The Fatal Lure of Assumed Linearity

Guest Post by Willis Eschenbach [note new Update at the end, and new Figs. 4-6]

In climate science, linearity is the order of the day. The global climate models are all based around the idea that in the long run, when we calculate the global temperature everything else averages out, and we’re left with the claim that the change in temperature is equal to the climate sensitivity times the change in forcing. Mathematically, this is:

∆T = lambda ∆F

where T is global average surface temperature, F is the net top-of-atmosphere (TOA) forcing (radiation imbalance), and lambda is called the “climate sensitivity”.

In other words, the idea is that the change in temperature is a linear function of the change in TOA forcing. I doubt it greatly myself, I don’t think the world is that simple, but assuming linearity makes the calculations so simple that people can’t seem to break away from it.

Now, of course people know it’s not really linear … but when I point that out, often the claim is made that it’s close enough to linear over the range of interest that we can assume linearity with little error.

So to see if the relationships really are linear, I thought I’d use the CERES satellite data to compare the surface temperature T and the TOA forcing F. Figure 1 shows that graph:

Figure 1. Land only, forcing F (TOA radiation imbalance) versus Temperature T, on a 1° x 1° grid. Colors indicate latitude. Note that there is little land from 50S to 65S. Net TOA radiation is calculated as downwelling solar less reflected solar less upwelling longwave radiation. Click graphics to enlarge.

As you can see, far from being linear, the relationship between TOA forcing and surface temperature is all over the place. At the lowest temperatures, they are inversely correlated. In the middle there’s a clear trend … but then at the highest temperatures, they decouple from each other, and there is little correlation of any kind.

The situation is somewhat simpler over the ocean, although even there we find large variations:

Figure 2. Ocean only, net forcing F (TOA radiation imbalance) versus Temperature T, on a 1° x 1° grid. Colors indicate latitude.

While the changes are not as extreme as those on land, the relationship is still far from linear. In particular, note how the top part of the data slopes further and further to the right with increasing forcing. This is a clear indication that as the temperature rises, the climate sensitivity decreases. It takes more and more energy to gain another degree of temperature, and so at the upper right the curve levels off.

At the warmest end, there is a pretty hard limit to the surface temperature of the ocean at just over 30°C. (In passing, I note that there also appears to be a pretty hard limit on the land surface air temperature, at about the same level, around 30°C. Curiously, this land temperature is achieved at annual average TOA radiation imbalances ranging from -50 W/m2 up to 50 W/m2.)

Now, what I’ve shown above are the annual average values. In addition to those, however, we are interested in lambda, the climate sensitivity which those figures don’t show. According to the IPCC, the equilibrium climate sensitivity (ECS) is somewhere in the range of 1.5 to 4.5 °C for each doubling of CO2. Now, there are a several kinds of sensitivities, among them monthly, decadal, and equilibrium climate sensitivities.

Monthly sensitivity

Monthly climate sensitivity is what happens when the TOA forcing imbalance in a given 1°x1° gridcell goes from say plus fifty W/m2 one month (adding energy), to minus fifty W/m2 the next month (losing energy). Of course this causes a corresponding difference in the temperature of the two months. The monthly climate sensitivity is how much the temperature changes for a given change in the TOA forcing.

But the land and the oceans can’t change temperature immediately. There is a lag in the process. So monthly climate sensitivity is the smallest of the three, because the temperatures haven’t had time to change. Figure 3 shows the monthly climate sensitivities based on the CERES monthly data.

Figure 3. The monthly climate sensitivity. 

As you might expect, the ocean temperatures change less from a given change in forcing than do the land temperatures. This is because of the ocean’s greater thermal mass which is in play at all timescales, along with the higher specific heat of water versus soil, as well as the greater evaporation over the ocean.

Decadal Sensitivity

Decadal sensitivity, also called transient climate response (TCR), is the response we see on the scale of decades. Of course, it is larger than the monthly sensitivity. If the system could respond instantaneously to forcing changes, the decadal sensitivity would be the same as the monthly sensitivity. But because of the lag, the monthly sensitivity is smaller. Since the larger the lag, the smaller the temperature change, we can use the amount of the lag to calculate the TCR from the monthly climate response. The lag over the land averages 0.85 months, and over the ocean it is longer at 2.0 months. For the land, the TCR averages about 1.6 times the monthly climate sensitivity. The ocean adjustment for TCR is larger, of course, since the lag is longer. Ocean TCR is averages about 2.8 times the monthly ocean climate sensitivity. See the Notes below for the calculation method.

