# Global Warming and Cooling After CO2 Shutoff at +1.5°C

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,
(https://manuelgarciajr.com/2020/06/13/living-with-global-warming/).

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

Global Warming, CO2 Shutoff

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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,

where:

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.

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# Global Warming After 1.5°C Without Emissions

If greenhouse gas emissions stop just as the temperature rise (relative to 1910) reaches 1.5°C, what is the projected trend of temperature rise (or fall) after that point in time (year)?

[This scenario presumes an infinite lifetime of CO2 in the atmosphere. So it is the extreme of pessimism. The effect of finite lifetimes of CO2 in the atmosphere is shown in later work, at https://manuelgarciajr.com/2020/06/20/global-warming-and-cooling-after-co2-shutoff-at-1-5c/]

If greenhouse gas emissions ceased entirely in the year 2047 (in 27 years), just as the relative temperature was nearly 1.5°C above that of 1910, then the subsequent trend of relative temperature would still be a rise but at a decreasing rate over time, and with an asymptote of 6.2877°C, which would essentially be achieved by the year 3160. Projections here are that for

year 2120 (in 100 years):
with same emissions rate after 2047, temperature rise = 3.2°C,
without any emissions after 2047, temperature rise = 2.75°C,

year 2185 (in 165 years):
with same emissions rate after 2047, temperature rise = 5°C,
without any emissions after 2047, temperature rise = 3.6°C.

The “no emissions” asymptotic temperature rise of ~6.29°C (by year 3160) would mean the average global temperature would be comparable to that of 55.5 million years ago at the very beginning of the upswing in temperature during the Paleocene-Eocene Thermal Maximum (PETM). The PETM began at a temperature about +4°C above that of our 1910 datum, and shot up to somewhere in the vicinity of +8°C to +12°C above it, and even possibly +16°C above it. It then took 200,000 years for the “excess” atmospheric CO2 to be cleared away by rock weathering, and the average global temperature to return to +4°C above our datum. This all occurred during the early Eocene geological epoch (which occurred between 56 to 33.9 million years ago).

In the post 2047 “no emissions” model used here, the albedo (the light reflectivity of the Earth) would still be higher than today because of increased reflective cloud cover, because of higher temperature.

Though the fallout of light-reflecting pollution grit would occur quickly in and after 2047, which is an albedo-reducing (warming) effect, it is not considered significant in relation to the reflective effect of the temperature-enhanced cloud cover (a cooling effect). The Earth’s albedo is dominated by cloud cover.

The temperature-enhanced reduction of ice cover (an albedo-reducing and thus warming effect) is always insignificant in comparison to the effect of cloud cover.

The infrared (heat) absorptivity (parameter F in the model) remains unchanged after 2047 because no new greenhouse gases are added to the atmosphere after that year (hypothetically), and because carbon dioxide (CO2) remains present in the atmosphere for a very long time (once the oceans are saturated with it), on the order of 150,000 years or more.

As noted previously (in “Living With Global Warming”), because of the immense thermal inertia of the biosphere and its climate system, the effect of an abrupt cessation of greenhouse gas emissions would come on slowly over the course of hundreds of years [an e-folding time of 240 years].

As will be evident from Figure 3, below, if we cared to limit temperature rise as much as possible for the sake of future generations, we could never cease emitting greenhouse gases too soon.

On the basis of the modeling described here, it seems impossible to ever limit the ultimate rise of temperature to below +2°C relative to 1910.

If we ceased all greenhouse gas emissions this minute in the year 2020, we might be able to keep the average global temperature from ever rising above +5.8°C, relative to 1910, in the distant future.

It will be interesting to see what the state-of-the-art supercomputer numerical models project as possible future “no emissions” temperature rises, as those models are further refined from today.

Technical Details

The technical details of how I reached these conclusions now follow. This discussion is a brisk and direct continuation of

Living With Global Warming
13 June 2020
https://manuelgarciajr.com/2020/06/13/living-with-global-warming/

For a description of the parameters used in my model, and their numerical values, see

A Simple Model of Global Warming
26 May 2020
https://manuelgarciajr.files.wordpress.com/2020/05/global-warming-model.pdf

The previous model of temperature rise relative to 1910 is called “example #5” because it was the 5th numerical example devised from the general solution of the relative temperature rate-of-change equation. For that model, at relative time =137 years (for year 2047, which is 137 years after 1910):

T = 1.4867°C, temperature rise relative to 1910,

A = 0.5226, albedo,

F = 0.5931, infrared (heat) absorptivity.

If greenhouse gas emissions cease entirely in year 2047 (at 137 years of relative time), then:

ap = 0, (grit pollution enhancement of albedo over time ceases),

fp = 0, (increasing greenhouse gas pollution enhancement of heat absorptivity over time ceases),

and the temperature change trend continues after t = 137years with:

T(at t=137) = 1.4867°C, (the “initial” relative temperature at t=137),

A = 0.5226 + 0.004286T, (albedo after t=137 is only dependent on relative temperature: clouds),

F = 0.5931, (heat absorptivity is unchanged after t=137, greenhouse gases persist, but none added),

alpha = 0.019919 °C/year, (new value),

beta = 0.004194 year^-1, (unchanged),

gamma = 0, (since strictly temporal increases/effects of pollution have ceased).

The relative temperature from t=137 on is now given by:

T(t≥137) = 1.4876°C + (4.801°C)[ 1 – exp(-0.004194[t-137]) ].

Figure 3: Relative Temperature Change after 2047 (1.5°C) w/o Greenhouse Gas Emissions

Note the following points on the “no emissions” relative temperature curve:

for t=210 (year 2120), T=2.75°C instead of 3.2°C,

for t=275 (year 2185), T=3.6°C instead of 5°C,

for t=1250 (year 3160), T=~6.28°C

The “no emissions” relative temperature curve after 1.5°C has an asymptote of 6.2877°C.

Note

For descriptions of the PETM, see:

Paleocene-Eocene Thermal Maximum
https://en.wikipedia.org/wiki/Paleocene%E2%80%93Eocene_Thermal_Maximum

Ye Cannot Swerve Me: Moby-Dick and Climate Change
15 July 2019
https://manuelgarciajr.com/2019/07/15/ye-cannot-swerve-me-moby-dick-and-climate-change/

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