In the initial stage of this project, we used the following formula (from this simple model) for greenhouse effect (clear-sky):
εA = 1− exp(−0.4634∙exp(18.382 − 5294/T) − 3.52∙CCO2)
This is simple to understand, in principle – there is an exponential variation in the emissivity of the atmosphere with respect to vapour concentration (H2O, which depends on dew-point temperature) and CO2 concentration (which is presumably constant globally). The remainder (clouds, dust, ozone, etc.) is considered constant here, which I consider a flaw, and we’ll see later what is actually constant and what isn’t (clouds are definitely not!).
Problems and Solutions
This simple model overestimates the effect of CO2 and water vapour on the full greenhouse effect, due to the way the exponential functions are put together. Every version of our models that uses this formula, however, for some reason as yet unknown, behaves inversely – CO2 causes cooling. Obviously, something is inverted. But, inverted, the formula produces a runaway greenhouse effect that would make Venus look like an ice-ball. We might be applying it wrong, but ultimately this formula seems to be flawed (or at least limited to a small range of temperatures).
Enter the new greenhouse effect formula, based partly on Schmidt et al…
gamma = (1+co2)*(1+h2o)*(1+others)*(1+ozone) – 1
Here gamma is the greenhouse effect (downward flux forcing) as a fraction of solar flux at the surface (Is). Thus, to calculate the surface temperature, all we have to do is:
Ts = fourth-root ( Is * (gamma + 1) / ( eS * sigma ) )
Where Ts is the temperature at the surface, eS is the emissivity of the surface (usually 0.9 dry to 0.96 wet), and sigma is the Stefan-Boltzmann constant of 5.67e-8.
The result should be close to 288K (15°C) at gamma = 0.654, Is = 237 W/m2 and eS = 0.96.
But the question is, what is gamma, physically? And how do we derive it from basic concentrations of various gases, each gas having a different proportional forcing effect?
Well, we have to use the actual absorption bands (spectra) of each of the gases. It turns out that water absorbs up to 31% of Earth’s emitted radiation at 8000 ppm, and CO2 up to 14% at 8000 ppm. The concentration is important because it affects our optical depth. Less concentration means a greater transparency. The ppm values are based on molar concentration (i.e. n mol CO2 / n mol of air = conc of CO2 in air). 8000 ppm is high enough concentration that we have nearly 100% of the effect that we’ll ever have.
Therefore, we have this simple code:
var co2conc = co2level;//ppm
var vapconc = Math.exp(18.38-5294/Math.max(dewpoint,205))*10000;//ppm
var lapseAvgCO2 = 0.875; //87.5% or up to 70 km in atmosphere is a VERY SLOW lapse rate! (is this correct ??)
var lapseAvgH2O = 0.07; //7% or up to 5.6 km in atmosphere
var co2 = 0.14*(1-Math.exp(-lapseAvgCO2*10000*co2conc/1e6)); //0.7% at 15 ppm to 14% at 8000 ppm;
var vap = 0.31*(1-Math.exp(-lapseAvgH2O*10000*vapconc/1e6)); //1.5% at 210K 15 ppm at 0.02 mb to 31% at 285K 8000 ppm at 13.5 mb
var others = 0.065; // aerosols, CH4 (clouds included in vap)
var ozone = 0.036;
return (1+co2)*(1+vap)*(1+others)*(1+ozone) - 1;
Note – the lapse values are used to correct the concentration to the atmospheric lapse rate of the gas. That is, at height H1, what is the concentration with respect to height H0 (sea level)? It turns out, with CO2 we have roughly the same concentration up to 70 km, then we drop off sharply. So we still have 380-400 ppm all the way up to the stratosphere. Using the lapse rate of the air, it is possible to calculate the equivalent concentration of air at altitude, arriving with a height of 54.7 km where the concentration of the air matches that of CO2. This is how we determine our “scale height” of the CO2 column. The same principle works with H2O, where it turns out the lapse rate is much more abrupt, leading to nearly zero concentration at 5.6 km (instead of 70 km for CO2). You’ll notice most places on Earth at 5.6 km altitude don’t have clouds or precipitation, so we are quite accurate. Based on this, we can conclude that most of the H2O greenhouse effect occurs in the lower troposphere, so adjusting to the entire column height of air, we get a lapse value for concentration of 7%. This means 8000 ppm at sea level is in reality 560 ppm across the entire column due to this sharp lapse rate.
You’ll notice the numbers 56 and 57 a lot, especially if you keep in mind that the solar temperature is around 5600 Kelvin to 5700 Kelvin and that the solar radiation wavelength is around 560 nm to 570 nm. All of this points to a potential holographic fine balance (a sort of numerological conservation) of mass-energy-information within the Earth-Sun-Moon system. But alas, we digress…
The model in this post suggests the CO2 height in the atmosphere has dropped in the last century, from about 57 km down to 54.7 km, a distance of about 2.3 km. This is no small feat, and indicates one of two things: (1) there are greater net emissions of CO2 causing the overall total mass of CO2 to increase, or (2) the total mass of CO2 has remained roughly the same, but the sun’s radiation at stratosphere heights has been cooling.
If the sun were cooling, things would be quite interesting. We would indeed observe a greater apparent sea-level concentration of CO2 simply by virtue of the fact that there is less energy available to heat and dissipate the CO2 molecules. Furthermore, this concentration would also increase due to the reduced excitation at the edge where the CO2 layer meets space, especially since less UV radiation would be hitting the CO2 molecules directly. In other words, today’s perceived high CO2 concentration may be due to the fact that the sun is emitting less shortwave radiation and more longwave radiation, equivalent to a Doppler red shift, or a cooling of the sun‘s radiation.
57 km is way above the ozone layer, where all three types of UV radiation come in. CO2 has absorption bands in the ultraviolet range, which means it should theoretically heat up and even leave Earth’s atmosphere and go out into space at rarefied concentrations. It would not cool down and freeze, since its partial pressure is so low. So obviously the increase in CO2 concentration is particularly troubling once you realize how it is connected to the energy level of the sun.
Long story short, if the sun were emitting longer wavelengths, at a lower temperature (say, 5500 K), we would notice that as an increase in apparent CO2 concentration and an even greater increase in the greenhouse effect. Even our regular daylight would change, to have more reds and yellows, and fewer blues and violets, and even less ultraviolet. With less ultraviolet available to break O2 into O3, we would also see a reduction in the total level of ozone directly related to the sun’s longer wavelengths. Thus, the ozone hole, which appeared since the late 1970s would also be connected to the rise in CO2 and the cooling of the sun.
Eventually, the cooling of the sun should overpower the greenhouse effect, which is relatively weak on Earth, leading to a substantial cooling. The magnitude of that cooling remains to be explored in a future article.