Sunday, March 18, 2012

A Better Way to Calculate Climate Sensitivity

Note added: The first commenter ("sylas") on my last post makes a very important point: this isn't the "climate sensitivity" I've calculated, because that technical term means after equilibrium has been reached, which, of course, is not the case -- more warming is "in the pipeline," as they say. (And it is a looooooong pipeline.) It's more the "transient climate response," and would be a lower bound on the climate sensitivity -- which, as he writes, you can't calculate from the data. Read his entire comment for more.

While I was driving down to Portland to take care of my nephew and niece for the evening, it occurred to me there's a better way to calculate climate sensitivity from the data, if you assume atmospheric CO2 levels are increasing exponentially with time t:

C(t) = C0 exp(at)

where "a" is some constant to be determined by the data. Then the climate sensitivity S is

S = [mT/a]*ln(2)

where mT = ΔT/Δt is the linear slope of temperature change with time. This avoids having to pick an N-year moving average and just uses the long-term trends in the data. It's what you'd expect for an exponentially increasing CO2 level, since then the time for CO2 to double (D) is just

D = ln(2)/a


S = D mT

which, of course, is just what you expect.

The constant "a" can be calculated by linear regression of ln(C/C0) versus time, and using the NOAA CO2 Mauna Loa data I find D = 166.0 years. The exponential assumption is a very good one (as Keeling discovered, of course,), since the Pearson correlation coefficient is R2 = 0.9914.

From the data I find:

mT(Hadley) = 0.126°C/decade

mT(UAH LT) = 0.135°C/decade

The difference is that the calculation for Hadley starts in 1958, when the CO2 data starts, and surface temperatures really didn't head upward strongly until the mid-70s when, shortly after, UAH started to record data. So

S(Hadley) = 2.1°C

S(UAH LT) = 2.2°C

You can calculate the statistical uncertainty too, since you know both of these from the data, which I'll do sometime soon.


Anonymous said...

I think one problem with this approach is that it assumes the change in radiative forcing is constant throughout the doubling, which is not the case. As an example using the standard CO2 RF formula - ln(C1/C0)*5.35:

280 -> 420ppm = 2.17 W/m^2

420 -> 560ppm = 1.54 W/m^2

Sensitivity is overestimated because we are currently in the first half of doubling (particularly so calculating from 1958), where RF change is greater than average.

It also ignores other GHGs, which don't have a simple exponential progression. CO2 represents about 65% of the long-lived GHG forcing ( and 50-60% of GHG forcing when you include Ozone. Potentially you're missing out half of the main cause of warming. And of course there is no accounting for aerosol forcing.

I realise you're trying to simplify things but I don't think such a simplification can provide a meaningful answer.

-Paul S

charlesH said...

" more warming is "in the pipeline,""

I have a hard time believing in this concept. Somehow the heat is stored somewhere and slowly heats up the oceans? Where is it stored?

Anonymous said...


Think about the same amount of energy being absorbed at every point on the globe. Now what happens to it?

On land, say a soil surface, energy can be transported downwards via conduction or cause a warming at the surface. Conduction is a very inefficient means of transport so mostly the energy warms the surface. You can test this by digging a hole in sand or soil. Dig down a metre or so and the soil at the bottom is considerably colder than at the surface.

On the oceans the energy can be transported downwards via convection. This is a much more efficient process so a considerable amount of the energy is vertically distributed in the oceans rather than simply heating the surface.

Let's go back to our simple situation in the first paragraph and say that the energy coming into the Earth = 1e ('e' being a fictional unit of energy). At a first (or zeroth) order we have an energy imbalance equal to 1e. Energy in = 1e, Energy out = 0e. Imbalance = ein - eout = 1e.

Now let's assume all the energy goes towards warming the surface. The surface will radiate almost immediately at an intensity that emits 1e out of the Earth, thus closing the imbalance.

In our more real world situation, where the oceans push down much of the energy but the land uses all energy to warm the surface, ee can see that the amount emitted from the overall surface warming will be lower. This means the energy emitted by the Earth will be lower than 1e. We now have a persistent imbalance whereby more energy is coming into the system than leaving. This imbalance will persist until the ocean surface warms sufficiently to emit enough energy out of the system.

To summarise, the reason there is warming "in the pipeline" is that the Earth has a heat capacity, mainly due to the oceans' ability to push down and store energy. This heat capacity means that the amount of warming needed to close an energy imbalance can't be realised immediately.

- Paul S