Into The Black Box

Guest Post by Willis Eschenbach

Through what in my life is a typical series of misunderstandings and coincidences, I ended up looking at the average model results from the Climate Model Intercomparison Project 5 (CMIP5). I used the model-by-model averages from each of the four scenarios, a total of 38 results. The common period of these results is 1860 to 2100 or some such number. I used the results from 1860 to 2020, so I could see how the models were doing without looking at some imaginary future. The CMIP5 analysis was done a few years ago, so everything up to 2012 they had actual data for. So the 163 years from 1860 to 2012 were a “hindcast” using actual forcing data, and the eight years from 2013 to 2020 were forecasts.

Figure 1. CMIP5 scenario averages by model, plus the overall average.

There were several things I found interesting about Figure 1. First was the large spread. Starting from a common baseline, by 2020 the model results ranged from 1°C of warming to 1.8°C of warming …

Given that horrible inter-model temperature spread in what is a hindcast up to 2012 plus eight years of forecasting, why would anyone trust the models for what will happen by the year 2100?

The other thing that interested me was the yellow line, which reminded me of my post entitled “Life Is Like A Black Box Of Chocolates“. In that post I discussed the idea of a “black box” analysis. The basic concept is that you have a black box with inputs and outputs, and your job is to figure out some procedure, simple or complex, to transform the input into the output. In the present case, the “black box” is a climate model, the inputs are the yearly “radiative forcings” from aerosols and CO2 and volcanoes and the like, and the outputs are the yearly global average temperature values.

That same post also shows that the model outputs can be emulated to an extremely high degree of fidelity by simply lagging and rescaling the inputs. Here’s an example of how well that works, from that post.

Figure 2. Original Caption: “CCSM3 model functional equivalent equation, compared to actual CCSM3 output. The two are almost identical.”

So I got a set of the CMIP5 forcings and used them to emulate the average of the CMIP5 models (links to models and forcings in the Technical Notes at the end). Figure 3 shows that result.

Figure 3. Average of CMIP5 files as in Figure 1, along with black box emulation.

Once again it is a very close match. Having seen that, I wanted to look at some individual results. Here is the first set.

Figure 4. Six scenario averages from different models.

An interesting aspect of this is the variation in the volcano factor. The models seem to handle the forcing from short-term events like volcanoes differently than the gradual increase in overall forcing. And the individual models differ from each other, with the forcing in this group ranging from 0.5 (half the volcanic forcing applied) to 1.8 (eighty percent extra volcanic forcing applied). The correlations are all quite high, ranging from 0.96 to 0.99. Here’s a second group.

Figure 5. Six more scenario averages from different models.

Panel (a) at the top left is interesting, in that it’s obvious that the volcanoes weren’t included in the forcing for that model. As a result, the volcanic forcing factor is zero … and the correlation is still 0.98.

What this shows is that despite their incredible complexity and their thousands and thousands of lines of code and their 20,000 2-D gridcells times 60 layers equals 1.2 million 3-D gridcells … their output can be emulated in one single line of code, viz:

T(n+1) = T(n)+λ ∆F(n+1) *(1-exp( -1 / τ ))+ ΔT(n) exp( -1 / τ )

OK, now lets unpack this equation in English. It looks complex, but it’s not.

T(n) is pronounced “T sub n”. It is the temperature “T” at time “n”. So T sub n plus one, written as T(n+1), is the temperature during the following time period. In this case we’re using years, so it would be the next year’s temperature.

F is the radiative forcing from changes in volcanoes, aerosols, CO2, and other factors, measured in watts per square metre (W/m2). This is the total of all of the forcings under consideration. The same time convention is followed, so F(n) means the forcing “F” in time period “n”.

Delta, or “”, means “the change in”. So ∆T(n) is the change in temperature since the previous period, or T(n) minus the previous temperature T(n-1). Correspondingly, ∆F(n) is the change in forcing since the previous time period.

Lambda, or “λ”, is the scale factor. Tau, or “τ”, is the lag time constant. The time constant establishes the amount of the lag in the response of the system to forcing. And finally, “exp (x)” means the number 2.71828 to the power of x.

So in English, this means that the temperature next year, or T(n+1), is equal to the temperature this year, T(n), plus the immediate temperature increase due to the change in forcing, λ F(n+1) *(1-exp( -1 / τ )), plus the lag term, ΔT(n) exp( -1 / τ ) from the previous forcing. This lag term is necessary because the effects of the changes in forcing are not instantaneous.

Curious, no? Millions of gridcells, hundreds of thousands of lines of code, a supercomputer to crunch them … and it turns out that their output is nothing but a lagged (tau) and rescaled (lambda) version of their input.

Having seen that, I thought I’d use the same procedure on the actual temperature record. I’ve used the Berkeley Earth global average surface air temperature record, although the results are very similar using other temperature datasets. Figure 6 shows that result.

Figure 6. The Berkeley Earth temperature record (left panel) including the emulation using the same forcing as in the previous figures. I’ve included Figure 3 as the right panel for comparison.

It turns out that the model average is much more sensitive to the volcanic forcing, and has a shorter time constant tau. And of course, since the earth is a single example and not an average, it contains much more variation and thus a slightly lower correlation with the emulation (0.94 vs 0.99).

So does this show that forcings actually rule the temperature? Well … no, for a simple reason. The forcings have been chosen and refined over the years to give a good fit to the temperature … so the fact that it fits has no probative value at all.

One final thing we can do. IF the temperature is actually a result of the forcings, then we can use the factors above to estimate what the long-term effect of a sudden doubling of CO2 will be. The IPCC says that this will increase the forcing by 3.7 watts per square meter (W/m2). We simply use a step function for the forcing with a jump of 3.7 W/m2 at a given date. Here’s that result, with a jump of 3.7 W/m2 in the model year 1900.

Figure 7. Long-term change in temperature from a doubling of CO2, using 3.7 W/m2 as the increase in forcing and calculated with the lambda and tau values for the Berkeley Earth and CMIP5 Model Average as shown in Figure 6.

Note that with the larger time constant Tau, the real earth (blue line) takes longer to reach equilibrium, on the order of 40 years, than using the CMIP5 model average value. And because the real earth has a larger scale factor Lambda, the end result is slightly larger.

So … is this the mysterious Equilibrium Climate Sensitivity (ECS) we read so much about? Depends. IF the forcing values are accurate and IF forcing roolz temperature … maybe they’re in the ballpark.

Or not. The climate is hugely complex. What I modestly call “Willis’s First Law Of Climate” says:

Everything in the climate is connected with everything else … which in turn is connected with everything else … except when it’s not.

And now, me, I spent the day pressure-washing the deck on the guest house, and my lower back is saying “LIE DOWN, FOOL!” … so I’ll leave you with my best wishes for a wonderful life in this endless universe of mysteries.


My Usual: When you comment please quote the exact words you are discussing. This avoids many of the misunderstandings which are the bane of the intarwebs …

Technical Notes:

I’ve put all of the modeled temperatures and forcing data and a working example of how to do the fitting as an Excel xlsx workbook in my Dropbox here.

Forcings Source: Miller et al.

The forcings are composed of:

  • Well mixed greenhouse gases
  • Ozone
  • Solar
  • Land Use
  • Snow Albedo & Black Carbon
  • Orbital
  • Troposphere Aerosols Direct
  • Troposphere Aerosols Indirect
  • Stratospheric Aerosols (from volcanic eruptions)

Model Results Source: KNMI

Model Scenario Averages Used: (Not all model teams provided averages by scenario)


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February 3, 2022 10:09 am

What! No solar activity forcing?

Reply to  Leif Svalgaard
February 3, 2022 10:15 am

In the voice of Arnold Schwarzenegger, “No, it’s kah-bun.”

Reply to  Leif Svalgaard
February 3, 2022 10:34 am

I envisage Ol Sol peeping up over the horizon every morning and spying the latest climate equation, which gives him a good laugh to start his day’s work.

He probably has another chuckle about climate equations before he shuts up shop for the day, knowing there will be a new one tomorrow morning to keep the comedy coming.

Because –
“That lucky old sun
has nothin’ to do
but rolll around heaven all day”

Reply to  Mr.
February 3, 2022 11:17 am

Absolutely, sun is the big dog, insolation energy + GCR control valve. Its minor detructor is the Earth itself via magnetic field and volcanic eruptions. Four variables none predictable, but all within a reasonably well defined range, keeping the planet’s climate relatively stable on the centenary to millennial range. On longer time scale Milankovic cycles and possibly galactic transit come to force.

Jeff Alberts
Reply to  Mr.
February 3, 2022 8:01 pm

Fortunately for us, the Sun never shuts up shop for the day.

Reply to  Willis Eschenbach
February 3, 2022 12:00 pm

But the ‘solar forcing’ does not match the modern sunspot record.

John Tillman
Reply to  Leif Svalgaard
February 3, 2022 12:40 pm

IMO, it does. Just not immediately.

The LIA was characterized by repeated solar minima.

The Medieval and Modern Warm Periods, and their antecedants, ie the Holocene Climatic Optimum, Egyptian, Minoan and Roman WPs, enjoyed solar maxima.

But I liked your initial comment.

Reply to  John Tillman
February 3, 2022 1:10 pm

The minima back then were not any deeper than now.
The radiation we get from the sun (TSI) is determined as the sum of dark sunspots and bright magnetic background fields. The latter dominates and overwhelm the sunspots. With no sunspots, it is very possible that TSI was higher during the LIA than now…

John Tillman
Reply to  Leif Svalgaard
February 3, 2022 1:59 pm

But TSI doesn’t rule.

