A tale of 13 degrees

 

What happens if we don’t stop burning the fossil fuels? Is it the end of the world? Nobody knows certainly but I wanted to tackle this question with another back-of-the-envelope calculation. Here’s what I found:

First, let’s take the same starting point as in the previous post, the climate equation

dT = aS - bT^4.

If the world is in a thermal equilibrium, dT = 0 and

aS = bT^4.

From this, we may solve how emissivity b affects the equilibrium temperature. (Remember, the effect of CO2 comes through the emissivity in this model.) We have

T = (aS/b)^0.25.

Thus, we may conclude that the equilibrium temperature is directly proportional to the power -0.25 of emissivity. In practice, this means that every 1% decrease in emissivity increases the equilibrium temperature by approximately 0.25%.

The next question is obviously, how much unlimited burning of fossil fuels affects the emissivity.

This is now hardcore speculation, but I chose to model this by using emission spectra. Above, the first image is the thermal spectrum of the Earth, calculated from Planck’s law. They say [1] that the blackbody temperature of the Earth is 250K and this figure is calculated accordingly. Some claim that CO2 blocks ”all” radiation beyond 13 microns [2]. Yet, they say, the spectral line is at 15 microns [3]. So, as a compromise, I removed the range of 13-17 microns from the emission spectrum. The result is the lower image. 

A soaring 21% decrease in emissivity!

If this was the end result of an unlimited carbon party, the equilibirum temperature of the Earth would rise by 14.5 degrees, of which they say, 1.5 degrees has already incurred [4]. Thus, my first guess is a 13 degrees’ rise in the global mean temperature.

Is this the end of the world? It will be distributed unevenly and very probably the true effect is something else becuase of various feedback mechanisms. Even so, as a thought experiment, 13 degrees is horrific. I’m certain that most of the world population lives in places where this much warming would be a catastrophe. For my country, it would bring the climate of… erm… Spain. We may take another example from New York City. The mean tempearture of New York is 12.9 Celsius. To find a coastal city with a mean temperature of some 26 degrees, you would need to go to somewhere in Southeast Asia. (To me, it seems rational to compare coastal cities to coastal cities and continental cities to continental cities. The tropical coastal cities might have more rainfall than the temperate ones? I don’t know… Yet, the rising mean temperature brings rising evaporation from the Oceans and a rising rainfall. Here, we are stepping into the realm of climate science.)


References

[1] https://www.oxfordreference.com/display/10.1093/oi/authority.20110803095509515

[2] https://www.meteor.iastate.edu/gccourse/forcing/spectrum.html

[3] Wei et al. (2018). Absorption coefficient of carbon dioxide across atmospheric troposphere layer. Heliyon, 4(10).

[4] https://ourworldindata.org/grapher/temperature-anomaly


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