Now you’ve handed me a quantitative model I’m going to indulge my curiosity :-)
You would have to bear in mind that most anthropogenic emissions occurred in recent decades, so we should still be in the “transient response” phase for the additional perturbation they impose...
I think we can account for this by tweaking equation 4.14 on your linked page. Whoever wrote that page solves it for a constant additional forcing, but there’s nothing stopping us rewriting it for a variable forcing:
dT(t)dt=Q(t)−T(t)λCs
where T(t) is now the change in temperature from the starting temperature, Q(t) the additional forcing, and I’ve written the equation in terms of my λ (climate sensitivity) and not theirs (feedback parameter).
Solving for T(t),
T(t)=e−tλCs⎛⎝constant∫etλCsQ(t)Csdt⎞⎠
If we disregard pre-1850 CO2 forcing and take the year 1850 as t = 0, we can drop the free constant. Next we need to invent a Q(t) to represent CO2 forcing, based on CO2 concentration records. I spliced together twoAntarctic records to get estimates of annual CO2 concentration from 1850 to 2007. A quartic is a good approximation for the concentration:
The zero year is 1850. Dividing the quartic by 280 gives the ratio of CO2 at time t to preindustrial CO2. Take the log of that and multiply by 5.35 to get the forcing due to CO2, giving Q(t):
Plug that into the T(t) formula and we can plot T(t) as a function of years after 1850:
The upper green line is a replication of the calculation I did in my last post—it’s the temperature rise needed to reach equilibrium for the CO2 level at time t, which doesn’t account for the time lag needed to reach equilibrium. For t = 160 (the year 2010), the green line suggests a temperature increase of 0.54K as before. The lower red line is T(t): the temperature rise due to the Q(t) forcing, according to the thermal inertia model. At t = 160, the red line has increased by only 0.46K; in this no-feedback model, holding CO2 emissions constant at today’s level would leave 0.08K of warming in the pipeline.
So in this model the time lag causes T(t) to be only 0.46K, instead of the 0.54K expected at equilibrium. Still, that’s 85% of the full equilibrium warming, and the better part of the 0.8K increase; this seems to be evidence for my guess that we wouldn’t have to wait very long to get close to the new equilibrium temperature.
Suppose you had no paleo data or detailed atmospheric physics knowledge, but you just had to choose between 1 degree and 3 degrees as the value of climate sensitivity, i.e. between the hypothesis that all the feedbacks cancel, and the hypothesis that they triple the warming, solely on the basis of (i) that observed 0.8K increase (ii) the elementary model of thermal inertia here.
If I knew that little, I guess I’d put roughly equal priors on each hypothesis, so the likelihoods would be the main driver of my decision. But to run this toy model, should I pretend the only variable forcing I know of is anthropogenic CO2? I’m going to here, because we’re assuming I don’t have ‘detailed atmospheric physics knowledge,’ and also because I haven’t run the numbers for other variable forcings.
To decide which sensitivity is more likely, I’ll calculate which value of λ produces a 0.8K increase from CO2 emissions by 2010 with this model and the above Q(t); then I’ll see if that λ is closer to the ‘3 degrees’ sensitivity (λ between 0.8 and 0.9) or the ‘1 degree’ sensitivity (λ = 0.3). For an 0.8K increase, λ = 0.646, so I’d choose the higher sensitivity, which has a λ closer to 0.646.
Now you’ve handed me a quantitative model I’m going to indulge my curiosity :-)
I think we can account for this by tweaking equation 4.14 on your linked page. Whoever wrote that page solves it for a constant additional forcing, but there’s nothing stopping us rewriting it for a variable forcing:
dT(t)dt=Q(t)−T(t)λCs
where T(t) is now the change in temperature from the starting temperature, Q(t) the additional forcing, and I’ve written the equation in terms of my λ (climate sensitivity) and not theirs (feedback parameter).
Solving for T(t),
T(t)=e−tλCs⎛⎝constant∫etλCsQ(t)Csdt⎞⎠
If we disregard pre-1850 CO2 forcing and take the year 1850 as t = 0, we can drop the free constant. Next we need to invent a Q(t) to represent CO2 forcing, based on CO2 concentration records. I spliced together two Antarctic records to get estimates of annual CO2 concentration from 1850 to 2007. A quartic is a good approximation for the concentration:
The zero year is 1850. Dividing the quartic by 280 gives the ratio of CO2 at time t to preindustrial CO2. Take the log of that and multiply by 5.35 to get the forcing due to CO2, giving Q(t):
Plug that into the T(t) formula and we can plot T(t) as a function of years after 1850:
The upper green line is a replication of the calculation I did in my last post—it’s the temperature rise needed to reach equilibrium for the CO2 level at time t, which doesn’t account for the time lag needed to reach equilibrium. For t = 160 (the year 2010), the green line suggests a temperature increase of 0.54K as before. The lower red line is T(t): the temperature rise due to the Q(t) forcing, according to the thermal inertia model. At t = 160, the red line has increased by only 0.46K; in this no-feedback model, holding CO2 emissions constant at today’s level would leave 0.08K of warming in the pipeline.
So in this model the time lag causes T(t) to be only 0.46K, instead of the 0.54K expected at equilibrium. Still, that’s 85% of the full equilibrium warming, and the better part of the 0.8K increase; this seems to be evidence for my guess that we wouldn’t have to wait very long to get close to the new equilibrium temperature.
If I knew that little, I guess I’d put roughly equal priors on each hypothesis, so the likelihoods would be the main driver of my decision. But to run this toy model, should I pretend the only variable forcing I know of is anthropogenic CO2? I’m going to here, because we’re assuming I don’t have ‘detailed atmospheric physics knowledge,’ and also because I haven’t run the numbers for other variable forcings.
To decide which sensitivity is more likely, I’ll calculate which value of λ produces a 0.8K increase from CO2 emissions by 2010 with this model and the above Q(t); then I’ll see if that λ is closer to the ‘3 degrees’ sensitivity (λ between 0.8 and 0.9) or the ‘1 degree’ sensitivity (λ = 0.3). For an 0.8K increase, λ = 0.646, so I’d choose the higher sensitivity, which has a λ closer to 0.646.