What matters is UV flux.

Reply to  John Tillman
February 3, 2022 2:56 pm

which is 10 times smaller…

John Tillman
Reply to  Leif Svalgaard
February 3, 2022 3:02 pm

Yet UV effect isn’t just energy, but quality of effect. It makes and breaks ozone and rule air ptessure, influencing the trade winds, which account for ENSO fluctuations.

John Tillman
Reply to  Leif Svalgaard
February 3, 2022 3:16 pm

Ten times on average, but it swings wildly.

John Tillman
Reply to  Leif Svalgaard
February 3, 2022 2:55 pm

Almost certainly not, but it doesn’t matter. What does matter is UV flux.

Joel O'Bryan
Reply to  Leif Svalgaard
February 3, 2022 6:04 pm

Now that thar’ is funny.

Reply to  Leif Svalgaard
February 3, 2022 12:54 pm

Hi doc
.. and why would anyone expect that the single partly subjective visual assessment (modern or historic) of one of a number of aspects of the solar activity, would always be a dominant factor of the planet’s climate quasi periodic oscillations.

Reply to  Vuk
February 3, 2022 1:11 pm

lots of people expect that…

Robert W Turner
Reply to  Willis Eschenbach
February 3, 2022 4:10 pm

Right now you can see on satellite how important a single geographic feature like the Isthmus of Tehuantepec can be to global climate. Positioning of the oceans/continents is one of the primary forcings, some of these others are noise.

Pat from kerbob
Reply to  Robert W Turner
February 3, 2022 8:25 pm

If we want to really experiment with geoengineering wouldn’t the be all and end all (likely “end all”) be to open the ithsmus of Panama?
Biggest earth moving project in history, completely alter ocean currents just to see what could happen?

What could go wrong.

Reply to  Pat from kerbob
February 5, 2022 7:59 am

Are you saying that the canal did that already? Any references, sorry if it it sounds like a rude challenge or something, but the canal is tiny in comparison to the landmasses around it.
The global warming panic, the climate emergency that is driving people batty is built on a lousy 1-2°C warming over the past ~200 years which is similar to the swing in temperatures from previous warm or cold periods, so basically, situation normal on planet Earth, regardless of the Panama Canal.

Now, if you were talking about REALLY opening wide the Isthmus of Panama so a significant opening is created, enough to affect weather and climate, well that would be interesting to talk about!

Reply to  Pat from kerbob
February 6, 2022 5:12 pm

I thought there was a difference in sea levels between the 2 sides . . could stick a few turbines in the flow and global energy problem solved.

Reply to  Leif Svalgaard
February 4, 2022 12:53 am

Solar forcing is coupled with clouds.
As Willis E well has described.
Solar brightening and solar dimming is a fact and not well explained.
We have 20% more sun(h) today compared to 1980

Dave Yaussy
February 3, 2022 10:16 am

Nice job, Willis. Like many WUWT denizens, I look forward to your articles, and I learn a lot from them.

I also learn quite a bit from the responses, even the critical ones. Some responses make me think. Some responses make me think the author is a pretentious prat. Takes all kinds.

Hope your back feels better soon.

John Tillman
February 3, 2022 10:21 am

Somehow models supposed to be skillful and fit for policy purposes out to AD 2100 missed the past six years’ global cooling and the 1998-2014 Pause.

John Tillman
Reply to  John Tillman
February 3, 2022 10:43 am

Thus, 21st century warming was due to Super El Niño of 2015-16.

Indeed warming since the 1977 PDO shift is largely the story of three Super Los Niños, ie 1982-83, 1997-98 and 2015-16. And they are driven by solar variations.

Reply to  John Tillman
February 5, 2022 8:04 am

Yes! That’s exactly what the more impartial satellite temperature record looks like: step change in temps after an El Nino followed by steady periods – not something expected from steady CO2 increases.

February 3, 2022 10:28 am

“That same post also shows that the model outputs can be emulated to an extremely high degree of fidelity by simply lagging and rescaling the inputs.”

We’ve been through this before. The forcings are not the inputs to the models. They are deduced from the outputs, or possibly from some intermediate steps. The “one single line of code” is very close to what was used to calculate F. So of course the output temperatures “follow” the forcings. It is just using the formula by which they were deduced in reverse.

Frank from NoVA
Reply to  Nick Stokes
February 3, 2022 10:48 am

“The forcings are not the inputs to the models. They are deduced from the outputs, or possibly from some intermediate steps.”

No? Then what are the inputs?

Reply to  Frank from NoVA
February 3, 2022 10:52 am

Gas concentrations, as per scenario. Plus, of course, aerosols, insolation, etc.

Frank from NoVA
Reply to  Nick Stokes
February 3, 2022 11:14 am

“Gas concentrations, as per scenario.”

Like CO2 and CH4? What’s the difference between inputting a forcing in terms of w/m^2 and inputting a change in gas concentration that the GCM, or some subroutine thereof , will directly convert to w/m^2?

Reply to  Frank from NoVA
February 3, 2022 1:53 pm

“What’s the difference between inputting a forcing in terms of w/m^2 and inputting a change in gas concentration that the GCM”

The important difference is that the former is not done. The reason is that it can’t be. The forcing cited here is a global aggregate. What a GCM needs is a distribution to determine each cell value. With gases you can do that, because they are well-mixed. Maybe not perfectly, but any corrections needed are small. A number for global aggregate forcing can’t provide cell values. 

Reply to  Willis Eschenbach
February 3, 2022 5:02 pm

Nick is playing the usual deflection games .. ignore him.

Reply to  Willis Eschenbach
February 3, 2022 7:40 pm

” In that it is clear that many of the forcings are NOT calculated by the model”

No, the link (Miller et al) makes clear that they are. Here is their familiar plot of forcings

comment image

And here is what they say about how the plot was calculated. 

comment image

In Table 2 they even list the model runs that were used.

Reply to  Willis Eschenbach
February 4, 2022 12:16 am

Yes, but MAGICC is not a GCM. It is a simple model using hemispheric averages. So it doesn’t help your contention that “(GCM) model outputs can be emulated to an extremely high degree of fidelity by simply lagging and rescaling the inputs”.

Reply to  Willis Eschenbach
February 5, 2022 8:16 am

Top notch job as always! Love it when I see a new article from you is posted on WUWT.

Joel O'Bryan
Reply to  Nick Stokes
February 3, 2022 10:52 am

If it is purely circular in construct, then why does the CMIP model ensemble average run 1.5ºC to 3.0ºC higher than the emulation Willis shows here?

Joel O'Bryan
Reply to  Joel O'Bryan
February 3, 2022 10:56 am

hint: I know the answer. Are you willing to admit it Nick?

Reply to  Joel O'Bryan
February 3, 2022 11:06 am

This is how it so often goes here:
“Look how a forcing formula emulates temperature so well!”
“But that is because forcings were calculated from that formula”
“So why does the formula emulate temperature so badly?”

In fact Willis’ formula determines temperature differences. It does not determine the offset. 

Frank from NoVA
Reply to  Nick Stokes
February 3, 2022 11:37 am

“Look how a forcing formula emulates temperature so well!”

  • That’s rather the point.

“But that is because forcings were calculated from that formula”

  • Who said this?

“So why does the formula emulate temperature so badly?”

  • Inconsistent with the first quote, above. Again, who said this?

PS – I see you’re getting downvotes. For the record, I appreciate your willingness to debate the issues on WUWT, but understand why some might find your responses a bit slippery.

John Tillman
Reply to  Frank from NoVA
February 3, 2022 11:55 am

I do, too. Nick’s comments are important to this blog, IMHO.

Reply to  Frank from NoVA
February 3, 2022 12:54 pm

“Who said this?”
“Again, who said this?”
Joel: “why does the CMIP model ensemble average run 1.5ºC to 3.0ºC higher than the emulation Willis shows here?”

Reply to  Willis Eschenbach
February 3, 2022 7:22 pm

I was paraphrasing the to and fro

Joel O'Bryan
Reply to  Nick Stokes
February 3, 2022 6:14 pm

The smoking gun, so to speak, of why the models run too hot (an ECS 2X to 4X higher than observed) and thus ‘why they fail’ is diagnosed by the lack of observation of the CMIP ensemble predicted mid-tropospheric hotspot. That failed prediction is diagnostic of the fact that all the important water vapor physics of cloud formation and precipitation related to convective heat transport in the models is fudged with hand tuning to get the ECS answer the modellers expect.

They might be able to fool the general public and lapdog media journalist, but they can’t fool Mother Nature (a.k.a. the real world).

Reply to  Nick Stokes
February 3, 2022 11:44 am

“The forcings are not the inputs to the models.”

They effectively are when they’re built into the parameterisations.

Reply to  Nick Stokes
February 3, 2022 3:49 pm

This is all devils sitting on the head of a pin. The relevant forcings can be found by asssuming the system is stochastic and fitting a regression model to various observed time series of global average temperature, greenhouse gas concentrations etc. My paper on this was recently rejected by Proc. Roy. Soc. A. because it was “too simple”. Evidently they had never heard of Occam’s Razor. See .

Reply to  John Reid
February 3, 2022 5:21 pm
Mark BLR
Reply to  Nick Stokes
February 4, 2022 7:47 am

The forcings are not the inputs to the models.

There is a wide range of climate models, from the simplest “0-D energy balance models” to the largest and most complex 3-D AOGCM “climate models”.

You have failed to specify which variant(s) of “the models” you had in mind when you were typing your OP here.

