In some comment thread at Climate Audit, Willis Eschenbach estimated the sensitivity of Hansen’s GISS II model by fitting a straight line to the data. I commented, that, at least if the climate behaved like a simple lumped parameter “thing”, that fit would give misleading results for the sensitivity of Hansen’s GISS model.
Now, it may well be that the climate does not behave like a simple lumped parameter, “thing”, but these sorts of simple models can be useful for understanding how to look at both real data and model predictions. (Note: feel free to skim the equations– all the important stuff is shown in the charts. )
How does a simple lumped parameter climate work?
If we model the climate as a lumped parameter, we would basically claim the climate has one characterisitic temperature anomaly, θ, with a single time constant τ and heat capacity α-1. The equation for conservation of energy is linearized about an average temperature, and we expect for small excursions of temperature, we can predict the anomaly using this ordinary differential equation:
Where α-1 is the the effective heat capacity per unit area, τ is the time constant for the climate and q is a forcing (a heat flux.)
If we know the temperature anomaly at time zero, θ(0), and we know forcing between a time period 0 and t, we can obtain the temperature at time t by integrating over a dummy variable for time, as follows:
where integration is performed for values of t’ ranging from 0 to t. In principle, if we know the magnitude of τ and α and the forcing at all times after t=0, we can predict the temperature.
If the we know the the temperature at time=0 is equal to zero (θ(0)=0) and forcing is held constant at qo for a period after t=0, the solution to for the temperature obeys this equation:
For the 99.99% of the world who would rather see a graph than imagine the shape of the solution, for a value of (α τ qo ) =1C and &tau=10 years, temperature as a function of time is illustrated by the orange curve in the figure below:
The yellow line illustrates the “equilibrium” temperature. Because I said I was illustrating a sudden increase in the forcing at time =0, that line t is equal to 1 after t=0, but 0 before time =0. That’s equivalent to saying the equilibrium temperature suddenly jumped at time=1.
Because people always want to fit lines to these things, I’ve also illustrated what would happen if you fit a straight line to this curve. Notice the line doesn’t look much like the shape of the temperature trend over time.
(Oh, and in this problem, if we imagine that the forcing has changed in some way that can’t or wont’ be undone, the distance between the orange line– real temperature– and the yellow line– “equilibrium temperature” is the temperature change that’s “in the pipeline”.)
Now for the bit I said I’d explain to Willis!
Linear Ramp Function:
Suppose instead of thinking the forcing increased all at once, one thought the forcing increased linearly with time; that is q(t)= At.
Sort of the way it looks in the yellow curve below:
Well, in this case,
It turns out the solution to this equation is:
In most real problems, A and τ would just be numbers.
To illustrate a point about fitting lines, I plotted the solution (4) for particular values of A and &tau. It happens that τ = 10 years in this curve, but initially, I want you to ignore the numbers and focus on the shape of the curve.
Notice that near time=0, the temperature increases slowly. In fact, it’s possible to do expand the solution represented by equation (4) in a Taylor series, and show the temperature increases line a quadratic near time=0. However, at larger times, the temperature starts to obey a linear relationship, following the red line which has a slope of 0.1C /year.
Notice the blue line? That’s the line you would get if you fit a linear regression to that curve after collecting data for only 20 years: the regression line says temperature increased at a rate of 0.06 C/year. And, in fact, that is the average rate of increase during that time.
But, if you use the value of 0.06C/year to estimate the sensitivity of the planet, in this case you will underestimate the sensitivity which is, in this illustrative problem, know to be 0.1C/year. (I know this because I picked the product A &tau = 0.1C/year!)
Notice also, that I illustrated a yellow line: that’s “the equilibrium temperature”. If, at some time “t” forcing stopped increasing, and remained constant, the yellow line would then switch to a horizontal line at that time. What would happen to the real temperature? The real temperature (orange line) would begin to approach that yellow line, following an exponential decaying curve.
What does this have to do with Willis’s curve fit to Hansen’s GISS II model data?
