**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.

if you were to do a best fit to the data, placing today's warming at 1.1C with a total accumulated forcing of 2.5 W/m^2 but with an assumed lag time to reach full warming potential of ~8.5 years, what would your TCR be?

ReplyDeletet=137 years

forcing = 2.5/3.7 (percent of doubling CO2)

temp response = 90% of full (reaching full of 1.3 in 2025 - if all emissions halted today)

I can pretty much guarantee your TCR goes above 2.5

John, You are closer to the truth than others.

DeleteThe most important point to get across is to reinforce the idea of an

effectiveTCR and ECS for CO2 doubling.Who cares what the sensitivity of CO2 alone is when we all know that all the other GHGs either get multiplied by (i.e. H2O) or get dragged along with the CO2 (the other industrial pollutants, including methane).

It's obvious that the effective ECS is around 3 just by looking at the data. And that hasn't changed for years.

The TCR that corresponds to an ECS of 3 is closer to 1.6. This actually has changed quite a bit since the last AR from the IPCC. In addition, the rapid warming observed in the Arctic these last 3 years indicates that even these recent projections are biased low.

DeleteWithout being able to account without being able to account for 20th century aerosols, It’s not possible to get a good feature for TCR using or warming to-date.

DeleteI think you can estimate the ECS just by looking at warming on land only. Since there is less of a thermal capacitance (or heat sink) on land, it will respond quickly so in that case ECS~TCR.

DeleteTrue, that this doesn't take into account any recent warming.

@whut,

DeleteI don't think that's necessarily true. There's a reasonable expectation that it's not just the heat capacity, but other things like changes in the patterns of warming. Kyle Armour has a bunch of papers on this topic, here's a good place to start:

http://faculty.washington.edu/karmour/papers/Armour_etal_JClim2013.pdf

There are good reasons to expect that the ECS will look bigger than anything you'd get from a simple linear model right now. I haven't actually checked, but it might be true that land warming in models gives you their ECS, but that must just be coincidental if true!

"Since there is less of a thermal capacitance (or heat sink) on land, it will respond quickly so in that case ECS~TCR."

DeleteI don't think you can isolate the land from the oceans; both heat and humidity flow between them. So, directly, the temperatures will be "trying" to equalize between land and ocean which mitigates the effects of the temperature differential. And then I'm sure there are feedback effects from water vapor and clouds related to this differential being larger pre-equilibrium, too.

I'd not assume that the land warming approximates ECS.

I did do something like a proportional land/ocean model here: http://contextearth.com/2014/01/25/what-missing-heat/

DeleteI did put in place the depositing of latent heat from the ocean's moisture being deposited over land, which is this figure:

https://i0.wp.com/imageshack.com/a/img854/8775/l7g.gif

Alas, this no longer interests me too much, as it is just a single number that has way too much wiggle room with unknowns.

Much more important to work on ENSO, where it is far easier to make progress and to eliminate the unknowns with respect to variability.

"...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."

ReplyDeleteThis error, conflating the short-term temperature-forcing connection with the equilibrium temperature-forcing connection, also underlies Pat Frank's repeated errors about cloud feedback uncertainty.

He uses an equilibrium model to relate temperatures to forcing. Then adds cloud feedback uncertainty into that model. So far, so good; this would give you a static uncertainty in temperatures at equilibrium.

And then he integrates over time.

How you get time units into a time-insensitive equilibrium model, beats me. But you can't integrate over what's already been integrated over. Equilibrium models already integrate over temporal changes in forcings to reach equilibrium temperature.

Australian+USA team of 11 claim long-term ENSO behavior derived from coral proxy data is

ReplyDelete"reproduced once orbital forcing is taken into account"and that it"is not consistent with the unforced model simulations"https://www.stevenphipps.com/presentations/egu2015_enso.pdf

One of the most obvious disconnects in climate models is the reliance of orbital forcing to explain long-term paleo records, but the almost complete neglect (apart from the annual cycle) in including orbital forcing, in the form of precise lunar and solar path, to explain ENSO and related short-term variability.

DeleteCan see this weighting if you do a Google Scholar search on "orbital forcing"

Including lunar forcing should really be considered the null hypothesis

Nick,

ReplyDeleteYes ∆T = λ∆F is only true if you wait long enough for the climate to stabilise. The climate we experience currently is never in equilibrium so the global temperature in 2017 is a time dependent integral of all previous ∆Fs working through the 'system'.

That's what Nick wrote.

DeleteAnother approach is to use the climate carbon response function of Matthews et al, Nature 2009

ReplyDeletehttps://www.nature.com/articles/nature08047

which *is* linear after a few decades:

change in global mean temperature = 1.0–2.1 °C per trillion tonnes of carbon (Tt C) emitted (5th to 95th percentiles).

It has a fairly large uncertainty, but I believe it's the calculation done by policy makers when they say we have X amount of carbon we can emit before the world reaches Y °C of warming.

Here I disagree with you. How can logarithmic forcing of CO2 become linear? Well you need Earth Systems Models to perform this type of magic. There seems to be an inbuilt assumption in ESMs that the carbon cycle is going to saturate and that the airborne fraction of CO2 emissions will increase from 0.44 towards 1.0, in such a way that it exactly cancels out the logarithmic CO2 forcing.

DeleteThe carbon budget idea is a nice simple idea but unfortunately it's wrong as Miler et al. recently discovered. According to ESMs there should be more CO2 in the atmosphere than there actually is, given the current rate of emissions. Global temperatures are lower than predicted and the carbon budget left to reach 1.5C has increased.

The simple fact is that the airborne fraction has remained constant at ~0.44 so forcing is still logarithmic with carbon emissions. As Myles Allen put it “We haven’t seen that rapid acceleration in warming after 2000 that we see in the models. We haven’t seen that in the observations.”

If atmospheric CO2 is an exponential function of time (Keeling Curve), and forcing is a logarithmic function of CO2, then forcing will be a linear function of time.

DeleteCO2 forcing has remained close to linear so far:

https://www.esrl.noaa.gov/gmd/aggi/aggi.fig3.png

That maybe true but that is a different argument. Forcing is not linear with total Carbon budget.

DeleteQuote from Miller et al.

"A second important feature of ESMs is the increase in airborne fraction (the fraction of emitted CO2 that remains in the atmosphere after a specified period) over time in scenarios involving substantial emissions or warming (Friedlingstein et al., 2006). An emergent feature of the CMIP5 full complexity ESMs appears to be that this increase in airborne fraction approximately cancels the logarithmic relationship between CO2 concentrations and radiative forcing, yielding an approximately linear relationship between cumulative CO2 emissions and CO2-induced warming."

However their paper then found that AR5 carbon cycle models could not actually reproduce the Mauna-Loa CO2 concentrations from the historic emissions data. In short they needed to use a lower and stable airborne fraction to reproduce historic CO2 levels.

http://clivebest.com/blog/wp-content/uploads/2017/10/Millar-1.png

No one at last month's AGU meeting presented anything on oil depletion or really anything related to fossil fuel resource depletion. And overall, no one seems to have a handle on understanding how to convolve a CO2 emissions profile with a

Deletedispersed diffusionalCO2 sequestering response.Clive Best wrote:

ReplyDelete"...appears to be that this increase in airborne fraction approximately cancels the logarithmic relationship between CO2 concentrations and radiative forcing, yielding an approximately linear relationship between cumulative CO2 emissions and CO2-induced warming."

That's exactly what the Matthews et al paper I cited says.