My understanding (which may well be wrong in the details !) is that “the (climate) models” can be classified as either :
1) “Full-blown” 3-D AOGCMs, which use GHG emissions as inputs and then “calculate” atmospheric concentrations as outputs (from which RF values are only a simple algebraic formula away…); or
2) “Models of intermediate complexity”, typically 2-D or 3-D, which use atmospheric GHG concentrations as inputs; or
3) “Simple” climate models, from 0-D to 2-D, which use RF numbers “directly” as inputs.

– – – —

Yes, but MAGICC is not a GCM.

I repeat, “the models” is not limited to “GCMs”.

Instead of Willis’s standard “When you comment please quote the exact words you are discussing” request a famous quote (by Voltaire ?) may well be applicable :

If you would debate me, first define your terms.

– – – —

PS – I see you’re getting downvotes. For the record, I appreciate your willingness to debate the issues on WUWT

PS : I completely agree with this sentiment (by “Frank from NoVA” below).

Reply to  Nick Stokes
February 5, 2022 8:14 am

Reading comprehension is a necessity for this site – Willis is not saying that the models use one line of code, but one gets the same result as all those thousands of lines of code running for hours on supercomputers by using his one formula based on lagging the inputs. Just read the article again, his meaning was plainly indicated and everyone else here got the same idea.

Joel O'Bryan
February 3, 2022 10:28 am

Again we see support for ECS in the range of 1.4ºC to 1.5ºC per CO2 doubling. Turn-off the alarms. There is no climate crisis. As HL Mencken would put it, it’s is an imaginary hobgoblin meant to scare the public in order to concentrate power in the hands of a few.

John Garrett
Reply to  Joel O'Bryan
February 3, 2022 10:47 am

It’s all bullsh*t.

I’ve seen this time and time and time again. The modelling boffins think they know but what they don’t know (yet) is that they don’t know. Climate is a complex (possibly chaotic), dynamic, non-linear, multi-variate system. The clowns don’t even know what all the damn independent variables are.

The modellers don’t have a frickin’ clue.

Rud Istvan
Reply to  John Garrett
February 3, 2022 12:10 pm

All nonlinear dynamic systems are mathematically chaotic (a well defined term with several characteristics including but not limited to sensitive dependence on initial conditions and strange attractors in N-1 Poincare space).
That climate is nonlinear is proven by feedbacks like water vapor.
That climate is dynamic is proven by the fact that feedbacks operate with some time lag.

Izaak Walton
Reply to  Rud Istvan
February 3, 2022 1:24 pm

That is just not true. Firstly it is easy to show that no two dimensional system can be chaotic. Secondly there are plenty of nonlinear systems that are strongly dissipative and thus are not chaotic since all trajectories will evolve to some fixed point. Then even if you restrict yourself to driven dissipative systems then often chaos can only be found in restricted parameter regions. The Lorenz system for example is only chaotic for a restricted set of parameter values.

Clyde Spencer
Reply to  Izaak Walton
February 3, 2022 3:02 pm

Firstly it is easy to show that no two dimensional system can be chaotic.

Alright, if it is easy, it shouldn’t be asking too much for you to show it.

Robert W Turner
Reply to  Clyde Spencer
February 3, 2022 4:24 pm

It’s hard to show you from flat land.

Izaak Walton
Reply to  Clyde Spencer
February 3, 2022 4:55 pm

It is an immediate consequence of the Poincare-Bendixson theorem which was proved in 1901. Have a look on wikipedia.

Clyde Spencer
Reply to  Izaak Walton
February 3, 2022 6:26 pm

I’m not very bright so I was hoping that you could explain it in a manner that I could understand, not just provide a link.

Were you perhaps referring to the statement, “However the theorem does not apply to discrete dynamical systems, where chaotic behaviour can arise in two- or even one-dimensional systems.”?

Izaak Walton
Reply to  Clyde Spencer
February 3, 2022 7:50 pm

The simple non technical answer is that there is just not enough room in 2 dimensions for chaos to occur.
A 2D dynamical system is defined by a set of equations (dx/dt = f(x,y), dy/dt=g(x,y)) where f and g are smooth continuous functions. There is thus a single trajectory that passes through any point and so in phase space (x,y) trajectories cannot cross.

Thus in 2D, trajectories correspond to paths you can draw on a piece of paper that never self intersect. Now try to draw a curve that remains inside a finite square is smooth (i.e. no sharp turns) and which never crosses itself. If you do this you will see that either the curve approaches a fixed point or a periodic orbit. There is no other option.

The only other option is that your curve goes off to infinity.

In 3D the situation becomes different since when a curve comes close to self intersection it can swerve around itself in the 3rd dimension.

Adding additional
dimensions to a problem allows new dynamics to arise. In 1D for example solutions to d x/d t = f(x)
can not be periodic and must either go to infinity or
to a fixed point.

Mike McMillan
Reply to  Clyde Spencer
February 3, 2022 5:28 pm

comment imagev

Clyde Spencer
Reply to  Mike McMillan
February 3, 2022 6:18 pm

Am I to assume that this is an example of chao in 2 dimensions since even a double pendulum swings in a plane?

Izaak Walton
Reply to  Mike McMillan
February 3, 2022 7:51 pm

That is a 4 dimensional system. There are 4 degrees of freedom — the angle of each pendulum and its velocity. Thus the motion can be chaotic.

Reply to  Izaak Walton
February 3, 2022 10:02 pm

He is playing stupid word games like Nick Stokes does with “dimension” he is using the mathematical version.

Reply to  LdB
February 4, 2022 12:06 am

He’s using the correct version. Chaos requires at least three degrees of freedom.

Tim Gorman
Reply to  Izaak Walton
February 4, 2022 9:43 am

How is this any different than the ocean or the atmosphere? Neither of these are 2 dimensional. In both you have at least x, y, z, and t (time) as dimensions.

Reply to  Izaak Walton
February 3, 2022 4:03 pm

no two dimensional system can be chaotic.

But thin films can – don’t forget the Belousov-Zhabotinski reactions. Plenty of chaos and emergent pattern there. There’s a large literature on chaotic and fractal dynamics in thin films, e.g.

You can hardly deny that turbulence – that is, high dimensional chaos, is not a prominent, even dominant feature of ocean circulation, which is where climate comes from (in case you were wondering).,0.00,128

But what is more interesting than chaos per se in climate is where chaos decreases in dimensionality from turbulence toward borderline chaos, the regime where the interesting emergent spatiotemporal patterns and oscillations arise.

Two things can give chaotic climate fluid flow systems the dimensional haircut that brings them toward the borderline chaos regime of emergent pattern: internal feedbacks, and external periodic forcings. There are plenty of these both in the earth system.

Ocean driven oscillations based on oceanic feedbacks (often associated with upwelling or downwelling) together with astrophysical forcing include the ENSO on short time scales and “stadium wave” oscillations such as PDO and AMO on longer ones.

Izaak Walton
Reply to  Phil Salmon
February 3, 2022 8:02 pm

Looking at the papers you cite. For the first the system is three dimensional not two and so chaos is possible. The second paper describes solving a nonlinear partial differential equation which has an infinite number of degrees of freedom and so again doesn’t count.

I am not denying that nonlinear dynamics plays an important role in lots of areas. Again my point was that the fact that a system is nonlinear does not imply that it is chaotic. There are a lot of nonlinear systems out there that are not chaotic. Just as there are a lot that are.

Reply to  Izaak Walton
February 3, 2022 5:27 pm

“Firstly it is easy to show that no two dimensional system can be chaotic”

So there are 3 and 4 dimensional (Gravity and EM ) non-linear chaotic systems and analysis which don’t exist according to you.

So you are clearly using special Izaak Walton physics and probably you need to take that dribble to a physics forum.

Izaak Walton
Reply to  LdB
February 3, 2022 7:32 pm

Giving an example of a 3 or 4 dimensional system that is chaotic does not disprove the claim that 2 dimensional systems cannot be chaotic.

Tim Gorman
Reply to  Izaak Walton
February 4, 2022 9:45 am

Exactly what in the system we know as Earth is two-dimensional?

Reply to  Izaak Walton
February 3, 2022 11:50 pm

Firstly it is easy to show that no two dimensional system can be chaotic.”

What kind of strawman is this?

You know the context in which the statement was made, right?

Old Cocky
Reply to  Izaak Walton
February 4, 2022 1:13 pm

You’ve got me there. Non-linear implies > 1 dimension, and dynamic implies another.

Reply to  Izaak Walton
February 5, 2022 8:30 am

The Earth and the biosphere are at least 4 dimensions.

Frank from NoVA
Reply to  Joel O'Bryan
February 3, 2022 11:02 am

It’s worse than you think. The temperature anomalies that the models are ‘tuned’ to hindcast are based on spatially sparse temperature records that have been heavily tampered to show warming since the onset of the industrial era.

Reply to  Frank from NoVA
February 3, 2022 1:01 pm

THIS a 1000 gazillion times

Reply to  Frank from NoVA
February 4, 2022 9:03 am

What happens when they have to “correct” the past temperature record upon which their hindcasted models are based?

Reply to  Joel O'Bryan
February 5, 2022 8:28 am

Heck, even if it were 5°C per doubling CO2, it wouldn’t be a bad thing. W.E. has given good evidence that tropical temperatures won’t change much which supported by IPCC as well, so warmunists have no case. And a much warmer north polar region would be a gift to the biosphere there, regardless of any rising of water levels – the extra water is much needed in the biosphere elsewhere. The south polar region would barely notice I think, though the penguins would appreciate the improvement in their living conditions and survival rates.
Anyways, historically the Earth has been warmer than now on average, and it’s insanity to think we could stop it returning to that level.