When Willis fit a curve to Hansen’s data, he hoped to estimate the time constant. To do that, he assumed he could get the sensitivity by first fitting a line and getting the change in temperature as a function of time.
But, unfortunately, if you eyeball Hansens’s model curves, you’ll notice the temperature dependence looks more like a quadratic near time =0. And this isn’t just statistical uncertainty– we know that Hansen’s scenarios increased forcing either linearly after time=0 (1958) or more rapidly.
So, if you fit a line to the model predictions in this particular case, you will under estimate the sensitivity. Sad… but alas, true. How badly will you underestimate the sensitivity? That depends on
a) the time constant in Hansen’s model and
b) whether Hansen’s forcing is simply linear or whether it increases more rapidly than linear. (That is, as a quadratic, an exponential etc.)
The larger the time constant in his model, the more you’ll underestimate the sensitivity. If his forcing increases quadratically, or as an exponentially, you’ll under estimate the sensitivity more than otherwise.
But, unfortunately, for all three cases, you tend to underestimate!
lucia, thank you for discussing this issue. A quick point. You say:
In fact, I did the analysis neglecting the time constant entirely, looking merely to estimate the climate sensivity (degrees C per Watt/metre^2).
More later, you have an interesting analysis, I need to consider the math …
w.
PS – can we post images here?
Transient model forcings result in a climate sensitivity range of 1-3 K/2xCO2.
http://www-pcmdi.llnl.gov/model_appraisal.pdf
A few long range models gave a response time of typically 200-500 years, here the 50% IPCC add on originates: 1 to 3 transient –> 1.5 to 4.5 equilibrium.
http://www.grida.no/climate/ipcc_tar/wg1/fig9-1.htm
Graphically it works like this,
what you also see in the same graph is that empirical data all points to a low climate sensitivity.
—-
Edited: I inserted graphs.
Willis– yes you can post images. I need to add an image link to the quick tags to help people who don’t know the html.
I’m going to edit Hans comment to turn the image into an image.
Hans–
I fit this simple lumped parameter earth to GISS Land/Ocean data and right now, I get a climate sensitivity near 1.9-2.0. (It depends on what I think about using surface albedo as a forcing or feedback. Ice melts when the temperature rises, so it may be that for the purpose of this physically based curve fit, it’s a feedback. The “feedback” idea makes the sensitivity 2.0.)
Setting surface albedo to “feedback”, I get a time constant near 15-30 years. The time constants all shift around as I fix my modules for slight errors– or write small VBS codes to make it easier to get Excel to converge quickly, my time constants move around. (The time constants move more than the sensitivity!)
Anyway, I’m in that “between” stage, where I think I know my main results, but I really want to check things.
Lucia, when you say “climate sensitivity”, do you mean per doubling or per watt/m2?
Thanks,
w.
Willis– I usually try to mean use it as sensitivity to doubling of CO2, which I guess is supposed to give 3.6 W/m^2 before feedbacks.
With regard to this model, the product The equilibrium temperature = α τ q
Where q is an added forcing. I still haven’t found boo-boos in my spread sheet, and I’m getting for 1 watt/m^2 of forcing, an equilibrium temperature increase of about 0.6 C. So, for 3.6 W/m^2 I get about 2C. (This is after taking surface albedo increases out of the forcings Gavin pointed me to. I did that on the assumption they are, at their core, feedbacks in the sense that ice melts because the earth’s temperature rises. I get less sensitivity if I leave those in as forcing.)
Of course, I also get a noticeable amount heat “in the pipeline”, since forcing has been increasing. Still, not a huge amount “in the pipeline” because I’m getting a time constant near 8-9 years, not 10-20 years as suggested by some others.
I haven’t tried to run the Hansen ABC scenarios through this with their exact pattern of forcings. It might be worth a shot to see if I get close the the sensitivity Hansen reported for his model–but I figure I’ll do that at some point.
Thank you, lucia. What sensitivity did Hansen report?
I get the following best fits with the Hansen scenarios.
