 °C, the black-body-like i.e. fourth-power-law value of the energy flows gave us about 15 W/m^2.

If you use the exponent being 1.3, the energy flow will be (1/2)^1.3 = 40.6 percent of 235 W/m^2 i.e. 95 W/m^2. That's more than six times higher than our "black body value". Is it physically possible?

The answer is a resounding No. But we must be a little bit more careful about the black bodies.

So far, I have been using the term "black body" for any object that respects the fourth power, Stefan-Boltzmann relationship between the temperature and the emitted thermal energy. However, the overall coefficient didn't have the proper black-body value.

If you apply the Stefan-Boltzmann law to the temperatures we mention, you obtain:
288 K i.e. 15 °C: 390 W/m^2
144 K i.e. -129 °C: 24 W/m^2
You see that these energy flows are less than twice larger than the "black-body" figures we had derived from the known figures at 288 K and from the extrapolation by the fourth power.

However, at -129 °C, the exact black-body radiation produces 24 W/m^2. This is actually the ultimate upper limit of the thermal energy radiated by a -129 °C object because the emissivity can never exceed one!

I should explain some insights about the physics of radiation by non-black bodies. The general bodies at temperature T emit less than the black body.

The ratio (real radiation over the black body one), called the emissivity, turns out to be exactly equal to the absorptivity - the percentage of electromagnetic radiation absorbed at a given temperature, angle, and wavelength. This equality is called the Kirchhoff's law of thermal radiation and has already been known since the years of classical physics, namely from 1859. It follows from the detailed balance - a kind of properly calculated time-reversal symmetry (symmetry of the microscopic laws between the past and the future) applied to the radiation phenomena - and from the equilibrium.  
The more a piece of material reflects a given type of light, the less it will radiate thermal radiation through this light. But just like a material can't absorb more than 100% of the incoming radiation, it can't emit more than 100% of the radiation that the equally warm black body would emit at the same temperature.

While the thermal radiation at -129 °C can't exceed 24 W/m^2 because of the most basic and universal laws of physics, the power-law needed to reproduce the IPCC mean climate sensitivity required us to believe that the thermal radiation actually gives almost four times as much at this temperature!

This discrepancy is huge, indeed. I went to 144 K i.e. -129 °C because the numbers were simpler: the absolute temperature was reduce to 1/2 of the current value. However, there's nothing wrong about these temperatures. After all, most of the atmosphere is composed of nitrogen. Its boiling point is 77 K i.e. -196 °C: so even at those 144 K, there would be a similar atmosphere to ours, a gassy one.

However, the discrepancy between the IPCC value and the basic laws of physics was so huge that it should be obvious to you that I didn't really have to go to -129 °C. So let's add one more curve to our graph: the exact black-body curve that defines the upper limit of thermal emissivity.

Because emissivity can't be greater than one, the whole black region in the middle of the graph - and anything above the upper boundary of the purple region - is actually impossible. Because I have already drawn a couple of graphs, it's useful to mention the Mathematica code for the graph above so that you can check what I am doing:
Plot[{235*(x/288)^4, 390*(x/288)^4, 235*(x/288)^1.3}, {x, 0, 288}, Filling -> {1 -> {2}}]
If you calculate where the forbidden black region starts (I mean its corner on the right, warmer side), it's at 239 K i.e. -34 °C. That's a pretty imaginable temperature! In fact, The Catlin Arctic Survey had to work in the wind chill of -50 °C today, another heat wave (well-known for them by now) that was apparently caused by global warming. ;-)

The challenge for the IPCC is simple: try to determine the amount of thermal radiation by the Earth when the global mean temperature drops to -34 °C (imagine a thought experiment in which the Earth is moved slightly away from the Sun). Or try to draw the whole function that should replace the grey body of mine.

I claim that their energy flows don't decrease sufficiently quickly when the temperature drops (because they don't increase sufficiently quickly when the temperature goes up) - and the problem is so bad that already at -34 °C, their emissivity has to exceed the maximum value allowed by the laws of physics.

Needless to say, the problem becomes even worse when you try to get the climate sensitivity around 4 or 5 °C. The IPCC's "flat line" actually becomes inverse U-shaped. It will be even more horizontal in the physical region and it will be even harder for the IPCC curve to stay within the allowed purple strip.

