I have done further analytical modeling of global warming, using the same general method described earlier.
The question addressed now is: what is the trend of temperature change after an abrupt shutoff of all CO2 emissions just as the net temperature rise (relative to year 1910) reaches +1.5°C, given the lifetime of CO2 in the atmosphere?
For this problem, it is assumed that when the temperature rise (relative to 1910) reaches ~+1.5°C, that:
– all greenhouse gas emissions cease;
– pollution grit (which scatters light) falls out of the atmosphere “instantly” (a few weeks);
– CO2 (greenhouse gas) concentration decays exponentially after emissions shutoff;
– for CO2 lifetimes [e^-1] in years: 20, 50, 100, 238.436, 500, 1,000, 10,000, 100,000;
– temperature sensitivities of cloud cover, ice cover and albedo are as in the previous model;
– all other fixed physical parameters are as in the previous model,
In general, for the 8 cases calculated, the temperature increases at a diminishing rate after the emissions shutoff, reaches a peak, then trends downward.
The longer the lifetime of carbon dioxide in the atmosphere, the later and higher is the temperature peak, and the longer it takes to cool back down to the baseline temperature of 1910, which is 1.5°C below the starting temperature for this problem.
The 4 figures below show the calculated results.
Figure 1: °C change vs. years after shutoff, for lifetimes: 20, 50, 100, 238.436 years.
Figure 2: °C change vs. years after shutoff, for lifetimes: 20, 50, 100, 238.436, 500, 1,000 years.
Figure 3: °C change vs. years after shutoff, for lifetimes: 238.436, 500, 1,000, 10,000 years.
Figure 4: °C change vs. years after shutoff, for lifetimes: 1,000, 10,000, 100,000 years.
It is evident from the figures that if the lifetime of carbon dioxide in the atmosphere is greater than 500 years, that a temperature overshoot above +2.0°C (relative to 1910) will occur before cooling begins.
If the lifetime of carbon dioxide in the atmosphere is greater than about 250 years, it will take over a century for the eventual cooling to reduce average global temperature to its baseline temperature (which is for 1910 in this model).
If the lifetime of carbon dioxide in the atmosphere is greater than 10,000 years, the temperature overshoot will take global warming past +4.0°C (above our 1910 datum) for hundreds to thousands of years, and cooling back down to the temperature at our datum would take millennia.
The clearing of carbon dioxide from the atmosphere is a slow process. The absorption of CO2 by the oceans, and the subsequent dissolution of seafloor sediments (acidifying the oceans) occur over decades to centuries. The uptake of carbon dioxide by weathering reactions in carbonate and silicate soils and rocks occurs over millennia to many tens of millennia.
It took about 200,000 years to clear away the CO2 that caused the +8°C to +12°C global warming spike that occurred 55.5 million years ago, which is known as the Paleocene-Eocene Thermal Maximum (PETM).
Beyond its intrinsic scientific interest, this study confirms what has long been known as the needed remedy: anthropogenic emissions of greenhouse gases must permanently cease as soon as possible in order to limit the ultimate extent and duration of unhealthy global warming.
My notes on the mathematical solution of this problem are available through the following link