Frank from NoVA
February 3, 2022 10:42 am

“T(n+1) = T(n)+λ ∆F(n+1) / τ + ΔT(n) exp( -1 / τ )”

Isn’t the ability to emulate the output of the “complex” GCMs as a simple function of the “forcings” similar to Pat Frank’s findings? If so, would you agree with his conclusion that the models are essentially meaningless because the temperature impact from errors in cloud forcing at all time scales far exceeds the modeled change in surface temperature?

John Tillman
Reply to  John Tillman
February 3, 2022 11:56 am

Pat and I have differed on some issues, but my admiration for him and his work on error propagation, so devastating to the CACA “consensus”, is unlimited.

John Tillman
Reply to  John Tillman
February 3, 2022 12:17 pm

PS: The man is fearless and honest as the day is long, traits sorely lacking in the CACA cabal.

Reply to  John Tillman
February 6, 2022 5:47 pm

Thanks for your kind words, John.

Reply to  John Tillman
February 3, 2022 1:07 pm

If I recall this past low value discussion, here is error propagation: You can’t keep your car on the road. Because all steering input contain errors. So at any time because of error propagation, you are in the ditch. But here’s the problem with that failed theory. The system feeds back. Everytime something is off you don’t veer into a ditch. Sorry. That’s the way it works.
Another example: There is no life on Earth. Because all errors have led us and everything else to extinction. Life and certain models can feedback to keep things stable enough.
Another example: Goats actually cannot walk on the side of steep mountains. Because every step has an error in it. So they all died centuries ago. Wait, there’s feedback to make up for that.

John Tillman
Reply to  Ragnaar
February 3, 2022 2:29 pm

Not at all comparable to Pat’s work on CACA.

Reply to  John Tillman
February 3, 2022 3:22 pm

Maybe you’re right. Here’s Spencer: “The reason is that the +/-4 W/m2 bias error in LWCF assumed by Dr. Frank is almost exactly cancelled by other biases in the climate models that make up the top-of-atmosphere global radiative balance. It doesn’t matter how correlated or uncorrelated those various errors are with each other: they still sum to zero, which is why the climate model trends in Fig 1 are only +/- 0.10 C/Century… not +/- 20 deg. C/Century. That’s a factor of 200 difference.”
What I am saying is the models act as if they feedback. For instance, they still disburse El Nino events and recover from huge volacanoes. They feedback like the climate does, like your body does. Like my goat on a mountain does. Spencer says more or less, apply the right math. I predict you can’t keep any car on the road with my Dr. Frank math. Because all steering inputs have errors. Spencer says also, if they didn’t the CMIPs would show like 20 C warming. You might be technically right. But I am on the right path I think.

Dave Miller
Reply to  Ragnaar
February 3, 2022 5:19 pm

The map is never the territory.

Reply to  Ragnaar
February 5, 2022 1:12 pm

“I predict you can’t keep any car on the road with my Dr. Frank math. Because all steering inputs have errors”

I think the analogy with driving on a road needs to start with…the road is the path that is unknown.

So imagine driving in a massive car park and making whatever steering corrections you like and then afterwards put in the “road” as the path in that car park and see how well you did driving down it.

It’s almost certain the car quickly left where the road was and went on its own merry way around the car park.

Reply to  Ragnaar
February 6, 2022 7:29 pm

Roy Spencer says the propagation errors don’t occur because there are compensating errors in the models. Oh, so it’s all good then?
No, it’s not! That just makes it so much worse. That is an admission that there so many errors in the model that aren’t even being uncovered and measured.

Joel O'Bryan
Reply to  Ragnaar
February 3, 2022 6:45 pm

What you describe is how weather forecasting is done with weather models. Weather forecasters make new model runs every 6 or 12 hours using the latest set of observations, that is they are constantly corrected with observables.
This is not how climate modeling is done. Climate models frequently run into the ditch because they rely on estimating non-observables (via parameters and finite precision measured inputs). Those in-the-ditch runs just never see the light of day beyond the trash can in the climate team computer center.

Frank from NoVA
Reply to  Ragnaar
February 3, 2022 7:28 pm

Bad analogies to the issue of error propagation. Taking the first example to illustrate; yes, any competent driver can easily compensate for the tracking error in a typical car, because this error is relatively minor compared to the lock-to-lock range of the car’s steering mechanism. But tell me honestly, would you drive, or allow a loved one to drive, a car in which the steering error was so large that a mere second of driver inattention would allow the car to rapidly veer into the oncoming lane or a concrete bridge abutment? Maybe you or your loved one consider yourselves skilled enough drivers to compensate for the errors of such a defective steering system, but why would either of you voluntarily take the risk?

The fact that a GCM can somehow ‘offset’ errors in cloud cover, the temperature equivalent forcing impact of which far exceeds that of the GHG additions during the same period, against other model errors doesn’t mean these errors don’t matter. While they don’t show up, or rather, are not allowed to cause the model’s temperature outputs to attain clearly unphysical values, they are still inherent to the model, which means that these outputs still have no physical basis no matter how reasonable they may seem.

Reply to  Frank from NoVA
February 4, 2022 7:37 am

Frank from NoVA:
It doesn’t happen right away. But it happens. Then we see it happening, and adjust. AI vehicles can do the same thing. But the answer is not you move 1 foot to the right or left every 15 minutes. I am assuming this skill the models have. But if this problem does happen, the accident reports would show us that. When we drive, we iterate. It’s mostly corrections. How about this: The CMIPs model a heat engine. The front end impacts the back end. If a signal from the front end misfires, the thing doesn’t blow up. The back end ramps up once it actually sees what really happened at the front end.

Reply to  Ragnaar
February 4, 2022 2:36 am

I think the point Pat Frank is making isn’t about errors in the result, its about the propagation of uncertainty that comes about with the errors propagating with each iteration of the model’s run.

Yes, GCMs are constrained to stay within believable results (when they’re not blowing up that is) but the propagation of errors (ie accumulating uncertainty) means that whatever result is returned, its within the propagated error limits which have become enormous.

So the models can return any value they like (they’re tuned to return believable ones) and its within their enormous error range.

It makes any result returned by a GCM utterly useless as a projection.

Tim Gorman
Reply to  TimTheToolMan
February 4, 2022 10:00 am

You pretty much nailed it. The problem with the driving example is that it doesn’t cover the situation where the uncertainty of the input to the steering mechanism is additive. A better analogy would be trying to correct the direction of the car while skidding on ice. Usually any corrections are additive to the amount of skidding, the car just keeps on swapping direction in an ever increasing amount, left and right, till it spins out of control and where you wind up is no longer under your control.

An even better example would a tank slapper wobble on a motorcycle. Once that wobble starts almost anything you do with the handlebars will make it worse till you “slap the ground with your tank”! With each iteration of the wobble whatever you do just makes it worse. (there is one thing you can try but I’m not going to give away the secret here)

Reply to  Tim Gorman
February 4, 2022 12:08 pm

I think a clear example would be modelling the tattslotto balls rolling around in their cage. Very quickly the accumulating errors in their modelled positions mean that the modelled results no longer reflect reality and the modelled result won’t be the same as the actual result.

Reply to  Ragnaar
February 4, 2022 10:36 am

So at any time because of error propagation, you are in the ditch. But here’s the problem with that failed theory. The system feeds back.
That is incorrect because the feedback is in the future and not available at the start of the journey.

The models are trying to predict the traffic and potholes on the road in the future using averages from today.

The same problem affects all inertial navigation systems. They need mid course corrections based on future data that is not available in the present.

Reply to  ferdberple
February 5, 2022 7:14 am

See the plot:
comment image
This is not a plot of the possible GMST outcomes of a CMIP run.
The plot looks like chaotic bifurcations.

Carlo, Monte
Reply to  Ragnaar
February 5, 2022 4:24 pm

Uncertainty is not error!

Reply to  Ragnaar
February 6, 2022 5:46 pm

That plot has nothing to do with error propagation, Ragnaar. It merely shows that Roy Spencer got it completely wrong.

It prima facie conveys an impossibility. Look at the two lines. Roy has them predicting temperatures that coincidentally both rise and fall.

Somehow he didn’t notice that impossibility built right into his figure. And neither did you.

See my analytical comment here:

Reply to  Frank from NoVA
February 3, 2022 2:14 pm

Here’s another link.

February 3, 2022 11:45 am

Pondering. Has anyone attempted to construct a GCM without GHG components? Not saying GHGs don’t influence but since none of the current models work all running too hot, why not try to make one the runs too cool? Edison knew 999 ways how not to make a light bulb. He also made a light bulb because of that.

Frank from NoVA
Reply to  Rob_Dawg
February 3, 2022 11:54 am

You mean, if someone took a fully parameterized GCM and then zeroed out the future increases in GHG concentrations, would the earth then cool over time? I’ve wondered that myself.

Reply to  Frank from NoVA
February 3, 2022 1:25 pm

The answer that I will offer to your question is;

Nope, no cooling.
Same end result, Same “ball court” result finale, as all others… a projection, of Atmospheric thermal expansion at ~3C warming equivalent, with a 200-220 ppm CO2 upswing correlation.
But with a Delta Time reaching at some 2K years+ (or even more),instead of decades or a century or two.
The main thing that actual GCM simulations (free of extra extrapolation) diverge from each other, is Time factor. Basically GCM simulations perform and do the same steps and cycles, almost, but in a different delta T.

That is my understanding, which you should not take it for granted…obviously.


old engineer
Reply to  Rob_Dawg
February 3, 2022 1:11 pm


In Hason’s 1988 paper he ran a 100 year baseline (starting in 1958) with the GHG’s held at constant 1958 values. The results are presented as Figure 1 in the paper. It surprises me that you don’t see this figure quoted more often. If I were more computer savvy, I would include it here, but alas I am not.

