If we open an introductory atmospheric physics textbook, we find that the temperature profile in the troposphere (lower atmosphere) is mostly explained by convection. (See for example, Things Climate Science has Totally Missed? – Convection)
We also find that the temperature profile in the stratosphere is mostly determined by radiation. And that the overall energy balance of the climate system is determined by radiation.
Many textbooks introduce the subject of convection in this way:
- what would the temperature profile be like if there was no convection, only radiation for heat transfer
- why is the temperature profile actually different
- how does pressure reduce with height
- what happens to air when it rises and expands in the lower pressure environment
- derivation of the “adiabatic lapse rate”, which in layman’s terms is the temperature change when we have relatively rapid movements of air
- how the real world temperature profile (lapse rate) compares with the calculated adiabatic lapse rate and why
We looked at the last four points in some detail in a few articles:
Density, Stability and Motion in Fluids - some basics about instability
Potential Temperature - explaining “potential temperature” and why the “potential temperature” increases with altitude
Temperature Profile in the Atmosphere – The Lapse Rate - lots more about the temperature profile in the atmosphere
In this article we will look at the first point.
All of the atmospheric physics textbooks I have seen use a very simple model for explaining the temperature profile in a fictitious “radiation only” environment. The simple model is great for giving insight into how radiation travels.
Physics textbooks, good ones anyway, try and use the simplest models to explain a phenomenon.
The simple model, in brief, is the “semi-gray approximation”. This says the atmosphere is completely transparent to solar radiation, but opaque to terrestrial radiation. Its main simplification is having a constant absorption with wavelength. This makes the problem nice and simple analytically – which means we can rewrite the starting equations and plot a nice graph of the result.
However, atmospheric absorption is the total opposite of constant. Here is an example of the absorption vs wavelength of a minor “greenhouse” gas:
So from time to time I’ve wondered what the “no convection” atmosphere would look like with real GHG absorption lines. I also thought it would be especially interesting to see the effect of doubling CO2 in this fictitious environment.
This article is for curiosity value only, and for helping people understand radiative transfer a little better.
We will use the Matlab program seen in the series Visualizing Atmospheric Radiation. This does a line by line calculation of radiative transfer for all of the GHGs, pulling the absorption data out of the HITRAN database.
I updated the program in a few subtle ways. Mainly the different treatment of the stratosphere – the place where convection stops – was removed. Because, in this fictitious world there is no convection in the lower atmosphere either.
Here is a simulation based on 380 ppm CO2, 1775 ppb CH4, 319 ppb N2O and 50% relative humidity all through the atmosphere. Top of atmosphere was 100 mbar and the atmosphere was divided into 40 layers of equal pressure. Absorbed solar radiation was set to 240 W/m² with no solar absorption in the atmosphere. That is (unlike in the real world), the atmosphere has been made totally transparent to solar radiation.
The starting point was a surface temperature of 288K (15ºC) and a lapse rate of 6.5K/km – with no special treatment of the stratosphere. The final surface temperature was 326K (53ºC), an increase of 38ºC:
The ocean depth was only 5m. This just helps get to a new equilibrium faster. If we change the heat capacity of a system like this the end result is the same, the only difference is the time taken.
Water vapor was set at a relative humidity of 50%. For these first results I didn’t get the simulation to update the absolute humidity as the temperature changed. So the starting temperature was used to calculate absolute humidity and that mixing ratio was kept constant:
The lapse rate, or temperature drop per km of altitude:
The flux down and flux up vs altitude:
The top of atmosphere upward flux is 240 W/m² (actually at the 500 day point it was 239.5 W/m²) – the same as the absorbed solar radiation (note 1). The simulation doesn’t “force” the TOA flux to be this value. Instead, any imbalance in flux in each layer causes a temperature change, moving the surface and each part of the atmosphere into a new equilibrium.
