Guest Post by Willis Eschenbach
“Climate sensitivity” is the name for the measure of how much the earth’s surface is supposed to warm for a given change in what is called “forcing”. A change in forcing means a change in the net downwelling radiation at the top of the atmosphere, which includes both shortwave (solar) and longwave (“greenhouse”) radiation.
There is an interesting study of the earth’s radiation budget called “Long-term global distribution of Earth’s shortwave radiation budget at the top of atmosphere“, by N. Hatzianastassiou et al. Among other things it contains a look at the albedo by hemisphere for the period 1984-1998. I realized today that I could use that data, along with the NASA solar data, to calculate an observational estimate of equilibrium climate sensitivity.
Now, you can’t just look at the direct change in solar forcing versus the change in temperature to get the long-term sensitivity. All that will give you is the “instantaneous” climate sensitivity. The reason is that it takes a while for the earth to warm up or cool down, so the immediate change from an increase in forcing will be smaller than the eventual equilibrium change if that same forcing change is sustained over a long time period.
However, all is not lost. Figure 1 shows the annual cycle of solar forcing changes and temperature changes.
Figure 1. Lissajous figure of the change in solar forcing (horizontal axis) versus the change in temperature (vertical axis) on an annual average basis.
So … what are we looking at in Figure 1?
I began by combining the NASA solar data, which shows month-by-month changes in the solar energy hitting the earth, with the albedo data. The solar forcing in watts per square metre (W/m2) times (1 minus albedo) gives us the amount of incoming solar energy that actually makes it into the system. This is the actual net solar forcing, month by month.
Then I plotted the changes in that net solar forcing (after albedo reflections) against the corresponding changes in temperature, by hemisphere. First, a couple of comments about that plot.
The Northern Hemisphere (NH) has larger temperature swings (vertical axis) than does the Southern Hemisphere (SH). This is because more of the NH is land and more of the SH is ocean … and the ocean has a much larger specific heat. This means that the ocean takes more energy to heat it than does the land.
We can also see the same thing reflected in the slope of the ovals. The slope of the ovals is a measure of the “lag” in the system. The harder it is to warm or cool the hemisphere, the larger the lag, and the flatter the slope.
So that explains the red and the blue lines, which are the actual data for the NH and the SH respectively.
For the “lagged model”, I used the simplest of models. This uses an exponential function to approximate the lag, along with a variable “lambda_0” which is the instantaneous climate sensitivity. It models the process in which an object is warmed by incoming radiation. At first the warming is fairly fast, but then as time goes on the warming is slower and slower, until it finally reaches equilibrium. The length of time it takes to warm up is governed by a “time constant” called “tau”. I used the following formula:
ΔT(n+1) = λ∆F(n+1)/τ + ΔT(n) exp(-1/ τ)
where ∆T is change in temperature, ∆F is change in forcing, lambda (λ) is the instantaneous climate sensitivity, “n” and “n + 1” are the times of the observations,and tau (τ) is the time constant. I used Excel to calculate the values that give the best fit for both the NH and the SH, using the “Solver” tool. The fit is actually quite good, with an RMS error of only 0.2°C and 0.1°C for the NH and the SH respectively.
Now, as you might expect, we get different numbers for both lambda_0 and tau for the NH and the SH, as follows:
Hemisphere lambda_0 Tau (months)
NH 0.08 1.9
SH 0.04 2.4
Note that (as expected) it takes longer for the SH to warm or cool than for the NH (tau is larger for the SH). In addition, as expected, the SH changes less with a given amount of heating.
Now, bear in mind that lambda_0 is the instantaneous climate sensitivity. However, since we also know the time constant, we can use that to calculate the equilibrium sensitivity. I’m sure there is some easy way to do that, but I just used the same spreadsheet. To simulate a doubling of CO2, I gave it a one-time jump of 3.7 W/m2 of forcing.
The results were that the equilibrium climate sensitivity to a change in forcing from a doubling of CO2 (3.7 W/m2) are 0.4°C in the Northern Hemisphere, and 0.2°C in the Southern Hemisphere. This gives us an overall average global equilibrium climate sensitivity of 0.3°C for a doubling of CO2.
