*The ‘airborne fraction’ (atmospheric increase in CO2 concentration/fossil fuel emissions) . From 1959 to the present, the airborne fraction has averaged 0.55. the terrestrial biosphere and the oceans together have consistently removed 45% of fossil CO2 for the last 45 years.
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.”Basically, in equilibrium, the amount of dissolved inorganic carbon (DIC) in the ocean determines the partial pressure of CO2 and, hence, the atmospheric CO2 concentration via Henry’s Law”
This law works both ways.
In other words the partial CO2 pressure determines the DIC as well.
Which is important for this discussion..
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The amount of extra CO2 added to the atmosphere by human activity, while significant, and lets say cumulative to some degree, is still a small fraction of the total atmospheric CO2 720 GT and the 137 times greater DIC [136,800 GT of CO2.]
[ we’re dealing with a coupled system, so if you add new material to one of the reservoirs, it will rise in all reservoirs]
Atmospheric CO2 720 GT at 400 PPM, which is 1/182 of the ocean equivalent.
If you increased to 560 ppm 1008 GT, a 25% increase the amount of CO2 in the ocean would have to increase by 2.5% OR 3420 GT.
At 30 GT a year human contribution that would take 100 years and providing that the DIC did stay in solution and not precipitate out in part.
Did a mere 600 GT raise the DIC of the oceans from
120 ppm rise suggests a*

. However, there is a more formal way to show this. I recently worked through the ocean carbonate chemistry. It turns out that there is a factor called the Revelle factor, which is simply the ratio of the fractional change in atmospheric CO2, to the fractional change in total Dissolved Inorganic Carbon (DIC) in the oceans:

R = \dfrac{\Delta pCO_2/pCO_2}{\Delta DIC/DIC}.

The Revelle factor is about 10, which means that the fractional change in atmospheric CO2 will be about 10 times bigger than the fractional change in DIC. What this tells you straight away is that you can’t change the amount of CO2 in the oceans without also change the amount in the atmosphere; stabilising emissions will not stabilise concentrations.

The residual airborne fraction increases from about 15% for emissions of 100s of GtC (we’ve already emitted 600 GtC) to almost 30% if we were to emit as much as 5000 GtC.

Now, maybe if the fractional change in DIC is small enough, the fractional change in pCO_2 might also be small enough to essentially stabilise concentrations. However, we know the quantities in the various reservoirs, and we’ve already emitted enough CO2 to change the DIC by 1 – 2%, and – hence – the atmospheric CO2 concentration by 10 – 20%. If we stabilise emissions, we could easily change the DIC by a further 1 – 2%. In fact, we have sufficient fossil fuels to change it by more than 10% and, therefore, enough to change the atmospheric concentration by more than 100% (i.e., to, at least, double atmospheric CO2).

There is, however, something I’m slightly glossing over, so will try to clarify a little more. The above is based on an equilibrium calculation. In other words, it is the changes once the system has retained a quasi-steady equilibrium. Our emissions are continually pushing the system out of equilibrium and so the fractional change in atmospheric CO2 is actually greater than what the Revelle factor would suggest. Given what we’ve already emitted, we would expect about 20% of our emissions to remain in the atmosphere, but it’s currently more like 45%. This is because the timescale for ocean invasion is > 100 years, and so the system hasn’t yet had time to return to equilibrium.

hypergeometric says:

“”he internal variability effect? Is it buried within the albedo effect, or the partial of outgoing longwave with respect to temperature?”

As you imply Internal variability is due to a multitude of factors. Some one off, some repeatable, some cyclical. If we knew enough about the actual causes to model them correctly we could remove some of them from the larger Natural Variability uncertainty range.

Your two examples show why there may not be a paradox to Willard. The atmospheric changes with 2 different GHG both increasing but only one with pure absorptive/ radiative properties, the other, H2O, being unique in that it causes increasing reflectivity with increasing concentration [decreased albedo] means that it is not a simple case of gas absorption emission physics with positive feedbacks but a complex reducing external input as internal energy retention goes up.

It is this dynamic that enables one to propose two Climate sensitivities. One with high variability at a low CO2 level and one with reducing/reduced CS when CO2 levels double.

The insistence that the CS plus positive feedback stays relatively the same at all CO2 levels is not held by most scientists ie people here have argued for some variability of CS with different conditions as a reason for it being hard to pin down, but most assume it must be relatively the same at different levels of increasing CO2. Take away this assumption and the paradox disappears.

Thanks for the link to your site, looks interesting.