Interestingly, the temp anomaly peaks around 2008 and then decreases for the next 30 years.

John Tillman
Reply to  old engineer
February 3, 2022 3:13 pm

Today that result would be streng verboten!

Frank from NoVA
Reply to  old engineer
February 3, 2022 9:15 pm

Thanks for the reference. It (Figure 1) looks like a random walk. Was the so-called ‘control run’ a single iteration of Hansen’s model or the average of many runs?

Reply to  old engineer
February 4, 2022 7:57 am

old engineer

Technically speaking, GCM(s) do not do either the absence or the constant of GHG’s.
Even when water vapor, the most potent GHG, due to parameterizing could end up and considered constant at some given point… to a given degree.

February 3, 2022 11:58 am

Eschenbach demonstrates the net forcings factor and constant has no physical basis and is a simple function of the temperature series. An empirical fit with many (unlimited) degrees of freedom. The limit is only bound by the analysts imagination. This statistical relationship has no need for knowledge of mechanisms and so we have learned nothing about how the climate works.

It is irrelevant what anyone may believe about the physics of greenhouse theories or the partitioning of forcings, or what they believe about feedbacks. Radiation budget diagrams are imaginary, like cartoons, and so is anything gleaned from their computational GCMs. The output is merely a series impressive pictures, figments.

Regardless of what anyone believes might be happening the atmosphere has only one degree of freedom, the ratio of absorbed solar to non-radiation flux density. LW radiation equilibrium is not a free variable. The ratio of surface upward flux and OLR has not changed as predicted by current forcing hypotheses, in fact, it has barely changed at all (and in the wrong direction).

Ulric Lyons
February 3, 2022 12:08 pm

“IF the temperature is actually a result of the forcings”

The AMO doesn’t directly follow forcings, it is cooler when the solar wind is stronger and warmer when the solar wind is weaker.

Max Dupilka
February 3, 2022 12:12 pm

You have basically done a curve fitting using the known data. This is rather easy for any set of data. But how does that curve fit help in predicting the future path? Whether the models are good or bad, that is their function, to try to predict the future.

Frank from NoVA
Reply to  Max Dupilka
February 3, 2022 12:40 pm

The fact that curve fitting is easier than writing millions of lines of code is immaterial here. All forecasts have errors, which is why in the real world developing cost functions are important. Whether it’s WE’s emulator, or a GCM, you don’t ‘bet the ranch’ on a forecast unless you can afford the likelihood that it’s wrong.

Max Dupilka
Reply to  Frank from NoVA
February 3, 2022 1:24 pm

I am a professional meteorologist and forecaster. I know full well about model error. I have written atmospheric models. We always use them with healthy skepticism. That is what I get paid for. Unlike climate modelers who seem to think their models are truth.

Rud Istvan
February 3, 2022 12:20 pm

WE, a most interesting analysis. An observation and a question.

The Berkeley Earth result in your last figure is well within the error bounds of the revised Lewis and Curry energy budget method, which also relies on observed past temperatures. So that is in a way an observational triangulation.

The question is why the last figures Model ECS (1.4) is less than half of the CMIP5 average (3.2). My speculation is that Pat Frank’s compounding error issue is a root cause, as the models play out to 2100 and we know from the 8 year predictive results to 2020 that CMIP5 has a built in warm bias from unavoidable parameterization that drags in the natural variation attribution problem. Your method of deriving lambda and tau largely avoids that compounding error issue.

Smart Rock
February 3, 2022 12:45 pm


I got stuck for a while on your fig. 6 left panel. The caption says “Berkeley Earth Average Temperature…” and the legend in the figure says “CMIP Average Temperature”. I assume (after pondering it for the duration of a tea break) that the caption is right and that the legend is actually inherited from Fig.3.

Any chance that you could redo this exercise using actual temperatures from the CMIPs rather than anomalies?

I seem to recall another post a few years back saying that you could reproduce the CMIP ensemble with a hand-held calculator. Was it Pat Frank? But just think of the prestige attached to having a supercomputer in your lab! Who cares if it generates meaningful results? It probably impresses the heck out of the graduate students…

February 3, 2022 12:47 pm

I have never received an answer to these questions from any Climate Scare Alarmist: What should the earth’s perfect temperature be and has it ever been and for how long; and what should the correct level of CO2 be and do you believe like some that CO2 is a dangerous pollutant, and most important, if so, who amongst us should be forced to hold our breath to stop CO2 from getting into the atmosphere???(considering that we inhale 400ppm and exhale approximately 20,000ppm)
Last but not least, is there a published or otherwise empirical paper or experiment linking CO2 to the Earth’s temperature? I think
To my limited knowledge the last glaciation ended about 12,000 years ago and according to scientific records the earth’s temperature has been declining (negative trend line) with a number of ups and downs ever since and will continue to do so until the next glaciation.
Anyone think we can stop it?
I read this blog for fun and education,. I am 77 years old and as yet no one can answer my questions. If given the chance I would love to confront Mann, Kerry, Gore,et all

Reply to  JontheTechnologist
February 3, 2022 1:00 pm

“ What should the earth’s perfect temperature be and has it ever been and for how long; and what should the correct level of CO2 be and do you believe like some that CO2 is a dangerous pollutant, and most important, if so, who amongst us should be forced to hold our breath to stop CO2 from getting into the atmosphere???(considering that we inhale 400ppm and exhale approximately 20,000ppm)”

Good luck

Chris Hanley
Reply to  JontheTechnologist
February 3, 2022 1:10 pm

I understand John Tyndall was the first to demonstrate the ‘greenhouse effect’ in a laboratory experiment in 1861.
I’d also like to know of other more recent lab experiments that demonstrate the effect, the ‘science guy clown’ Bill Nye excepted.

Reply to  Chris Hanley
February 5, 2022 11:25 pm

Doesn’t matter if the effect is demonstrated or not – that’s just a limited lab experiment. Showing that CO2 can absorb and re-emit, in any random direction, wavelengths/frequencies of infrared radiation in a lab, is a long way from showing it does anything meaningful in the real world atmosphere – filled with much more water vapour absorbing and emitting at the same and more wavelengths, amongst other variables in the climate.

Climate scientists are fixated on this CO2, and can’t be bothered to think about anything else.

Reply to  JontheTechnologist
February 3, 2022 5:00 pm

The earth’s perfect temperature depends on what you are and what you want from life.
If you are a thermophile and want to hang around in hot springs you probably don’t want a temperature lower than 40°C. If you are a polar bear you probably don’t want the temperature to be above 0°C for long periods. If you want a temperature where people compose Baroque chamber music a range from about -10°C to 25°C is probably OK.

Pat from kerbob
Reply to  JontheTechnologist
February 3, 2022 8:36 pm

All excellent questions because there are no answers

Reply to  JontheTechnologist
February 4, 2022 9:25 am

Last but not least, is there a published or otherwise empirical paper or experiment linking CO2 to the Earth’s temperature? I think

There is this from NASA’s climate change site:

  • Climate models that include solar irradiance changes can’t reproduce the observed temperature trend over the past century or more without including a rise in greenhouse gases.“

So, they are saying that they have developed climate models that they know are correct because when they increase GHGs, the past temperature history is replicated. Isn’t this circular reasoning?

February 3, 2022 1:00 pm

ja. ja .
I told you.
Eventually you come where I am….
Click on my name

Pat from kerbob
Reply to  HenryP
February 3, 2022 8:36 pm

Is that like “pull on my finger”?

February 3, 2022 1:33 pm


February 3, 2022 1:56 pm

Still TCR, not ECS.

February 3, 2022 2:19 pm

Climate models are simply driven by the input forcings. So the result of a climate model is completely dependent on the input forcings. And nothing else. The climate models add no information whatsoever to our understanding of climate.

How do we know?

(a) because the mean climate model output can be linearly constructed from the input forcings (as Willis so ably demonstrates here) and
(b) if we subtract the mean model output from the individual model runs the residuals are uncorrelated random noise.

And the fact that AR6 (and all preceding AR’s) show the mean model shows the above to be a true and valid exercise. Basically averaging all the models simply returns the low frequency input constraint aka the input forcings.

I could run off and cite acres of literature on inverse problems and stochastic literature but its so blindingly obvious I shouldn’t need to. Climate models do nothing more than convert the input forcings via a linear transform and lag to temperature. And the only reason they can do that is that the input forcings are a linear match for temperature a priori.

Its not physics, its bollocks.

Reply to  ThinkingScientist
February 3, 2022 2:57 pm


Gabriel Oxenstierna
February 3, 2022 2:36 pm

Many thanks Willis for sharing the Excel file and the data!

This is the level of documentation and transparency that’s needed.

February 3, 2022 3:15 pm

So does this show that forcings actually rule the temperature? Well … no, for a simple reason. The forcings have been chosen and refined over the years to give a good fit to the temperature … so the fact that it fits has no probative value at all.

This is quite a significant revelation. Despite the pseudo sophistication of the models, the outcome appears to be tuned by the selection of forcings. And knowing those forcings, the outcome can be simply replicated.

Well if climate modelling goes out of fashion, the modellers needn’t worry – they can probably find work as piano 🎹 tuners.

Reply to  Phil Salmon
February 4, 2022 12:33 am

The forcings have been chosen and refined over the years to give a good fit to the temperature … so the fact that it fits has no probative value at all”

This is the only reason why climate models appear to “work” and reproduce temps.