Comments and criticisms gladly accepted, this is how science works. I put my ideas out there, and y’all try to find holes in them.
w.
NOTE: The spreadsheet used to do the calculations and generate the graph is here.
NOTE: I also looked at modeling the change using the entire dataset which covers from 1984 to 1998, rather than just using the annual averages (not shown). The answers for lambda_0 and tau for the NH and the SH came out the same (to the accuracy reported above), despite the general warming over the time period. I am aware that the time constant “tau”, at only a few months, is shorter than other studies have shown. However … I’m just reporting what I found. When I try modeling it with a larger time constant, the angle comes out all wrong, much flatter.
While it is certainly possible that there are much longer-term periods for the warming, they are not evident in either of my analyses on this data. If such longer-term time lags exist, it appears that they are not significant enough to lengthen the lags shown in my analysis above. The details of the long-term analysis (as opposed to using the average as above) are shown in the spreadsheet.
Willis writes: “This is because more of the NH is land and more of the SH is ocean …”
The wording could be confusing to some. There’s more land in the Northern Hemisphere than Southern Hemisphere but the oceans cover a greater surface area (approx 60%) in the Northern Hemisphere than land.
Excellent Willis, just how it should be done.
Love that Lissajous! Using it forces us to feel the permanence and repeating nature of a cycle. (Hint: it would be even better to display it as a GIF, moving like a real o’scope. Then the direction of change would also be feelable.)
The usual statistical stuff (eg scattergram) omits time entirely … in other words, omits EVERYTHING THAT MATTERS.
Thank you for the simplest, most elegant model I have seen so far. This is simple engineering test at its best. The result corresponds very well with the thermostat hypothesis and with the Svensmark research on clouds. So, for me, the result is in:
0.3 C per doubling of CO2, very little change in the tropics and more warmup as we reach the poles. This is good for the environment, increases crop yields and result in a calmer weather.
More CO2 please to feed a starving world!!
Sensitivity to forcing depends on wave length and on whether it falls on land or on sea. Short-wave (solar) radiation goes down into water to the depth of 2 dozen meters and is absorbed completely, heating ocean. Long-wave (greenhouse backscattering) penetrates into water by several microns and almost all is spent on evaporation. In tropics and mid-latitudes it can not heat near-surface air, since here air is warmer than surface. In high latitudes water vapoure is condenced in near-surface air and warms it. That is why Arctic is more sensitive to greenhouse effect than tropics and mid-lattitudes. All radiation falling on land is absorbed, that is why Northern hemisphere, where most of landmasses are, is more sensitive to greenhouse forcing.
Hi Willis,
I think a source of discrepancy between what you have done and many other analyses is that there is not a single time constant, but a broad range An annual forcing cycle is too short a period to see the longer term responses. The simplest example is the difference between land and water. The much faster land leads the water by quite a lot because it is so much lower in heat capacity, but heat transfer between the land and ocean slows down the response of the land somewhat. (with less slowing in the north than the south, of course). The response is a complicated mixture of the two response times, but the influence of the very fast land yields a love value for apparent response. Any single constant model is going to hide the longer response times (which extend to many years for the ocean) and so lead to a very low calculated sensitivity. A more informative analysis can be done by looking at the response to Pinatubo, which extends over several years (You have to account for ENSO influence to see the Pinatubo effect clearly, which I think you have said you object to). This kind of analysis does not rely on seasonality and shows at least some of the longer term response.
F is the total so-called forcing.. If one works out the doubling of CO2 from the formulae developed (from empirical data and experimentation) by the late Prof Hoyt Hottel of MIT (who understood more about radiant heat transfer than all the so-called climate scientists including sceptics put together) then absorption is only just over 1.0 watts/m2 but that does not mean an increase in equivalent temperature. There will be an increase in potential and kinetic energy and radiation to space. No one knows the proportion of these. So the net effect on atmospheric temperature of CO2 increase could well be close to zero. Heat and mass transfer is an engineering discipline which no one connected with the IPCC understands. because a) they have no qualifications in engineering disciplines (including thermodynamics) and b) they have no experience with heat transfer equipment (eg furnaces, heat exchangers, refrigeration etc)
Interesting, but I don’t see causation between CO2 changes and temp changes. What I see is CO2 increasing and temperature changing, that’s it. I see no CAUSATION between the two. Correlation is not evidence of causation.