Figure 4 shows what happens when we put the lag information together with the monthly climate sensitivity. It shows, for each gridcell, the decadal climate sensitivity, or transient climate response (TCR). It is expressed in degrees C per doubling of CO2 (which is the same as degrees per a forcing increase of 3.7 W/m2). The TCR shown in Figure 4 includes the adjustment for the lag, on a gridcell-by-gridcell basis.

Figure 4. Transient climate response (TCR). This is calculated by taking the monthly climate sensitivity for each gridcell, and multiplying it by the lag factor calculated for that gridcell. [NOTE: This Figure, and the values derived from it, are now updated from the original post. The effect of the change is to reduce the estimated transient and equilibrium sensitivity. See the Update at the end of the post for details.]

Now, there are a variety of interesting things about Figure 3. One is that once the lag is taken into account, some of the difference between the climate sensitivity of the ocean and the land disappears, and some is changed. This is particularly evident in the southern hemisphere, compare Southern Africa or Australia in Figures 3 and 4.

Also, you can see, water once again rules. Once we remove the effect of the lags, the drier areas are clearly defined, and they are the places with the greatest sensitivity to changes in TOA radiative forcing. This makes sense because there is little water to evaporate, so most of the energy goes into heating the system. Wetter tropical areas, on the other hand, respond much more like the ocean, with less sensitivity to a given change in TOA forcing.

Equilibrium Sensitivity

Equilibrium sensitivity (ECS), the longest-term kind of sensitivity, is what would theoretically happen once all of the various heat reservoirs reach their equilibrium temperature. According to the study by Otto using actual observations, for the past 50 years the ECR has stayed steady at about 130% of the TCR. The study by Forster, on the other hand, showed that the 19 climate models studied gave an ECR which ranged from 110% to 240% of the TCR, with an average of 180% … go figure.

This lets us calculate global average  sensitivity. If we use the model percentages to estimate the equilibrium climate sensitivity (ECS) from the TCR, that gives an ECS of from 0.14 * 1.1 to 0.14 *2.4. This implies an equilibrium climate sensitivity in the range of 0.2°C to 0.3°C per doubling of CO2, with a most likely value (per the models) of 0.25°C per doubling. If we use the 130% estimate from the Otto study, we get a very similar result, .14 * 1.3 = 0.2°C per doubling. (NOTE: these values are reduced from the original calculations. See the Update at the end of the post for details.]

This is small enough to be lost in the noise of our particularly noisy climate system.

A final comment on linearity. Remember that we started out with the following claim, that the change in temperature is equal to the change in forcing times a constant called the “climate sensitivity”. Mathematically that is

∆T = lambda ∆F

I have long held that this is a totally inadequate representation, in part because I say that lambda itself, the climate sensitivity, is not a constant. Instead, it is a function of T. However, as usual … we cannot assume linearity in any form. Figure 5 shows a scatterplot of the TCR (the decadal climate sensitivity) versus surface temperature.

Figure 5. Transient climate response versus the average annual temperature, land only. Note that the TCR only rarely goes below zero. The greatest response is in Antarctica (dark red). 

Here, we see the decoupling of the temperature and the TCR at the highest temperatures. Note also how few gridcells are warmer than 30°C. As you can see, while there is clearly a drop in the TCR (sensitivity) with increasing temperature, the relationship is far from linear. And looking at the ocean data is even more curious. Figure 6 shows the same relationship as Figure 5. Note the different scales in both the X and Y directions.

Figure 6. As in Figure 5, except for the ocean instead of the land. Note the scales differ from those of Figure 5.

Gotta love the climate system, endlessly complex. The ocean shows a totally different pattern than that of the land. First, by and large the transient climate response of the global ocean is less than a tenth of a degree C per doubling of CO2 (global mean = 0.08°C/2xCO2). And contrary to my expectations, below about 20°C, there is very little sign of any drop in the TCR with temperature as we see in the land in Figure 5. And above about 25°C there is a clear and fast dropoff, with a number of areas (including the “Pacific Warm Pool”) showing negative climate responses.