Is there any conceivable way to defend the possibility of a higher climate sensitivity comparable to the IPCC mean value against my new devastating argument?

Clearly, in order to keep the high sensitivity, the "flat line" in the graph above has to keep the same slope on the right side of the picture because this slope is what defines the climate sensitivity parameter. However, at the same moment, as you reduce the temperature, you must keep the curve within the purple region. As you're going to the left, you must brutally bend the curve to the bottom. Can such a resulting curve be defensible or even realistic?

I don't think so.

Recall that in order to get the climate sensitivity at (or above) 3 °C, the climate sensitivity parameter must be found at (or above) 0.8 °C / (W/m^2). For a little while, the right curve has to mimic the "flat line" in the graph above. However, as you move towards cooler temperatures, you're not allowed to jump above the upper boundary of the purple region.

So when you get to the global mean temperature being something like 239 K i.e. -34 °C, the slope of your curve already has to agree with the slope of the "exact black body". What is this slope? Well, it's again close to 0.3 °C / (W/m^2), the very same value of the climate sensitivity parameter I assumed to be correct everywhere, anyway.

So the realistic "gray body" estimate claims that the climate sensitivity parameter is 0.3 °C / (W/m^2) at the current temperature of 288 K. It's actually higher at lower temperatures. At the beginning, we calculated "d T / d RF" and it goes like "1/T^3". So at 239 K, the climate sensitivity parameter calculated from the realistic black-body-like fourth power is
λ = 0.3 / (239/288)3 = 0.53 °C/(W/m^2)
So at 239 K, the climate sensitivity parameter should actually be 1.75 times greater than what it is at 288 K.

On the other hand, for the unknown IPCC function to produce high sensitivity of 3 °C but to avoid a contradiction with the limit on emissivity, the value of λ at 288 K has to be 0.8 °C / (W/m^2) while at 239 K, it has to drop to 0.3 °C / (W/m^2) which is 2.6 times higher!

So the ratio of λ's calculated at 15 °C and -34 °C is 1.75 for the black body but it is 1/2.6 for the curve needed to defend the IPCC. So the double ratio is 2.6 * 1.75 = 4.55! For you to believe that the climate sensitivity is 3 °C or so, you have to believe that there exists an additional temperature-dependent correction to the "grey body factors" describing the Earth that changes by a factor of 4.55 when you go from -34 °C to 15 °C.

Something that dictates how the Earth's thermal-radiative behavior deviates from the very fourth law dictated by the black body - and universal dimensional analysis - has to brutally change in this modest interval.

Yes, I think that the probability of something like that occurring to the laws of physics in this "mundane" interval of temperatures is extremely tiny. For the climate sensitivity parameter to "speed up" in this brutal way around 15 °C, the required effects would probably have to be linked to the freezing of ice.

However, -34 °C is not too far from the freezing point, either. So it's hard to believe why this "acceleration" of the climate sensitivity parameter by a factor of 4.55 should affect the Earth near 15 °C but not near -34 °C (global mean temperatures). Moreover, only a tiny fraction of the Earth remains frozen - or covered by snow. So if the hypothetical acceleration were linked to some kind of ice-albedo effect, such an effect couldn't really influence most of the Earth's surface.

My argument above can also be reverted and used as an argument against too low a sensitivity. The only major change I would have to do is to study the temperature above 15 °C rather than below 15 °C.

For example, if you wanted to believe that the climate sensitivity parameter is 0.15 °C / (W/m^2), one half of the value calculated from the fourth law, something needed to give you 0.5 °C of the climate sensitivity, you would need something like the 8th power instead of the 1.3th power we used for the IPCC guess. And this power would exceed the tolerated exact black-body limit on the emissivity near 327 K i.e. 54 °C as well as higher temperatures. A pretty dramatic "fudge factor" changing the behavior from 15 °C to 54 °C would also be needed.

That's also pretty bad, I think. I don't think that the Earth's behavior "qualitatively changes" between 15 °C and 54 °C.

So I believe that the powerful argument above may be used against any hypothesis claiming that the climate sensitivity parameter is too different from 0.3 °C/(W/m^2), or that the climate sensitivity to CO2 doubling is too different from 1.1 °C.