“A change in forcing means a change in the net downwelling radiation at the top of the atmosphere, which includes both shortwave (solar) and longwave (“greenhouse”) radiation”
There is an immediate problem with that definition. If we are the top of the atmosphere (TOA) then there will be no downwelling radiation from greenhouse gases for the obvious reason that there are no greenhouse gases above the TOA.
Strictly speaking, radiative forcing is defined as the change in net downward radiative flux at the tropopause.
Interesting approach and graphics.
David Stockwell of Niche Modeling has evaluated the impact of the solar cycle. See his Solar Accumulation theory. He finds a theoretical Pi/2 (90degree) lag or 2.75 years for an 11 year cycle. The annual cycle equivalent would be 3 months.
I strongly suspect that if Willis keeps up this clear, focused analysis on the climate, he will have earned a PhD in Science. This continues his outstanding contributions to understanding and knowledge of our climate control mechanism.
M Seward says:
May 29, 2012 at 3:30 am
A brilliant post that gets to the heart of this charade.
Webster, Clayson and Curry have measured the various fluxes, wind speed, rainfall and surface temperature in the Western Pacific region.
http://www.arm.gov/publications/proceedings/conf05/extended_abs/webster_pj.pdf
The temperature change during the diurnal cycle at the surface skin (which is what satellites measure) is quite large 5.8 degrees and this is drive by a change in flux of 778 W/m2. Thus, one degree on the surface is 134 w/m2.
There can be no lag in the > y-1 range, of sea temperature down to 5 meters, as the response of temperature changes of up to 5.8 degrees at the surface, or delta 778 W/m2, are instantaneous using a yearly temporal window.
The analogy is the beating heart and blood pressure, the energy input is cyclical, blood flow has a dampened pulsating dynamic profile, and ‘average’ pressure (max+min)/2 is not an adequate measure of the system. Maximum pressure can increase due to arterial dilation/vaso-constriction or due to increased heart rate.
A simple test of whether Eschenbach’s method is junk is to apply it to climate model output. The climate sensitivity of the models are known, so if Eschenbach’s massively underestimates it in the model, we can be confident that it will massively underestimate it in reality.
A second test is to consider if the climate forcing during the LGM could plausibly have been sufficiently large to give a global cooling of a few degrees with so low a climate sensitivity.
That is a perfect encapsulation of the situation, cementafriend. Wouldn’t you think a powerful force that can increase the Earth’s surface temperature by 33C could be directly measured by replicable experiments in a lab?
As an aside, over the weekend, i had the ‘pleasure’ of hearing a replay of an interview with Michael E. Mann by Michael Smerconish. Mann is smooth, you gotta give him that–he completely rolls Smerconish. He has his storyline down pat. As a friendly suggestion, don’t listen to this before you’ve had your breakfast.
http://soundcloud.com/smerconishshow/dr-michael-mann-the-hockey
I don’t believe this is a valid assessment of sensitivity.
By comparing only shortwave, you are missing the amount of thermal energy that’s either going
into or coming our of the oceans and the amount of shortwave that’s either going into the oceans, or heating the atmosphere.
On a global average, after all, the earth is warmest at aphelion and coolest at perihelion.
So it turns out that the oceanic buffers are almost completely out of phase with the solar forcing – over the seasonal variation.
NREL still has data posted with 30 year averages (1961 – 1990) of the measured total solar insolation at ground level, and measured local average temperature, by month, for a long list of locations in the U.S. While this data is useful for estimating solar PV performance, it also provides a useful database to calculate climate sensitivity, but just for the U.S. The website is here-
http://rredc.nrel.gov/solar/old_data/nsrdb/1961-1990/redbook/sum2/state.html
Back in 2007, I wanted to verify one of Idso’s natural experiments (#3) in his 1998 paper. For about 60 locations, one or two from each state, I calculated the maximum difference in temperatures between winter and summer, and divided by the maximum difference in TSI between winter and summer. Note that the maximum temperature differential occurs a month or two after the maximum TSI differential.
