On Climate sensitivity and nonsense claims about an "IPCC model"

I have been arguing recently in threads where people insist that there is an "IPCC model" that says that the change in temperature is linearly related to change in forcing, on a short term basis. They are of course mangling the definition of equilibrium sensitivity, which defines the parameter ECS as the ratio of the equilibrium change in temperature to a sustained (step-function) rise in forcing. So in this post I would like to explain some of the interesting mathematics involved in ECS, and why you can't just apply its value to relate just any forcing change to any temperature change. And why extra notions like Transient Climate Sensitivity are needed to express century scale changes.

The threads in question are at WUWT. One is actually a series by a Dr Antero Ollila, latest here, "On the ‘IPCC climate model’". The other is by Willis Eschenbach "Delta T and Delta F". Willis has for years been promoting mangled versions of the climate sensitivity definition to claim that climate scientists assume that ∆T = λ∆F on a short term basis, giving no citation. I frequently challenge him to actually quote what they say, but no luck. However, someone there did have better success, and that made it clear that he is misusing the notion of equilibrium sensitivity. He quoted the AR5

"“The assumed relation between a sustained RF and the equilibrium global mean surface temperature response (∆T) is ∆T = λRF where λ is the climate sensitivity parameter.”"

Words matter. It says "sustained RF" and equilibrium T. You can't use it to relate any old RF and T. A clear refutation of this misuse is the use also of transient sensitivity (TCS). This typically is defined as the temperature change after CO₂ increased by 1% per year, compounding over 70 years, by which time it has doubled. CS is normally expressed relative to CO₂ doubling, but it is linked to forcing in W/m² by
∆F = 5.35* ∆log([CO2])
which implies ∆F = 3.7 W/m² for CO₂ doubling. Both ECS and TCS express the same change in forcing. But one is an initial change sustained until equilibrium is reached; the other is temperature after that 70-year ramp. And they are quite different numbers; ECS is usually about twice TCS. So there can't be a simple rule that says that ∆T = λ∆F for any timing.

A much better paradigm is that ∆T is a linear function of the full history of forcing F. This would be expressed as a (convolution) integral

∆T(t)=∫w(t-τ)*F(τ) dτ, where the integral is from initial time to t.

The weight function w is unknown, but tapers to zero as time goes on. I've dropped the ∆ on F, now regarding F as a cumulative change from the initial value, but a similar integral with increments could be used (related by integrating by parts).

This is actually the general linear relation constrained by causality. w is called the impulse response function (IRF). It's the evolution (usually decay) that you expect of temperature following a heat impulse (F dt), in the absence of any other changes. That pulse F*dt at time τ would then have decayed by a factor w(t-τ) by time t, and by linear superposition, the effect of all the pulses of heat that would approximate the continuous F over the period adds to the stated integral.

I gave at WUWT the example of heating a swimming pool. Suppose it sits in a uniform environment, and has been heated at a uniform burn rate, and settles to a stable temperature. Then you increase the heating rate, and maintain it there. The response of the pool might be expected to be something like:



I'm actually using an exponential decay, which means the difference between current and eventual temperature decays exponentially. That is equivalent to the IRF being an exponential decay, and the curve has it convolved with a step function. The result, after a long time, is just the integral of w. This might be called the zeroth order moment. It is what I have marked in blue and called the ECS value. The linearity assumed is that this value varies linearly with the forcing jump. It doesn't mean that T is proportional to F. How could it be, if  F is held constant and T is still warming?

So let's look at that ECS convolution integral graphically:



Red is the forcing, blue the IRF, purple the overlap, which is what is integrated (IOW, all of it).

But suppose we are just interested in shorter periods. We could use a different forcing history, which would give a number relevant to the shorter term part of w. I'll use the linear ramp of TCS, this time just over the range 4-5:



I could have omitted the 0-4 section, since there is zero integrand there. However, I wanted to make a point. The integral now is weighted to the near end of the IRF. Since the IRF conventionally has value 1 at zero, it could be represented over the range of the ramp by a linear function, and the resulting integral would give the slope. That is why I identified TCS with the slope at zero of the response curve. But it isn't quite, because w isn't quite linear.

So this shows up a limitation of TCS. It doesn't tell about the part of w beyond its range. That's fine if we are applying it to a forcing known to be zero before the ramp commences. But usually that isn't true. We could extend the range of the TCS (if T data is available) but then it's utility of giving the derivative w'(0) wuld lessen. That means that it wouldn't be as applicable to ranges of other lengths.

Because of the general noise, both of observation and GCM computation, this method of getting aspects of the IRF w by trial forcings is really the only one available. It gives numbers of some utility, and is good for comparing GCMs or other diagnostics. And in the initial plot I tried to show that if you have the final value of the evolution and the initial slope, you have a fair handle on the whole story. That corresponds to knowing an ECS and a short term TCS.

But it isn't a "model" of T versus forcing. That model is the IRF convolution. The sensitivity definitions are just extracting different numbers that may help.