I have done further analytical modeling of global warming, using the same general method described earlier (https://manuelgarciajr.files.wordpress.com/2020/05/global-warming-model.pdf).
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.
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
Climate System Response Time
The parameter “beta” is a reaction rate, or frequency, or inverse response time of the biosphere and its climate system. By my calculation, that rate is 1.329×10^-10 seconds^-1, or 0.004194 years^-1, or a response time of 238.436 years. Of course I am not saying the precision of this estimate is as suggested by all the decimal places shown, it’s just that these are the numbers that come out of my calculations, and these numbers are kept to remind me of what choices I made to eventually arrive at this result.
The parameter beta is the product:
beta = (S•a1)/C = [S•(a-cloud – a-ice)]/C,
S = the insolation on the entire disc area of the Earth (1.7751×10^17 Watts),
a-cloud = the temperature sensitivity of the albedo because of the extent of cloud cover (1/°C),
for a positive quantity of: increase of albedo for a given temperature rise (5.715×10^-3 1/°C),
a-ice = the temperature sensitivity of the albedo because of the extent of ice cover (1/°C),
for a negative quantity of: decrease of albedo for a given temperature rise (1.429×10^-3 1/°C),
C = the heat capacity of the biosphere (5.725×10^24 Joules/°C).
A better determination of a-cloud and a-ice would improve the estimate of beta. I chose these quantities to be in the ratio of 4:1, as is the ratio between the cloud reflection portion of the albedo (24%) to the Earth surface portion of the albedo (6%) for the total pristine (pollution free, pre-global warming) albedo (30%).
So, beta incorporates physical parameters that characterize: solar energy, atmospheric and Earth surface reflectivity of light, and the thermodynamics of the mass of the biosphere.
Events and inputs to that Earth climate system are recognized and responded to on a timescale of 1/beta. Events and inputs with timescales less than 1/beta are blips whose impact will become evident much later, if they are of sufficient magnitude and force. Events and inputs of timescales longer than 1/beta are “current events” to the biosphere’s thermodynamic “consciousness,” and act on the climate system as it reciprocally acts on them over the course of the input activity.
Turning a large ship around takes advanced planning and much space because it’s large inertia tends to keep it on its original heading despite new changes to the angle of its rudder. Even more-so, changes in the direction of Earth’s climate, which may be sought with new anthropogenic rudder angel changes — like drastic reductions of greenhouse gas emissions — will require fairly deep time because of the immense thermodynamic inertia of that planetary system.
This means that the climate system today is responding to the “short time” impulses it was given over the previous two centuries or more; and that both the more enlightened and most stupid impulses that we give it today could take several human lifetimes to realize their full response. We are dealing with Immensity here, and our best approach would be one of respect and commitment.