Scenario, Climate Sensitivity (°C/W-m2), Time Constant tau (Years)
a, 0.5, 8
a & b, 0.6, 14
a & c, 0.5, 10
b, 1.0, 38
b & c, 0.6, 19
c, 0.5, 19
all, 0.5, 13
Scenario “B” is obviously overfit. This is a problem with local minimization algorithms. I suspect the true value of B is pretty close to the values of C, since “B & C” is about the same as “C”.
That’s what I can tell you about planet GISSII.
w.
luica, I have what is very likely an ill-posed comment/question
It seems to me that whatever the quantity called Temperature, T, is, it must have a thermodynamic connection with the quantity called Energy, q, in your nomenclature.
Additionally I think that thermodynamics requires that whenever Energy is added into a material the Temperature increases, and when Energy is removed the Temperature decreases. My interpretation of the situation is as follows. The Temperature reported in some of the plots is some kind of rough approximation of the Temperature within the atmosphere. I’ll try to get back to those plots that have some kind of average for the Land + Ice + Ocean. But the Temperature being measured is not an accurate reflection of the radiative-equilibrium balance approach. The 0-D radiative-equilibrium model energy balance equation cannot capture the physical phenomena and processes that dominate and control the quantity measured as T. The question of equilibrium is a whole nother open issue imo.
The plots of the quantity T do not show a monotonic dependence of the Temperature with Energy; taking the Energy to be monotonically increasing as time increases. The smoothing and/or neglecting of the oscillatory nature of the Temperature does give a more or less monotonic increasing of Temperature. But maybe the time periods for which the Temperature is decreasing are trying to tell us something. Plus something deep in the recesses keep saying to me that the specific heats must be positive numbers. The variability in the plotted Temperature, when decreasing, means that Energy has been removed from the system, if the Temperature is the physical quantity associated with the Energy. My understanding is that there is seldom an actual net reduction of Energy in the Earth system. More is incoming than outgoing; let me know if that is not correct. So when a decrease in the Temperature is measured, that actually means that the phenomena and processes down here have taken control of the Temperature. The Energy already added into the system has caused/been-a-part-of some transport/storage processes that result in changes in the Temperature. Neglecting for a moment the cases for which Energy does in fact get blocked/reflected back. By the same token, when the Temperature is increasing that very likely is not an indication that an excess Energy addition has occurred in contrast to that processes here that control the Temperature have simply changed to other processes.
Whatever the case, the radiative transport problem/model in no ways reflects the actual physical system. The media through which the radiation, in both directions, passes is an interacting media, as you well know. Plus after the energy gets to the surface the surface is not a purely radiative body. All the radiative transport properties of the interacting surface vary all over the map (you might say). Covering the full ranges for about 0.0 to about 1.0. The energy is stored and transported in all the stuff here on the surface in addition to a part acting in a radiative-energy-transport way.
Hey I said that this was probably an ill-posed comment/question.
The Thermodynamic phenomena and processes present/undergoing down here are a heat engine in which the Energy additions to the atmosphere provide the driving potentials required to move fluids from regions of higher Temperature (Energy) to regions of lower Temperature (Energy). Typically from the tropics toward the poles. The poles, being at lower Temperature level, cannot reject all the Energy transported to those ares. And here I’ll guess that the greater Temperature increases at the North compared to the South is a reflection of the larger amounts of liquid and solid phases of water in the South. The liquid form, of course, has a high specific heat (and there’s tons of it around) and the solid form can absorbed Energy at constant temperature after it reaches the melting temperature. Getting it up to the melting temperature might also require significant amounts of energy, I haven’t made an approximation.
I think the Temperature measured in the atmosphere is more likely a function of states of this heat engine at the locations where the measurements are made than a function of the Energy additions to the system. There are of course simple systems for which Energy additions act solely to increase the Temperature; the Earth is not such a system. Phase-change and Energy storage and transport processes within the Earth system dominate the Temperature here, I think.