If your prior is already reproducing your output then your climate model is adding no information. This result is well known in forward and inverse modelling. Without the input forcings climate models would not do anything.

Reply to  Phil Salmon
February 4, 2022 1:44 am

There is a further glaring problem as well – the model forcings post-1950s are 3x larger than the forcings over the period 1910-1945 and yet the warming in that early C20th is almost identical in magnitude to the warming 1975-2010.

This is why in AR6 they fudge the fit of CMIP6 to the temperature data by fitting it along the whole time period 1850 – present. If you look closely you can see the fit in the C20th century is poor – it undershoots for a long time. If you re-baseline the CMIP6 output to the 1961-1990 period (previously used for AR5) the problem becomes very obvious. See picture below. Original from AR6 left panel, re-baselined comparison (plus UAH) right panel.

Gary Pearse
February 3, 2022 3:22 pm

I’m (guardedly) impressed that “the physics” can be reduced down to such a solid simple expression – it has the look of reverse engineering from a result that is supposed to be the object of the research.

I recall another mathematical finding of yours that revealed the iconic climate equation could be replaced by a simple linear equation.

Could it be that the models are input into the algorithm that is used to continuously make temperature adjustments to fit the desired result. Also over time, do they indicate what recalcitrant stations are identified for cutting out of the loop or marked for a station “move”.

Robert W Turner
Reply to  Gary Pearse
February 3, 2022 4:30 pm

And it’s reverse engineered to the adjusted “data”.

February 3, 2022 4:09 pm

Willis, I don’t think your equation, meaning this:

T(n+1) = T(n)+λ ∆F(n+1) / τ + ΔT(n) exp( -1 / τ )

proves anything, or invalidates the models (not that I am a fan of them).

It is a clever way to produce the output with a lot less fuss, but because you get the same answer as the other models doesn’t mean the other models are wrong. You both get the same answer!

Reply to  Willis Eschenbach
February 4, 2022 12:40 am

Absolutely true.

And is also why the fit to temperature only appears to look good when the models are averaged together: “the ensemble average”.

Each model output is basically forcings+random fluctations. By averaging the model outputs the random part is cancelled out and, lo and behold, the model output is revealed which matches temperature. But the priors (input forcings) already do that.

The further test is to subtract the model mean from the individual model runs. And then you will find that the model residuals around the collective mean are simply random and uncorrelated.

Climate models are not physics, they are bollocks.

Reply to  Willis Eschenbach
February 4, 2022 1:56 am

To demonstrate that Model Output = inputForcings + randomNoise

Here are the residuals of the 39 models in CMIP5 ensemble. The models have the mean model subtracted from them to show what’s left over after you take out what is effectively just the input prior (the forcings).

Note (a) the uncorrelated random noise of every model and (b) the huge spread of the models (+2.0 to -1.5 degK). I haven’t done this yet for CMIP6 but note John Christy has pointed out the new models have even greater internal variance.

X-axis is year, Y-axis is temperature residual in degK (or degC!).

paul courtney
Reply to  Tom.1
February 4, 2022 8:55 am

Mr. .1: Mr. E’s article clearly indicates what is proven, why are you so determined not to get it?

Carlo, Monte
February 3, 2022 4:36 pm

OK, now lets unpack this equation in English. It looks complex, but it’s not.

The equation assumes T_(n+1) is a function of T_n — what if this assumption is not accurate?

David L. Hagen
Reply to  Carlo, Monte
February 3, 2022 7:09 pm

Then you are down the rabbit hole following Alice in Wonderland.

February 3, 2022 6:46 pm

I appreciate Willis’ very nice demonstration of how and why CO2 warming looks straight-forward and believable, even though the models’ physics are entirely backward.

What is really going on is the rise in CO2 is a function of the ocean CO2 outgassing temperature threshold and area ≥25.6°C, which grew warmer and bigger since the 1850s, enhancing CO2 outgassing and limiting CO2 sinking of both man-made and natural CO2.

comment image

The 12m∆ in ML CO2 lags the 12m∆ in SST≥25.6°C by 5 months, r=.84. The 12m∆ in ML CO2 also has a period of 3.6±1 yr, in the range of Willis’ models tau values.

comment image

The warmists’ trick is to ignore the real 5 month lag of CO2 vs SST shown in my model by smoothing it over in various ways, especially if they didn’t know about the lag and believed the opposite, this can give the appearance of ‘models that work’, like Willis has shown, when in fact they are physically, causally backwards, temporally, by ignoring Henry’s Law.

My CO2 outgassing model has an r=.99 vs ML CO2, and r=.96 vs JMA OHC because it’s based on ocean temperature leading CO2 outgassing/sinking via Henry’s Law of Solubility.

The observed increase in SST≥25.6°C would drive an increase in convection and clouds, according to Willis’ evaporation/clouds work, however I am at a loss to understand how the thunderstorm SST thermo-regulation system he proposes would by itself give rise to the temperature rise in SST≥25.6°C when ostensibly more clouds are thereby generated.

The SST≥25.6°C growth was a response to high solar activity, driving CO2 growth.

Pat from kerbob
Reply to  Bob Weber
February 3, 2022 8:46 pm

I’m not a scientist, but from what I know it seems impossible that a 1 degree rise in atmospheric temperature can raise the temperature of the ocean in anything less than geologic timescale if even then.
Certainly not in a couple decades.

My own simple observation seemed to prove that two summers ago. We have a lake house in southern Saskatchewan and 2020 was awful, cold and crappy right through 3rd week of July, rarely above 20c even though the lake temp rose to its normal 76F as always

Then it turned hot and we had one of the most beautiful Augusts in memory, hot days many +30c or more, warm nights, perfect lake time.
And near the end of that hot period the lake was down to 63F.
Because the sun keeps getting lower.

For what little it’s worth.

Reply to  Willis Eschenbach
February 4, 2022 5:26 am

I don’t understand why people think that the CO2 we’re putting into the atmosphere isn’t the cause of the increase of CO2 in the atmosphere.

Willis says “Bob, if you look at the ice core data, it’s obvious (since CO2 lags temperature) that the increase in CO2 ppmv is due to ocean outgassing.”

And I want to add my own opinion that historically warming will have outgassed, as you say, and the thriving biosphere will have locked some of that up each season and ratcheted up the CO2 levels behind the warming. It’s not only about the outgassing IMO.

Mark BLR
Reply to  TimTheToolMan
February 4, 2022 8:09 am

I don’t understand why people think that the CO2 we’re putting into the atmosphere isn’t the cause of the increase of CO2 in the atmosphere.

Your misunderstanding arises from the fact that “people” don’t think that (anthropogenic) CO2 emissions are “the cause” (singular) of the current (since 1750-ish) rise in atmospheric CO2 levels, some (/ many ?) of us “think that” they are “one factor among several, exact proportions still TBD“, instead.

Reply to  Mark BLR
February 4, 2022 12:27 pm

Historically the CO2 was from outgassing and I think the mechanism I described, and it naturally lagged temperature increase. Today we’re putting the CO2 out there ahead of the warming induced outgassing.

So given the warming, the CO2 would have got there eventually without us adding it – although it would have been much less and much more slowly.

We’ve directly caused the bulk of the additional CO2 we’re seeing. A small amount will be natural as a result of warming induced outgassing.

Reply to  Willis Eschenbach
February 4, 2022 10:50 am

I believe you have assumed I thought the atmospheric CO2 increase wasn’t also related to manmade emissions. It’s a factor, but not the major factor by far.

Your calculations must also include the warm ocean area changes and the CO2 atmospheric residence time. The atmosphere holds CO2 for hundreds of years while the ocean either allows more CO2 to stay in the atmosphere or sinks more depending on temperature, like a temperature controlled ball valve, within months of temperature changes – two vastly different time scales.

When the ratio of ocean CO2 outgassing to sinking Temp*area grows higher with a warmer ocean, the sinking of manmade and natural CO2 is reduced, allowing for more manmade emissions and natural CO2 to remain in the atmosphere, creating the upward CO2 trend.

However, I did look at two correlations of manmade emissions vs CO2, and found they aren’t very much related. The ML CO2 derivative trend is 5.7x MME, with the MME derivative lagging ML by 2 years (annual data).

comment image

Large-scale bulk changes are completely out of phase, with MME lagging again:

comment image

Therefore I conclude that MM CO2 is a smaller portion of the total CO2 increase.

I don’t use ice core data for ML CO2 analysis, they are temporally incompatible.

You have to do better than ‘no way’ in explaining how the r=.84 happens from the 12m∆ in SST≥25.6°C and how it isn’t related.

Amazing how you can believe in your evaporation SST thermo-regulation system but not the CO2 SST thermo-regulation system when presented with such solid evidence, and you still haven’t answered how the ocean warmed while producing more clouds.

I never said it was only about outgassing, the other r=.16 came from somewhere.

Clyde Spencer
Reply to  Willis Eschenbach
February 6, 2022 7:17 pm

I suspect that the dominant source of CO2 changes over time, particularly with temperature and vegetation. Currently, it looks to me that respiration and bacterial decomposition is dominant during the Winter ramp-up phase, and photosynthesis is dominant during the Summer draw-down phase.

February 3, 2022 8:11 pm

Willis, wondering why you would use all scenarios when looking at the range of model outcomes?

I consider that there are two major sources of errors in the prognostications of the models. One is the model architecture itself. The other is the assumptions input into the model, controlled by the scenarios.

Would the four RCP scenarios not be considered mutually exclusive and splitting out each scenario a better way of determining the error range?

February 3, 2022 11:04 pm

Thanks a lot ! I was almost laughing when I read your post. I really liked it.