A bit more technically for interested readers.. For a given layer we sum:
- upward flux at the bottom of a layer minus upward flux at the top of a layer
- downward flux at the top of a layer minus downward flux at the bottom of a layer
This sum equates to the “heating rate” of the layer. We then use the heat capacity and time to work out the temperature change. Then the next iteration of the simulation redoes the calculation.
And even more technically:
- the upwards flux at the top of a layer = the upwards flux at the bottom of the layer x transmissivity of the layer plus the emission of that layer
- the downwards flux at the bottom of a layer = the downwards flux at the top of the layer x transmissivity of the layer plus the emission of that layer
End of “more technically”..
Anyway, the main result is the surface is much hotter and the temperature drop per km of altitude is much greater than the real atmosphere. This is because it is “harder” for heat to travel through the atmosphere when radiation is the only mechanism. As the atmosphere thins out, which means less GHGs, radiation becomes progressively more effective at transferring heat. This is why the lapse rate is lower higher up in the atmosphere.
Now let’s have a look at what happens when we double CO2 from its current value (380ppm -> 760 ppm):
Figure 6 – with CO2 doubled instantaneously from 380ppm at 500 days
The final surface temperature is 329.4, increased from 326.2K. This is an increase (no feedback of 3.2K).
The “pseudo-radiative forcing” = 18.9 W/m² (which doesn’t include any change to solar absorption). This radiative forcing is the immediate change in the TOA forcing. (It isn’t directly comparable to the IPCC standard definition which is at the tropopause and after the stratosphere has come back into equilibrium – none of these have much meaning in a world without convection).
Let’s also look at the “standard case” of an increase from pre-industrial CO2 of 280 ppm to a doubling of 560 ppm. I ran this one for longer – 1000 days before doubling CO2 and 2000 days in total- because the starting point was less in balance. At the start, the TOA flux (outgoing longwave radiation) = 248 W/m². This means the climate was cooling quite a bit with the starting point we gave it.
At 180 ppm CO2, 1775 ppb CH4, 319 ppb N2O and 50% relative humidity (set at the starting point of 288K and 6.5K/km lapse rate), the surface temperature after 1,000 days = 323.9 K. At this point the TOA flux was 240.0 W/m². So overall the climate has cooled from its initial starting point but the surface is hotter.
This might seem surprising at first sight – the climate cools but the surface heats up? It’s simply that the “radiation-only” atmosphere has made it much harder for heat to get out. So the temperature drop per km of height is now much greater than it is in a convection atmosphere. Remember that we started with a temperature profile of 6.5K/km – a typical convection atmosphere.
After CO2 doubles to 560 ppm (and all other factors stay the same, including absolute humidity), the immediate effect is the TOA flux drops to 221 W/m² (once again a radiative forcing of about 19 W/m²). This is because the atmosphere is now even more “resistant” to the escape of heat by radiation. The atmosphere is more opaque and so the average emission of radiation of space moves to a higher and colder part of the atmosphere. Colder parts of the atmosphere emit less radiation than warmer parts of the atmosphere.
After the climate moves back into balance – a TOA flux of 240 W/m² – the surface temperature = 327.0 K – an increase (pre-feedback) of 3.1 K.
Compare this with the standard IPCC “with convection” no-feedback forcing of 3.7 W/m² and a “no feedback” temperature rise of about 1.2 K.
Figure 7 – with CO2 doubled instantaneously from 280ppm at 1000 days
Then I introduced a more realistic model with solar absorption by water vapor in the atmosphere (changed parameter ‘solaratm’ in the Matlab program from ‘false’ to ‘true’). Unfortunately this part of the radiative transfer program is not done by radiative transfer, only by a very crude parameterization, just to get roughly the right amount of heating by solar radiation in roughly the right parts of the atmosphere.
The equilibrium surface temperature at 280 ppm CO2 was now “only” 302.7 K (almost 30ºC). Doubling CO2 to 560 ppm created a radiative forcing of 11 W/m², and a final surface temperature of 305.5K – that is, an increase of 2.8K.
