Thanks to Michael Tobis I discovered a new report this week from the Congressional Budget Office that has the most dramatic illustration I've seen of projections of temperature for the remainder of this century (right - figure 1 in the report). The PDF of the 33-page report is available for download from CBO.

What they've done here is to combine the historical annual global average temperature record from the Hadley center (going back to 1850) with a recent MIT report on likely temperature increases through 2100 under projected emissions scenarios. The citation for that is A.P. Sokolov et al, "Probabilistic Forecast for 21st Century Climate Based on Uncertainties in Emissions (Without Policy) and Climate Parameters", Report No. 169 (Cambridge, Mass.: MIT Joint Program on the Science and Policy of Global Change, 2009), and can be downloaded from MIT (pdf format).

The reason for the upward-turning curve is the projection that, without an international policy agreement to limit fossil carbon emissions, coal and other fossil fuel use will continue to grow, particularly with economic growth in developing nations. Even if that growth didn't happen, our current rate of CO2 emissions is enough to keep temperatures rising for a long time to come, though on a lower curve than this one. And if we completely halted all CO2 emissions now, temperatures would still rise for a while because Earth is presently out of equilibrium, and there is still some warming expected "in the pipeline". Nevertheless, the sooner we get an international agreement in place, the sooner we can start bringing that crazy curve down, at least a little bit. One policy option is illustrated in figure 4 of the CBO report, to the right of this paragraph.

The report rightly emphasizes the point that uncertainty should move us to swifter action, because uncertainty in effects includes uncertainty on the side of things being that much worse than we expect. The consequences of no action will be devastating to the world our children inherit.

## Comments

## I take it who ever wrote this

I take it who ever wrote this report doesn't believe in any kind of saturated response (e.g. Logarithmic).

## John - the logarithmic

John - the logarithmic "saturation" is overwhelmed by the exponential growth of our emissions in their scenario - this gives a superlinear curve because total GHG's goes like

current + A exp(r t)

so the forcing from CO2 (which is logarithmic in concentration) goes like

log(current + A exp(r t))

which is different from

log(A exp(r t)) = log A + rt

## Well: log(current + A exp(r

Well:

log(current + A exp(r t)) < current+log(Aexp(rt)) provided current > 1

But anyway, the logarithmic law I think is more like this:

Δ F=k log(CO2(t2)/CO2(t1))=k*log(exp(at))

## I'm not sure what you're

I'm not sure what you're trying to imply with your first inequality there. log(1 + ε) ≈ ε for small ε, so if ε is growing exponentially, the log(c + A exp(rt)) will also curve upwards until A exp(rt) is large compared with c. We're still in the upward curving part of the relationship for CO2 in the atmosphere (the fractional growth so far is about 40% from pre-industrial levels, and log(1.4) is about 0.34, not much reduced from 0.4).

You are correct that the law is based on the ratio of CO2 concentrations at two different times, but what's growing exponentially is our emissions, i.e the increment to CO2 in the atmosphere, not the total amount. That's growing more like (c + exp(rt)) as I said.

## "so if ε is growing

"so if ε is growing exponentially, the log(c + A exp(rt)) will also curve upwards until A exp(rt) is large compared with c. "

That would supprise me given that the second derivative is negative. Of course we can resolve this somewhat by plotting the function. Anyway, let me try my Algebra again. We both agree that:

Δ F=k log(CO2(t2)/CO2(t1))

But: CO2(t2) can be written as CO2(t2)=CO2(t1)exp(a(t2-t1))

In which case one gets:

Δ F=k log(exp(a(t2-t1)))

## John - no, CO2(t2) is not

John - no, CO2(t2) is not equal to CO2(t1) exp(a(t2-t1)). That's the whole point of my response to you on this.

The exponentially growing part of CO2 is our

incrementto the pre-industrial level. The pre-industrial level of CO2 was around 280 ppm, which is much larger than what we've added to this point. We have added about 40% now. It is the increment we are adding that is growing exponentially. The 40% was just a 20% increment back in 1978, and only a 10% increment back in 1932, so for the 20th century historically we had a roughly 50 year doubling time early in the century, decreasing to around 30 year doubling more recently. That exponential growth continues so the 40% would become an 80% increment by 2040 (business as usual). So that's 3 doublings from the human-caused CO2 increment of 1932, but we still haven't reached double the original pre-industrial level by then. Only the increment is growing exponentially, the total CO2 is growing in a different fashion, which has, for now, upward curvature in the logarithm. Go ahead and plot it, it's pretty clear (and essentially what the CBO did here).## Okay, you're right. I did

Okay, you're right. I did plot these figures (see my third figure here:)

http://earthcubed.wordpress.com/2009/09/19/logco2-and-scary-graphs/

(I should have realized this from the start)

Anyway, I calculated the doubling time for the CO2 concentration and found it to be about 100 years, and I also calculated with my fit that from 1997 to 2100 the CO2 will only increase by a factor of 1.2. This means that over a 100 year period that the change in equilibrium due to warming will be less then the CO2 doubling sensitivity by a factor of 0.3. Additionally the upper bound for the IPCC CO2 sensitivity is four, which gives only a 1.2 degree change in the equilibrium temperature over 100 years. I need to check my math but the above figure from the government report looks very suspicious.