Seems as if IPCC can save a lot of money by selling all supercomputers and fire the modelers.

Since your “black box” model agrees extremely well with CIMP I guess that your equation actually catches the basic logic in the climate models.

February 4, 2022 12:03 am

What is the budget of the CMIP6?
May be you should claim a bit share of it and release supercomputing hours for something more useful.
Have you tried any other variable besides temperature?

Mark BLR
Reply to  jpg
February 4, 2022 9:05 am

What is the budget of the CMIP6?

I am going to deliberately “misunderstand / misinterpret” your comment (about monetary / funding issues ?), and assume that you are looking for where datasets of RF and GMST “projections” (from 2005 to 2100) can be found for the CMIP6 / SSP set of climate model runs.

NB : The 5 “standard” emission pathways used in AR6 are SSP1-1.9, SSP1-2.6, SSP2-4.5, SSP3-7.0 and SSP5-8.5.
The most frequently “gap fillers”, that I have come across at least, are SSP4-3.4 and SSP4-6.0 (for comparison with CMIP5’s RCP6.0 emission pathway).

1) Go to the IIASA website.
[ URL : ]

2) Click on the “login as guest” button.

3) Click on the “IAM Scenarios” tab.

4a) In the “(2.) Model/Scenarios” box, de-select the “SSP1 – Baseline”, “SSP2 – Baseline” and “SSP4 – Baseline” options.
NB : “SSP3 – Baseline” = SSP3-7.0 and “SSP5 – Baseline” = SSP5-8.5.

4b) Select the “SSP1 – 1.9”, “SSP1 – 2.6” and “SSP2 – 4.5” [ and “SSP4 – 3.4” and “SSP4 – 6.0” … ] options.

5) In the “(3.) Variable” box, click to open up the “Climate” option, and then the “Concentration”, “Forcing” and “Temperature” sub-options.

6a) Select (sequentially) the variables you are interested in, e.g. “Concentration : CO2”, “Forcing : Total” then “Temperature : Global Mean”.

6b) I found all three “Emissions (harmonized) : CO2” sub-options to be “interesting” as well ..

7) Use the mouse to select (all of !) the data in the “Query Results” table (as a “Guest” the “Output Options : Microsoft Excel” button won’t work), and copy the results to a text file.

8) Once completed for all “interesting” variables, import the final text file(s) into your favourite spreadsheet program.

NB : You only get one datapoint per decade (+ 2005) instead of annual values, but it’s a good start to getting a general feel for the CMIP6 / SSP model run datasets.

Mark BLR
February 4, 2022 7:17 am

Forcings Source: Miller et al.

Interesting link, I hadn’t come across that set of data before.

I also use KNMI for “ensemble mean” GMST data, but have a personal preference for monthly rather than annual time resolution.

– – – – –

At the risk of being accused of “Look at me !” syndrome, I mostly use the CMIP5 inputs data from Malte Meinshausen’s “RCP Concentration Calculations and Data” page at PIK (the Potsdam Institute) :

Note that they use the standard “Historical Data = up to 2005, use “per-RCP pathway” data from 2006 (even if more up-to-date numbers are available)” assumption, but provide annual data from 1765 (AD / CE) all the way to 2500.

Note also that the “Download all data [files, ASCII and Excel versions ] …” option is a ZIP file only 6MB in size.

Options are available for :
1) “Harmonised” GHG emissions (used as inputs to the full-blown 3-D AOGCM climate models, which then “calculate” atmospheric concentrations as outputs, from which RF values are only a simple algebraic formula away…)
2) “Mixing Ratios”, i.e. atmospheric concentrations / abundances (used as inputs to “models of intermediate complexity”)
3) “Radiative Forcing” numbers (used as inputs to a wide range of “simple” climate models)

For reference, the standard “header” for the RF files includes the following :
1      TOTAL_INCLVOLCANIC_RF  Total anthropogenic and natural radiative forcing
2      VOLCANIC_ANNUAL_RF     Annual mean volcanic stratospheric aerosol forcing
3      SOLAR_RF               Solar irradience forcing
4      TOTAL_ANTHRO_RF        Total anthropogenic forcing
5      GHG_RF                 Total greenhouse gas forcing (CO2, CH4, N2O, HFCs, PFCs, SF6, and Montreal Protocol gases).
6      KYOTOGHG_RF            Total forcing from greenhouse gases controlled under the Kyoto Protocol (CO2, CH4, N2O, HFCs, PFCs, SF6).
7      CO2CH4N2O_RF           Total forcing from CO2, methan and nitrous oxide.
8      CO2_RF                 CO2 Forcing
9      CH4_RF                 Methane Forcing
10     N2O_RF                 Nitrous Oxide Forcing
11     FGASSUM_RF             Total forcing from all flourinated gases controlled under the Kyoto Protocol (HFCs, PFCs, SF6; i.e. columns 13-24)
12     MHALOSUM_RF            Total forcing from all gases controlled under the Montreal Protocol (columns 25-40)
13-24                          Flourinated gases controlled under the Kyoto Protocol
25-40                          Ozone Depleting Substances controlled under the Montreal Protocol
41     TOTAER_DIR_RF          Total direct aerosol forcing (aggregating columns 42 to 47)
42     OCI_RF                 Direct fossil fuel aerosol (organic carbon)
43     BCI_RF                 Direct fossil fuel aerosol (black carbon)
44     SOXI_RF                Direct sulphate aerosol
45     NOXI_RF                Direct nitrate aerosol
46     BIOMASSAER_RF          Direct biomass burning related aerosol
47     MINERALDUST_RF         Direct Forcing from mineral dust aerosol
48     CLOUD_TOT_RF           Cloud albedo effect
49     STRATOZ_RF             Stratospheric ozone forcing
50     TROPOZ_RF              Tropospheric ozone forcing
51     CH4OXSTRATH2O_RF       Stratospheric water-vapour from methane oxidisation
52     LANDUSE_RF             Landuse albedo
53     BCSNOW_RF              Black carbon on snow.

I believe the phrase “horses for courses” is applicable here …

Peta of Newark
February 4, 2022 9:17 am

I ignored CO2 forcing. It’s nonsense.

Instead I tweaked ‘Land Use’

I considered land at 45 degress latitude where the average solar input across the whole year, adjusted for 30% cloud Albedo, amounts to 240Watts/sqm

Then I took 1.5E13 square metres of ploughed land to have an albedo of 0.10 at the present time.Going back to 1850 that would have been perennially green grassland with albedo 0.40

I also took another 1.5E13sqm of grazed pasture land to have an albedo of 0.20

Running through the sums there gets an increase across the whole globe of 13.8 Watts/sqm

Putting that into the Total Forcings column, all forcing the same except with WGHG = 0,
I get a temp rise of 3.4 deg Celsius – running it from 1850 to 2021

Just by adjusting the land use for ploughing, tillage and (over) grazing

There are some of us who might say,,,
Well OK, lets go with the 3.5W/sqm for CO2

That is for CO2 working at a wavelength of 15 microns
But, as per OCO2 Sputnik, CO2 absorbs at wavelengths corresponding to 800 Celsius also 400 Celsius
So CO2 re-radiates 3.5Watts coming from the ground (stops it reaching leaving Earth, would it not be fair to say that CO2, working at 400 and 800 Celsius is stopping 7.0 Watts/sqm from reaching the ground

And why not, that is the very signal that OCO2 uses to measure the concentration of the CO2

So lets re-run the spreadsheet with that and get a temp rise of 1.5 Celsius.

In a nutshell, run it with Land Use going from zero up to 13.8 Watts and CO2 going from zero to minus 7 Watts gives a temp rise of 1.5 Celsius from 1850 to present
How does that compare with anything?

Or ignoring all the other forcings, a result of 1.7 Celsius

edit to PS
I should have increased where it adjusts for atmospheric dust. All that tillage has increased that and we know so, it is the root cause of the observed Global Greening – also the ice melt in the Arctic. In no small part the huuuuge amounts of de-icing salt now used on roads in the northern hemisphere.
We know the stuff, it makes the road go white when it dries out and the traffic pummels it down to an incredibly fine powder that could travels 000’s of miles.

Burgher King
February 4, 2022 9:30 am

Willis Eschenbach reply to Tom.1: “What it proves is that the model outputs are merely a simple transformation of the inputs. Period.”

Musical satirist and Harvard mathematician Tom Lehrer, in his 1958 comedy album, An Evening Wasted with Tom Lehrer, quoted the late Dr. Samuel Gall, inventor of the gall bladder, as saying this:

“Life is like a sewer. What you get out of it depends on what you put in to it.”

Reply to  Burgher King
February 4, 2022 7:47 pm

Ah yes Tom Lehrer, His “THAT WAS THE YEAR THAT WAS” is also classic Lehrer. “The garbage that we throw into the Bay, they drink at lunch in San Jose.”

Tom Morgan
February 4, 2022 9:46 am

Thanx for all your posts – they’re always interesting and informative, and get folks thinking, as is obvious from the comments here. I would like to ask the following question: In your equation there is a term involving F[n+1] for calculating T[n+1] – but that implies the forcing for the next period is already known prior to T[n+1] being calculated. That seems a bit non-causal. I’m wondering if F[n+1] could be replaced with F[n] , and still get a good model fit to the data.

February 4, 2022 10:46 am

We assume because time is everywhere we underatand it. The reality is that time is the least understood physical process. We routinely confuse the effects of time to fool ourselves.

The models are trying to predict the traffic and potholes on the road in the future using averages from today.

The same problem affects all inertial navigation systems. They need mid course corrections based on future data that is not available in the present.