Why is the surface temperature lower? Because in the “no solar absorption in the atmosphere” model, all of the solar radiation is absorbed by the ground and has to “fight its way out” from the surface up. Once you absorb solar radiation higher up than the surface, it’s easier for this heat to get out.
One of the common themes of fantasy climate blogs is that the results of radiative physics are invalidated by convection, which “short-circuits” radiation in the troposphere. No one in climate science is confused about the fact that convection dominates heat transfer in the lower atmosphere.
We can see in this set of calculations that when we have a radiation-only atmosphere the surface temperature is a lot higher than any current climate – at least when we consider a “one-dimensional” climate.
Of course, the whole world would be different and there are many questions about the amount of water vapor and the effect of circulation (or lack of it) on moving heat around the surface of the planet via the atmosphere and the ocean.
When we double CO2 from its pre-industrial value the radiative forcing is much greater in a “radiation-only atmosphere” than in a “radiative-convective atmosphere”, with the pre-feedback temperature rise 3ºC vs 1ºC.
So it is definitely true that convection short-circuits radiation in the troposphere. But the whole climate system can only gain and lose energy by radiation and this radiation balance still has to be calculated. That’s what current climate models do.
It’s often stated as a kind of major simplification (a “teaching model”) that with increases in GHGs the “average height of emission” moves up, and therefore the emission is from a colder part of the atmosphere. This idea is explained in more detail and less simplifications in Visualizing Atmospheric Radiation – Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions.
A legitimate criticism of current atmospheric physics is that convection is poorly understood in contrast to subjects like radiation. This is true. And everyone knows it. But it’s not true to say that convection is ignored. And it’s not true to say that because “convection short-circuits radiation” in the troposphere that somehow more GHGs will have no effect.
On the other hand I don’t want to suggest that because more GHGs in the atmosphere mean that there is a “pre-feedback” temperature rise of about 1K, that somehow the problem is all nicely solved. On the contrary, climate is very complicated. Radiation is very simple by comparison.
All the standard radiative-convective calculation says is: “all other things being equal, an doubling of CO2 from pre-industrial levels, would lead to a 1K increase in surface temperature”
All other things are not equal. But the complication is not that somehow atmospheric physics has just missed out convection. Hilarious. Of course, I realize most people learn their criticisms of climate science from people who have never read a textbook on the subject. Surprisingly, this doesn’t lead to quality criticism..
On more complexity – I was also interested to see what happens if we readjust absolute humidity due to the significant temperature changes, i.e. we keep relative humidity constant. This led to some surprising results, so I will post them in a followup article.
Note 1 – The boundary conditions are important if you want to understand radiative heat transfer in the atmosphere.
First of all, the downward longwave radiation at TOA (top of atmosphere) = 0. Why? Because there is no “longwave”, i.e., terrestrial radiation, from outside the climate system. So at the top of the atmosphere the downward flux = 0. As we move down through the atmosphere the flux gradually increases. This is because the atmosphere emits radiation. We can divide up the atmosphere into fictitious “layers”. This is how all numerical (finite element analysis) programs actually work. Each layer emits and each layer also absorbs. The balance depends on the temperature of the source radiation vs the temperature of the layer of the atmosphere we are considering.
At the bottom of the atmosphere, i.e., at the surface, the upwards longwave radiation is the surface emission. This emission is given by the Stefan-Boltzmann equation with an emissivity of 1.0 if we consider the surface as a blackbody which is a reasonable approximation for most surface types – for more on this, see Visualizing Atmospheric Radiation – Part Thirteen – Surface Emissivity - what happens when the earth’s surface is not a black body – useful to understand seeing as it isn’t..
At TOA, the upwards emission needs to equal the absorbed solar radiation, otherwise the climate system has an imbalance – either cooling or warming.