Correcting Contrarian Graphs on the Relationship Between CO2 and Warming

It's well-established physics that the Earth's surface is ~33K warmer than its effective temperature, and the relationship between increasing CO2 and radiative forcing can be approximated by the following logarithmic equation:

ΔF = 5.35*ln (C/Co) where,

• Co is an initial concentration of CO2 (preindustrial CO2 is generally regarded as 280 ppm). 
• C is the concentration of CO2 at any given time (currently 420 ppm).

The equation shows the change in the outgoing flux at the top of the atmosphere caused by a change in CO2 concentrations. Since CO2 concentrations have increased by 50%, we can say CO2 has caused a decrease in the outgoing flux of 5.35*ln (1.5) = 2.2 W/m^2. As a result of this decrease in outgoing flux, more energy enters the climate system than escapes into space, and so the planet's surface must warm until the outgoing flux equals incoming again. The relationship between a change in radiative forcing and temperature is linear, so

ΔT = λ*ΔF

So essentially the relationship between changes in CO2 and temperature is logarithmic, and each doubling of CO2 causes roughly the same amount of warming at equilibrium with the change in forcing. The central estimate for the amount equilibrium warming for doubling CO2 is 3 C (once rapid feedbacks are accounted for). If ECS = 3 C, then we can solve for λ (sensitivity) in the above equation to be 3/3.7 = 0.81 C/W/m^2. It's this logarithmic relationship that contrarians frequently exploit to create the false impression that continued increases in CO2 can have little to no effect on GMST. Here's one example I've seen floating around Twitter and Facebook recently.

The author of this graph, John Shewchuk, clearly doesn't understand what saturation is, but let's leave that aside for this post. What Shewchuk did is assume an irrationally low value for λ sensitivity of 0.15 C/W/m^2 (what would be an ECS of 0.56 C) and then assume the above equation for radiative forcing would hold at very small CO2 concentrations. The graph then creates the false impression that CO2 at 1200 ppm would only cause GMST to increase by about 5 C above the Earth's effective temperature. This is just plain silly.

But since I've seen these kinds of graphs many times in political tracts, social media, blogs, and YouTube videos, I thought it might be helpful to create a couple accurate graphs showing the amount of equilibrium warming above preindustrial levels (280 ppm CO2). The following graphs show the amount that the equilibrium GSMT anomaly would increase above preindustrial levels in 70 ppm (25% of 280 ppm) increments. In the first graph, I show this with CO2 concentrations on the x-axis. In the second, I show it with the ratio C/280 (rCO2) on the x-axis.

As you can see, with a rational estimate for ECS, we can expect about 7 C warming above preindustrial levels at 1400 ppm CO2 (or 5x preindustrial levels). If you find these graphs useful, feel free to use them as often as you like. I get so tired of responding to contrarians making crappy, deceitful graphs that I figured I'd do my part to help get accurate information out there that agrees with the best evidence we have for climate sensitivity. Have fun with them!