Plus, the various motions, large scale bulk motions in the atmosphere and oceans can, and do, affect the numbers reported to be the Temperature of the day at all locations. But, again, it is the motions and not the Energy additions that have caused the variations in the Temperature. Some of the motions are of course a result of the Energy additions. The Temperature is a function of which way the wind is blowing, jet stream, macro-scale motions in the oceans, etc. And the macro-scale conditions near the measuring stations will significantly affect the reported values of the Temperature. Again, functions of things other than Energy additions.
The quantity being measured near the surface is the Weather Temperature. It will always be the Weather Temperature. I think to take it to be the radiative-equilibrium Temperature is not the right thing to do. I guess the assumption is that long-term averages of the Weather Temperature are in fact the radiative-equilibrium Temperature. Is that assumption sound? If the Weather Temperature continues to decrease as well as increase while all the time the Energy content is increasing, I think the assumption needs to be examined in a little more depth.
The typical 0-D model energy balance equations do not account for the heat engine processes, constant-temperature phase-change processes, dissipation, work. I certainly understand the concepts on which such an approach is based. And maybe everyone is happy even when they see a Temperature decreasing, for whatever the time period, while the Energy content of a system is increasing.
I suggest improved 0-D modeling based on
1. A radiative-equilibrium balance written for its more appropriate physical system in the overall scheme of things. I don’t know what this equation might include nor where in physical space it should be applied.
2. 0-D model energy balance equations that account for the heat-engine processes occurring down here on the surface; storage, phase change, transport, work dissipation, and radiative Energy additions.
I’m certain that this is in fact the approach taken in the early modeling days. However I think a good argument can be made that deeper understanding of the system and its responses might in fact be more readily available through study of these more simple approaches. Additionally there is software that will fit parameter values appearing in ODEs to data. So as the number of ODEs increases to be beyond hand/analytical work, the software can save the day.
That’s as far as I’ve gone on this, so it’s a very rough draft.
Have at this; correct, respond, snip, delete, ignore.
oops, in my comment above under item 2. there should be a comma between ‘work, dissipation’
Probably a very minor thing compared to the possible incorrectos in the big picture concept.
Dan–
I’m still doing some consistency checks I haven’t explained. The involve what happens on average during the yearly cycle.
If this passes, I’ll be writing up how “Temperature” relates to actual physical quantities. But, it’s true that the surface temperature is not the radiating temperature, and that zero order models are a bit odd when you start to try to connect various things directly to local instantaneous physical processes we can think about in a more direct concrete way.
I’d say more, but quite honestly, your comment is good, but I need to answer with a full blog post with equations. That will take me about… two weeks to a month. (Unless I get lazy, then it will be longer, which would be my deficiency.)
Lucia, I have been doing something similar and interested in what you were doing here. I had a perplexing issue though with sensitivity and the forcing file. When the file has a net forcing of 1W/m2 for a specific year say, you can plot that against temperature increase for the century say, and you get a temperature increase of 0.6C say.
But, if you use an autoregressive AR1 model, then the net forcing required per time step is correspondingly orders of magnitude smaller. Say, 0.01 W/m2 is needed per year to get an AR1 series up to a change of 1C/century. But are you perhaps expressing this as a time delay?
So if temperature is AR1, and there is good evidence it is, then the forcings in the net forcings file represent the sum, or the integral of forcings over the entire period and not the instantaneous forcing per year, which is 100 times smaller. This seems really obvious by is confusing me as I havent seen anything about this. Do you follow?
Cheers
David–
Watts are already in units “Energy/time”. Since I’m a mechanical engineer, I always think dimensionally (before making dimensional), so when I integrate, the when I use “months”, the energy added is (Energy/time) * (time increment) = Energy. So, I think I properly account for energy added in a month. (Depending on which stage of “Lumpy” file, I did this explicitly. I did have to when I switched from month data to year data — because yes, 1 Watt/m^2 acting for 1 month is 1/12th that acting for a year.
I think often statistical packages turn things into “trend per increment in the series”. When they do, the user as to figure out the physical units, and if your increments are monthly, you need to convert to yearly.