I also see in passing that the 30°C limit on the temperatures observed in the open ocean occurs at the point where the TCR=0 …

What do I conclude from all of this? Well, I’m not sure what it all means. A few things are clear. My first conclusion is that the idea that the temperature is a linear function of the forcing is not supported by the observations. The relationship is far from linear, and cannot be simply approximated.

Next, the estimates of the ECS arising from this observational study range from 0.2°C to 0.5°C per doubling of CO2. This is well below the estimate of the Intergovernmental Panel on Climate Change … but then what do you expect from government work?

Finally, the decoupling of the variables at the warm end of the spectrum of gridcells is a clear sign of the active temperature regulation system at work.

Bottom line? The climate isn’t linear, never was … and succumbing to the fatal lure of assumed linearity has set the field of climate science back by decades.

Anyhow, I’ve been looking at this stuff for too long. I’m gonna post it, my eyeballs are glazing over. My best regards to everyone,

w.

NOTES

LAG CALCULATIONS

I used the Lissajous figures of the interaction between the monthly averages of the TOA forcing and the surface temperature response to determine the lag.

Figure N1. Formula for calculating the phase angle from the Lissajous figure.

This lets me calculate the phase angle between forcing and temperature. I always work in degrees, old habit. I then calculate the multiplier, which is:

Multiplier = 1/exp(phase_angle°/360°/-.159)

The derivation of this formula is given in my post here. [NOTE: per the update at the end, I’m no longer using this formula.]

To investigate the shape of the response of the surface temperature to the TOA forcing imbalance, I use what I call “scribble plots”. I use random colors, and I draw the Lissajous figures for each gridcell along a given line of latitude. For example, here are the scribble plots for the land for every ten degrees from eighty north down to the equator.

Figure N2. Scribble plots for the northern hemisphere, TOA forcing vs surface temperature.

And here are the scribble plots from 20°N to 20°S:

Figure N3. Scribble plots for the tropics, TOA forcing vs surface temperature.

As you can see, the areas near the equator have a much smaller response to a given change in forcing than do the extratropical and polar areas.

DATA AND CODE

Land temperatures from here.

CERES datafile requisition site

CERES datafile (zip, 58 MByte)

sea temperatures from here.

R code is here … you may need eyebeach, it’s not pretty.

All data in one 156 Mb file here, in R format (saved using the R instruction “save()”)

[UPDATE] Part of the beauty of writing for the web is that my errors don’t last long. From the comments, Joe Born identifies a problem:

Joe Born says:

December 19, 2013 at 5:37 am

My last question may have been a little obscure. I guess what I’m really asking is what model you’re using to obtain your multiplier.

Joe, you always ask the best questions. Upon investigation, I see that my previous analysis of the effect of the lags was incorrect.

What I did to check my previous results was what I should have done, to drive a standard lagging incremental formula with a sinusoidal forcing:

R[t] = R[t-1] + (F[t] – F[t-1]) * (1 – timefactor) + (R[t-1] – R[t-2]) * timefactor

where t is time, F is some sinusoidal forcing, R is response, timefactor = e ^ (-1/tau), and tau is the time constant.

Then I measured the actual drop in amplitude and plotted it against the phase angle of the lag. By examination, this was found to be an extremely good fit to

Amplitude as % of original = 1 – e ^ (-.189/phi)

where phi is the phase angle of the lag, from 0 to 1. (The phase angle is the lag divided by the cycle length.)

The spreadsheet showing my calculations is here.

My thanks to Joe for the identification of the error. I’ve replaced the  erroneous figures, Figure 4-6. For Figs. 5 and 6 the changes were not very visible. They were a bit more visible in Figure 4, so I’ve retained the original version of Figure 4 below.

NOTE: THE FIGURE BELOW CONTAINS AN ERROR AND IS RETAINED FOR COMPARISON VALUE ONLY!! 

NOTE: THE FIGURE ABOVE CONTAINS AN ERROR AND IS RETAINED FOR COMPARISON VALUE ONLY!!