This post covers a dull subject. If you are new to Science of Doom, the subject matter here will quite possibly be the least interesting in the entire blog. At least, up until now. It’s possible that new questions will be asked in future which will compel me to write posts that climb to new heights of breath-taking dullness.
So commenters take note – you have a duty as well. And new readers, quickly jump to another post..
In an earlier post – Why Global Mean Surface Temperature Should be Relegated, Or Mostly Ignored – we looked at the many problems of trying to measure the surface of the earth by measuring the air temperature a few feet off the ground. And also the problems encountered in calculating the average temperature by an arithmetic mean. (An arithmetic mean for those not familiar with the subject is the “usual” and traditional averaging where you add up all the numbers and divide by how many values you had).
We looked at an example where the average temperature increased, but the amount of energy radiated went down. Energy radiated out would seem to be a more useful measure of “real temperature” so clearly arithmetic averages of temperature have issues. This is how GMST is calculated – well not exactly, as the values are area-weighted, but there is no factoring in of how surface temperature affects energy radiated.
But in the discussion someone brought up emissivity and what effect it has on the calculation of energy radiated. So in the interests of completeness we arrive here.
Emissivity of the Earth’s Surface
Our commenter asked:
So what are the non-black body corrections required for the initial calculation 396W/sqm? And what are the corrections for the equivalent temperature calculation? And do they cancel out (I think not due to the non-linearity issue) ?
What’s this about? (Of course, read the earlier post if you haven’t already).
Energy radiated from a body, E=εσT4
where T is absolute temperature (in K), σ=5.67×10-8 and ε is the emissivity.
ε is a value between 0 and 1, and 1 is the “blackbody”. The value – very important to note – is dependent on wavelength.
So the calculations I showed (in the thought experiment) where temperature went up but energy radiated went down need adjustment for this non-blackbody emissivity.
How Emissivity Changes
Here we consult the “page-turner”, Surface Emissivity Maps for use in Satellite Retrievals of Longwave Radiation by Wilber (1999).
And yet more graphs at the end of the post – spreading out the excitement..
Note the key point, in the wavelengths of interest emissivity is close to 1 – close to a blackbody.
For beginners to the subject, who somehow find this interesting and are therefore still reading, the wavelengths in question: 4-30μm are the wavelengths where most of the longwave radiation takes place from the earth’s surface. Check out CO2 – An Insignificant Trace Gas? for more on this.
I did wonder why the measurements weren’t carried on to 30μm and as far as I can determine it is less interesting for satellite measurements – because satellites can see the surface the best in the “atmospheric window” of 8-14μm.
So with the data we have we see that generally the value is close to unity – the earth’s surface is very close to a “blackbody”. Energy radiated in 4-16μm wavelengths only account for 50-60% of the typical energy radiated from the earth’s surface, so we don’t have the full answer. Still with my excitement already at fever pitch on this topic I think others should take on the task of tracking down emissivity of representative earth surface types at >16μm and report back.
So we have some ideas of emissivities, they are not 1, but generally very close. How does this affect the calculation of energy radiated?
Not much effect.
I took the original example with 7 equal areas at particular temperatures for 1999 and show emissivities (these are arbitrarily chosen to see what happens):
- Equatorial region: 30°C ; ε = 0.99
- Sub-tropics: 22°C, 22°C ; ε = 0.99
- Mid-latitude regions: 12°C, 12°C ; ε = 0.80
- Polar regions: 0°C, 0°C ; ε = 0.80
The average temperature, or “global mean surface temperature” = 14°C.
And in 2009 (same temperatures as in the previous article):
- Equatorial region: 26°C ; ε = 0.99
- Sub-tropics: 20°C, 20°C ; ε = 0.99
- Mid-latitude regions: 12°C, 12°C ; ε = 0.80
- Polar regions: 5°C, 5°C ; ε = 0.80
The average temperature, or “global mean surface temperature” = 14.3°C.
The calculation of the energy radiated is done by simply taking each temperature and applying the equation above – E=εσT4
Because we are calculating the total energy we are simply adding up the energy value from each area. All the emissivity does is weight the energy from each location.
- With the emissivity values as shown, the 1999 energy = 2426 W/ arbitrary area
- With the emissivity values as shown, the 2009 energy = 2416 W/ same arbitrary area
So once again the energy radiated has gone down, even though the GMST has increased.
If we change around the emissivities, so that ε=0.8 for Equatorial & Sub-Tropics, while ε=0.99 for Mid-Latitude and Polar regions, the GMST values are the same.
- With the new emissivity values, the 1999 energy = 2434 W/ arbitrary area
- With the emissivity values as shown, the 2009 energy = 2442 W/ same arbitrary area
So the temperature has gone up and the energy radiated has also gone up.
Therefore, emissivity does change the situation a little. I chose more extreme values of emissivity than are typically found to see what the effect was.
The result is not complex or non-linear because emissivity simple “weights” the value of energy making it more or less important as the emissivity is higher or lower.
In the second example above, if the magnitude of temperature changes was slightly greater in the polar and equatorial regions this would be enough to still show a decrease in energy while “GMST” was increasing.
More Emissivity Graphs
Emissivity in the wavelengths of interest for the earth’s radiation is generally very close to 1. Assuming “blackbody” radiation is a reasonable assumption for most calculations of interest – as other unknowns are typically a higher source of error.
Because the earth’s surface has been mapped out and linked to the emissivities, if a particular calculation does need high level accuracy the emissivities can be used.
In the terms of how emissivity changes the “surprising” result that temperature can increase while energy radiated decreases – the answer is “not much”.