The nice feature of this approach is that small errors in either measurement have only a small impact on the calculated sensitivity. It does not rely on tiny differences between two large values, such as the top-of-atmosphere radiation imbalance.
The result is a short-term climate sensitivity of 0.08 C/W/m^2, or about 0.3 C for a doubling of CO2.
Sounds familiar.
The sensitivity calculated from coastal locations is lower than that calculated from inland locations. Small islands had the lowest sensitivity.
Sounds familiar again.
The sensitivity increases as a function of increasing latitude. The values range from <0.02 up to 0.11 C/W/m^2.
What happens when you try averaging the temperature and solar forcing in both hemispheres together, before diagnosing the sensitivity and time scale?
M Seward says:
May 29, 2012 at 3:30 am
I am not a practicing religious person but was bought up a church goer who walked away quite young. That said I am mindful of the advent of Protestant Christianity through the agency of Martin Luther and others…………….
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Except the historical Luther was much more like Michael Mann than any skeptic I know.
steve fitzpatrick says:
May 29, 2012 at 5:24 am
Hi Willis,
I think a source of discrepancy between what you have done and many other analyses is that there is not a single time constant, but a broad range An annual forcing cycle is too short a period to see the longer term responses….
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The longer term responses are slow and fairly small over short periods of time Over the short time period of 14 years where the annual swings in insolation and the response of temperature are large this is a good first approximation of the energy/temperature response of the climate.
This is really good Willis. There is a lot that can be done with this data.
I note that Hansen calculates his sensitivity as 0.75C per W/m2 change. Well, obviously this data is far, far below that value. But another interesting thing from the data is that the Northern Hemisphere Albedo value rises to 0.333 in the winter. Hansen, calculates his 0.75C per W/m2 by using an Albedo value in the ice ages of just 0.305 (in other words, he artificially understated the ice age Albedo so that he can get a greater GHG effect). There is no way, the current Northern Hemisphere winter has a higher Albedo than the annual average in the ice ages.
There is also the issue of the greenhouse effect which is about 150 W/m2 (Temp – solar forcing). But using this data, the value in the Northern Hemisphere changes from about 200 W/m2 in the winter to 100 W/m2 in the summer (now some of this is just due to absorption/emission from surfaces but it is somewhat unusual).
Even for a one-box model, your tau’s look quite low. If given a cyclical period and a response lag, the formula I use is:
tau=period*tan(2pi*lag/period)/2pi
For a 2 month lag of midlatitude SST’s on the annual 12 month cycle, I get tau=3.3; a lag of 2.5 months gives tau=7.12
For a 4 month lag on a 60 month ENSO cycle, I get tau=4.25
I’m not sure if the formula is correct, but the results seem reasonable.
I also suspect that if a one-box model was adequate, then you could estimate sensitivity by simply regressing temperature against ln(co2). I would think that the rate that heat was being extracted from the pipeline would quickly match the rate that it was being added. That is, the amount of heat in the pipeline would quickly approach an upper limit.
Willis,
how did you calculate dF and dT for the hemispheres?
Did you consider that heat is transported from one hemisphere to the other each day and each moment?
However, since we also know the time constant, we can use that to calculate the equilibrium sensitivity.
No you can’t. The only thing that time constant is telling you is that the year is a bit longer than 4*2.4 months (using SH). The response will never lag the forcing by more than 1/4 the cycle time, 3 months in this case. One year (or 15 years) is far too short for the ocean to reach equilibrium.
jrwakefield says:
May 29, 2012 at 6:03 am
Interesting, but I don’t see causation between CO2 changes and temp changes. What I see is CO2 increasing and temperature changing, that’s it. I see no CAUSATION between the two. Correlation is not evidence of causation.
Jeez. For the hundredth time, correlation is proof of causation. With the usual statistical caveats.
Use of the phrase ‘Correlation is not evidence of causation.’ is proof the utterer doesn’t understand science or statistics.