Although most people are by now convinced that climate change is a reality, a significant minority harbors the belief that man is somehow too puny to affect something as large as the atmosphere. Similar arguments were put forward for the influence of chlorofluorocarbons on the ozone layer and shown to be wrong. Of course, climate change is an altogether different phenomenon, but the same argument is used there too. The calculations are trivial, but only if done in metric units. Most of the science here is simple physics, understandable by anyone with a high school diploma.
Mass of the Atmosphere
The total downward force, F, of the atmosphere is the surface pressure, p, times the Earth's surface area.
F = 4πR²p
where R is the Earth's radius.
The downward force, F, is the weight of the atmosphere, so the mass is
M = F/g
where g is the acceleration due to gravity.
Emissions into the atmosphere
Emissions of carbon dioxide into the atmosphere have been measured carefully. The latest study indicates that about 50% of the emissions go into the atmosphere, with the other 50% going into the biosphere and the oceans.
Denoting the total emissions of CO2 due to mankind by E, the mass mixing ratio of the human induced CO2 is simply
h = 0.5 E/M
Concentrations of CO2 in the atmosphere
The concentrations of CO2 in the atmosphere are usually measure in terms of the volume mixing ratio, i.e., volume of CO2 per volume of dry air. The ratio of the mass mixing ratio of CO2 to the volume mixing ratio is approximately 44/29, so the human induced volume mixing ratio is
h_v = 0.5 x 29/44 x E/M
Using the above equations, we find that
h_v = 14.5Eg/(176πR²p)
Quantifying the change
Total emissions are currently recorded as tonnes of carbon, whereas E is in kg of CO2. The conversion factor from carbon to CO2 is 44/12 = 3.67, hence
E = 3.67 x 5.5 x 10^14 kg
g = 9.8 m/s²
R = 6.37 x 10^6 m
p = 9.85 x 10^4 N/m² (985 mbar)
Note that we are not measuring emissions in tons, or the acceleration of gravity in ft per second per second, or the Earth radius in miles, or the surface pressure in terms of the length of column of a biohazard (mercury). Everything is in proper units, known formally as the MKS (metre, kilogramme, second) system of units. There are no conversion factors, except those necessary to convert from one type of gas mixing ratio to another.
Putting in the numbers, we obtain,
h_v = 130 x 10^-6
So, according to these figures, humans have contributed 130 parts per million to the atmospheric CO2 concentration.
The greenhouse effect
The pre-industrial concentration of CO2 was about 280 ppmv (parts per million by volume) and the current value of CO2 is 390 ppmv, so the above simplified calculation is in error, but not by much. It does show that humans can change the CO2 concentration in the atmosphere by 39%, if we use the observed 110 ppmv change.
According to our understanding of the physics of the heating of the Earth the atmospheric greenhouse effect is about 33C, most of which is due to water vapor. Assuming that CO2 is the only greenhouse gas emitted by humans and contributes 10% to the total warming (which ignores the effect of CH4 and N2O, as well as secondary effects such as the increase in atmospheric water with global warming), an additional 39% in CO2 would give rise to an additional equilibrium greenhouse effect of about 3.3 x 0.39 = 1.3°C (2.3°F).
Detailed calculations are needed to account for the other greenhouse gases and to allow for the fact that the equilibrium temperature has not yet been reached. Nonetheless, the figure is similar to that actually observed, and demonstrates categorically that humans can and have influenced the climate.
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