Proving the extra CO2 is all from fossil fuels

I've been doing  an online course on climate change at the University of Exeter for the last couple of weeks.  It's an excellent introduction if you're a bit hazy on the science and I thoroughly recommend it.  Plus it's free.

As someone who has already done a lot of research on the climate crisis, some of the material was already familiar.  However, I've still discovered a lot of new stuff.  This week we covered ocean acidification - which was an area I knew very little about - and I've gained a lot of insight into the mechanisms behind it.

What about the CO2?

In the past I've had a go at calculating exactly how much one would expect the atmospheric concentration to have increased assuming we were responsible for all of it.  If this happened to match the actual increase in atmospheric CO2, then - I thought - it would be a very nice slam dunk the next time I get into an argument with someone claiming it wasn't down to us$^\dagger \ ^{\dagger_2}$.

Unfortunately my calculations always seemed to produce a result that was over by a factor of two.  However, I read something interesting on the course today

"Approximately one quarter of man-made carbon dioxide emissions are taken up by the oceans, with another quarter taken up by the land biosphere and the remaining half accumulating in the atmosphere"

Yippee - that explains the discrepancy.  So let's look at the arithmetical evidence

The chart above shows an increase of 370ppm to 415ppm in the 20 years from 2000 to 2020:

>>> actual_additional_co2_ppm_per_year = (415-370)/20.
>>> actual_additional_co2_ppm_per_year
 2.25

Now we can work out what we'd expect it to be given the radius of the Earth is about 6,500 km, the height of the atmosphere is about 10 km $^{\dagger_3}$, and our annual emissions are around 35 GTonCO2:

>>> co2_kg=35e9*1e3 # yearly emissions
>>> co2_vol = co2_kg/1.5 # since 50% heavier than N2 or O2
>>> area = 4 * 3.14 * (6500000**2)
>>> atmos_height = 10000
>>> atmos_vol = area *atmos_height
>>> additional_co2_ppm_per_year = (co2_vol/atmos_vol)*1e6
>>> additional_co2_ppm_per_year
 4.39

Finally, divide by two to account for the 50% of emissions that go into either the soil or the oceans rather than the atmosphere and you get:

>>> additional_co2_ppm_per_year/2
 2.19

... which is only about 2% off the real figure!

FOOTNOTES:

• ($\dagger$) Sadly these people do still exist.
• ($\dagger_2$) Although in the near future it is quite possible that the majority of additional CO2 added to the atmosphere will come from receding permafrost.  If the vegetation exposed is able to decompose aerobically this will result in CO2 emission; if it decomposes anaerobically it will result in methane emissions - which is about 25 times more powerful a greenhouse gas.  There's 1.6 trillion tons of carbon locked up in permafrost - about 10 times our budget for 1.5C- so we really really don't want this to be released as CO2, and we really really really don't want it released as methane.
  
• ($\dagger_3$) You can verify this using surface pressure, which is about $100kN/m^2$.  Surface pressure is equal to the weight of the column of air divided by cross-sectional area.  So each square metre must have about $10$ Ton of air above it.  The density of air is about $1kg/m^2$, so the column must be about $10km$ high.