And that's the memo.

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snail feedback (4) :

reader Colin said...

This was an interesting read.
I don't entirely agree with the thesis, as the author concerns himself with the behaviour of gases in isolation from the Surface and the additional confounding factor of state changes of water.

My argument runs:
1. The lower atmosphere is entirely heated by energy fluxes from the surface. These are Direct Conduction (24W/m^2), Radiation (26W/m^2) and Condensing Water Vapour(78W/m^2). (Figures from IPCC AR4, WG1, Chapter 1).
2. If the temperature of the air next to the surface drifts away from the temperature of the surface, the fluxes alter in a way which tends to oppose the drift. Basically the air near the surface is tied closely to the surface temperature.
3. Although the exact change in evaporation with temperature is in dispute, the consensus is that it is between 2.5% and 6.5% per DegC. If you calculate the sensitivity of the Surface, using the flux balance equation, you get between 0.095 and 0.15 DegC/W/m^2. (This is half the sensitivity calculated by the author of the article, and half the sensitivity of the atmosphere at the Tropopause).
4. Because the temperature of the air at the surface is tied to the temperature of the surface, its sensitivity must be the same as the surface.
5. Energy imbalances at the tropopause or anywhere high in the atmosphere due to increased concentration of CO2 do not affect the surface, which is thermally and radiatively isolated from the upper atmosphere. The high sensitivity of the cold thin upper atmosphere does not control or affect the low sensitivity of the warm dense surface air.

reader DeWitt said...

Are you sure you haven't oversimplified your model? The planet isn't an isothermal gray body. Temperature is expected to increase more rapidly at high latitudes where lapse rate feedback due to atmospheric expansion, water vapor pressure and ice/albedo feedback will be greatest. Night time temperature is also expected to increase more rapidly than during the day. The brightness temperature of the planet as observed from space is not going to change very much except for a small decrease due to the time lag from the higher heat capacity of the ocean compared to land resulting in a small (less than 1 W/m2) radiative imbalance. Like Steve McIntyre, I would like to see a detailed, engineering quality explanation of how one gets from ~0.3 to ~0.8 degrees/W/m2, but at least I can see how one might get there.

reader Stefan said...

Dea Lubos,
The derivation of the radiative forcing from a doubled CO2 (3.7 W/m^2 ) mentionned in your interstic memois based on the atmospheric lapse rate in the troposphere. The central, strongest line of CO2(lambda=14 microns) is emitted from layers in the stratosphere where the lapse rate is positve, this would lead to the negative radiative forcing, possible to the cancelling of the troposphere. In the Houghton`s textbook we read tha this leads to the cooling of the stratosphere. Most climatoligists assume that the troposphere and the statosphere cannot exchange the heat. Can the temperature difference between tropo- and stratospehre rise without limits when we increase and increase the CO2 concentration?

reader Luboš Motl said...

Dear Stefan, thanks for your comments. It takes time to answer everyone - and I haven't calculated your problem precisely so someone may have a better answer.

But: the troposphere and stratosphere don't exchange heat by "contact" i.e. heat conduction because the lapse rate i.e. the gradient of temperature is zero at their boundary called the tropopause.

This doesn't prevent the radiation from going in both ways but an equilibrium is imposed pretty quickly. But I am not quite sure why you think that the emission in the stratosphere affects the radiative forcing. At equilibrium, it's canceled by parts of the energy flow in the opposite direction.

Also, it doesn't affect my quantity which is IR from the Earth crossing the tropopause on the way out, so emission from the stratosphere is also - deliberately - removed from my quantity.

Yes, rising CO2 (or, more realistically, water) should be able to increase the stratosphere-troposphere temperature difference, but it also moves the tropopause. However, clouds - out of H2O - can also change it. It's a complex problem.

The difference surely can't be "unlimited". At various temperatures that are too low, gases start to condense etc. At too high temperatures, they do other things. But yes, you can get those few degrees from this effect. The required changes of greenhouse gases are exponential - the temperature changes are just logarithmic - so you won't ever get too far. Yes, if you multiply the greenhouse gases by 100,000, you may get close to Venus and heat the Earth by hundreds of degrees. It depends what you're talking about.


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