February 4, 2022 1:09 pm

“Pay no attention to the forcings behind the curtain”

Aside from WE’s exposure of the hollowness of the climate models, this focuses on the real issue in how to understand climate: passivity.
Is climate active or passive?

The alarmist mainstream takes an extreme hard-core position of the climate being entirely passive. It only changes from forcing – thus the word “forcing”. It doesn’t want to change and left to itself, never would.

Those that argue that all climate change is solar forced, are of the same passive opinion as the CO2 alarmists. Climate only changes by CO2 / methane / ozone / other atmospheric thing, or climate only changes by solar forcing. The position is the same, only the forcing is different.

These positions have in common that they ignore the ocean, or dismiss it as a passive puddle. It is not.

The climate is in fact active. The atmosphere-ocean system is an excitable medium and a dissipative heat engine. Spontaneous spatiotemporal pattern continually emerges on a wide, fractal range of scales, with the signature log-log distributions. This means that essentially it changes itself. As Richard Lindzen stated, even if the total radiation budget equilibrium at top of atmosphere were to remain in zero balance for a thousand years, the oceans contain enough energy and unstable nonequilibrium dynamic to serve up continual climate change for the whole millenium – and more. I’m happy to go with Lindzen on that one.

“Climate” and “climate change” have exactly the same meaning. Adding “change” to “climate” adds zero meaning, and is a redundant tautology, like “frozen ice” or “wet rain”. It’s possibly the most profoundly stupid phrase in the history of human symbolic language.

February 4, 2022 4:51 pm

I understand that Tau (the number of years taken for forcing to impact on temperatures) is variable, but is much less than ten years. This is consistent with Hansen et al 1988 Scenario C which assumed global emissions were reduced to effectively zero by 2000 and global warming stopped in 2006 or 2007. It is also consistent with Scenario A (emissions growth of 1.5% pa) and Scenario B (roughly constant emissions).

But here is the issue that I have. I might be due to some misunderstanding on my part. If it is then I trust that Willis can put me straight enhancing my understanding and those of others.

The IPCC warming projections in AR4 and SR1.5 were based on Transient Climate Response defined in IPCC TAR 2001 as the temperature change at the time of CO2 doubling. In their example with ECS = 3.5C & CO2 rises at 1% per annum. At this rate CO2 levels will double in 70 years with TCR = 2.0 when ECS = 3.5. 3C of warming is not achieved in even 500 years. On this basis, with ECS = 3.0 and pre-industrial CO2 levels at 280ppm then a CO2 level of 395ppm would eventually lead to 1.5C of warming.
According to Mauna Loa CO2 readings CO2 rose from 316ppm in 1959 to 416ppm in 2021 or less than 32% in 62 years or less than 0.5% pa increase. That implies a doubling of CO2 of >150 years. Methane and Nitrous Oxide have been increasing at lower rates.
If this is true then adding in the forcing from other GHGs effectively the planet is at 445ppm CO2 levels. If ECS = 3.0 then 2.0C of warming will occur in the distant future, or not so distant future depending on your assumptions.
I assume here that a doubling of CO2 is related to an increase in forcing measured in Antony Watts per square metre.
I discussed the apparent difference between Hansen et al 1988 & TCR definition in TAR 2001 in my post below, with relevant graphics.

Robert of Texas
February 4, 2022 8:15 pm

My background is in computers and programming. I have seen this before – not with climate models but other very complex code, that when I dug in and started sorting out what-did-what found the net sum was a relatively simple function for the gross result and tiny little caveats that amounted to almost nothing for the rest.

It seems each person that inherited the program added their own little parts and then “tunned” until it looked right…never mind if they had actually analyzed what they ended up adding was insignificant (maybe after the tunning).

Programs like this can be very intimidating and convincing, and their owners seldom have any idea how to actually test them in any consistent manner. They just accept the results as meaningful.

One very fast inspection is to look for “bumpers” in the code to keep a *result* from deviating too far from the expected value. When you see these in a model, they are ad-hoc “fixes” for either bad programming or misunderstood physics. It’s like a finding a “divide by zero” when working in quantum mechanics – somethings wrong but you don’t know what, so you “normalize” the result and move on.

February 4, 2022 11:38 pm

What about the clouds?…

Gerry, England
February 6, 2022 4:28 am

For a ‘settled’ science the models show a hilariously large range of predictions. You could argue the hindcast failures are even funnier.

February 6, 2022 7:33 pm

To clarify, are Lambda and Tau simply selected by the modeller to fit the model to the past data, or are they calculable?

Raimund Raimund Müller
February 7, 2022 4:46 am

Hello Dr. Eschenbach,
very interesting. My point is focused on your e-time τ. IPCC does have 5 different ones in the Bern Model. I made an approximation with the antropogenetic emissions : we have 131ppm against anual emissions of 4.5 ppm gives a τ = 37 years. Taking all CO2 emissions together with 198 GtC by biomass aganist an total biomass reservoir of 600 GtC we will have a τ = 2,5 years. τ is a fundamentel value of IPCC to predict the future scenario. What is the basis of your 4,1 up to 5.7 years?

Joe Born
February 14, 2022 6:29 am

Great work, but I wonder if the post might have benefited from a little more explanation of the model equation’s rationale. 

This probably doesn’t matter, because what you’ve is a black-box exercise, and the results are what they are. To me, though, “simply lagging and rescaling the inputs” suggests the following differential equation:

\dot{T}=(\lambda F-T)/\tau,

where the dot means time differentiation, so differentiating that equation yields


and the result I get when I solve that equation is a model equation slightly different from yours. 

Starting out, I get an integral in \dot{F}





Now, ordinarily I’d have \dot{F} change linearly throughout the integration interval, but to get a result as close to yours as possible I instead assumed that \dot{F} is a constant \dot{F}(1) in that interval:



When I make the substitutions \dot{T}(1)=T_{n+1}-T_n, \dot{T}(0)=\Delta T_n\equiv T_n-T_{n-1}, and \dot{F}(1)=\Delta F_{n+1}\equiv F_{n+1}-F_n, though, I get

T_{n+1}=T_n+\Delta T_ne^{-1/\tau}+\left(1-e^{-1/\tau}\right)\lambda\Delta F_{n+1}.

Instead of being divided by \tau, that is, \Delta F_{n+1} gets multiplied by \left(1-e^{-1/\tau}\right).

Again, your results are what they are, but I for one got stuck on the model equation.

Joe Born
Reply to  Willis Eschenbach
February 14, 2022 5:04 pm

You may want to check the spreadsheet, too.

Reply to  Joe Born
February 14, 2022 11:41 pm

Very nice analysis !

A reflection is that if one linearize Stefan Boltzmann´s law (around todays temperature) then the “starting” differential equation can be interpreted as:

Temperature change is proportional to difference between incoming and outgoing energy fluxes

Joe Born
Reply to  JonasW
February 15, 2022 7:00 am

Thanks for the kind words.

And your reflection sounds right. I mean, if we take the rate \dot{T} of temperature change to be proportional to the radiation imbalance R:


where R is the radiation imbalance, C is the effective heat capacity in joules per square meter per kelvin, \alpha is the albedo, \epsilon is emissivity, \sigma is the Stefan-Boltzmann constant, T is the departure of the global-average surface temperature from a reference value T_0, and T_{GH} is the greenhouse effect, and if forcing F is what the imbalance would be at the current greenhouse-effect level T_{GH} if the departure T from the reference temperature were zero:



\lambda=\dfrac{1}{\partial F/\partial T_{GH}}=\dfrac{1}{4\epsilon\sigma\left(T_0-T_{GH}\right)^3}, and


\tau=\dfrac{1}{\partial\dot{T}/\partial T_{GH}}=\lambda C.

If we assume \epsilon=1.0, T_0=288\,\mathrm{K}, and T_{GH}=32\,\mathrm{K}, then the equation above gives us \lambda=0.263\,\mathrm{W/m}^2\mathrm{/K}, which doesn’t differ wildly from the \lambda=0.258\,\mathrm{W/m}^2\mathrm{/K} that Mr. Eschenbach’s spreadsheet gets for Model 15. 

Reply to  Joe Born
February 16, 2022 1:34 am

Hi Joe,

Been reading your analysis but I am not sure that I follow you 100%.

My understanding is that you use the “start” differential equation and write T as function of F and R. Then differentiate T with respect to Tgh and set the differential equal to zero (resulting in the expressions for lambda and tau).

I do like your analysis.

A minor thing is that I think you got the unit of labda “upside down”. Should be m2*K / W.

I calculated C just for fun (using Mr Eschenbachs value 5 years). The value I got was about 6*10^8 J/m2/K. This roughly corresponds to 100 meter deep water.

Bottom line seems that a very simple model based on radiation imbalance can reproduce the results from CMIP. I think it is time for IPCC modelers to do something else.

Joe Born
Reply to  JonasW
February 16, 2022 6:15 am

Oops! Yes, I did flub the dimensions.

Anyway, we know that \lambda is the ratio that T bears to F when the system is in equilibrium, i.e., when \dot{T}=0.  My rationale for using \lambda=\partial T_{GH}/\partial F was that the change in T‘s equilibrium value equals the change in the greenhouse effect T_{GH}.

It may help to observe explicitly that my previous comment’s first equation is the differential equation that describes the system’s attempting to reach equilibrium and that its second equation is the algebraic equation that defines forcing F as the imbalance the forcing would cause if T were zero.

Reply to  Joe Born
February 16, 2022 10:47 pm

Thanks for your comments.

Yepp, I do follow your analysis (and agree).

I think it is beautiful. One equation (based on fundamental physics) can replace the whole CMIP computer code.

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