Guest Post by Willis Eschenbach (@WEschenbach on X, my personal blog is here.)
Well, for my usual lack of reasons, I got to thinking about updating the subject of my three previous posts relating the input and the output of climate models. To my surprise, the most recent of these was a shocking fifteen years ago … tempus fuggit, as they say …
In that post, I discussed the idea of a “black box” analysis. This is where you have both the input and the output of some unknown process, and the challenge is to figure out what is happening functionally inside the black box. Here was my illustration showing the CCSM3 climate model as a black box, with the input being the “forcing” and the output representing the global annual surface temperature.
And what is “forcing” when it’s at home? Well, per the IPCC, it’s short for ‘radiative forcing’, which they define as
The change in the net, downward minus upward, radiative flux (expressed in W m−2) at the tropopause or top of atmosphere due to a change in an (external) driver of climate change, such as a change in the concentration of carbon dioxide (CO2), the concentration of volcanic aerosols or the output of the Sun.
Note that the result of a black box analysis may be totally different from what is actually happening inside the black box. The only requirement is that it takes the same input and produces the same output.
In the post linked to above, I showed that you can almost perfectly emulate the model output that represents the global annual surface temperature … and not only that, you can do it with a bozo-simple, one-line equation with only two tuned parameters. All that the equation does is to resize and lag the input forcing, and like some magic trick, it gives you the output temperature.
Now, there has been an intermittent series of climate model comparisons under the umbrella of something called the Climate Model Intercomparison Program. At that time, they were on the third comparison, called “CMIP3”. Since we’re now up to CMIP6, I thought I should take another look to see what changes have occurred.
I’m going to present the equation, and for those of you who are allergic to math, here is the trigger warning—avert your eyes …
With the trigger warning out of the way, here is the equation that rescales and lags the forcing to emulate the output temperature:
========================== Math Averse, skip this section ==========================
T(n+1) = T(n)+λ ∆F(n+1) * (1- exp( -1 / τ )) + ΔT(n) exp( -1 / τ )
OK, now lets render 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 forcing, in watts per square metre. 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). ∆F(n), correspondingly, is the change in forcing since the previous time period F(n-1).
Lambda, or “λ”, is the scaling 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(-1/τ)” means the number 2.71828 to the power of -1/τ.
========================== End of the section to skip ==========================
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 ), which is the temperature increase from the previous forcing. This lag term is necessary because the effects of the changes in forcing are not instantaneous.
Now, regarding the choice of the computer model and the data. I used the GISS-E2.1 model for a simple reason. It’s one of the few where the modelers published the forcings that they used. Here are the forcing data and the model temperature output. The temperature output at that source (KNMI) shows five individual model runs and, as is the custom, I’ve used the average of the runs. There’s a discussion of the use of the GISS-E2.1 model in the study entitled “CMIP6 Historical Simulations (1850–2014) With GISS-E2.1“. The E2.1 model is described as “an updated and more skillful version of the GISS-E2 model used in CMIP5”.
The forcings that they used are divided into three groups: greenhouse gases (except water vapor), anthropogenic aerosols, and “natural” forcings. Here are those forcings, from the link above:
So without further ado, here is the result of my black box analysis.
As you can see, the simple one-line two-parameter equation above emulates the model results to a ludicrous degree of accuracy and precision. Even the smallest changes are captured by the black box emulation.
So … what can we conclude from this?
My conclusion is that despite the many hundreds of thousands of lines of code in the GISS E2.1 model, in reality, nothing makes a difference in the outcome except for the claimed forcings.
Modulation of the North Atlantic Oscillation; changes in the frequency and strength of the La Nina Pump; variations in the Gulf Stream; alterations in cloud type, cover, and time of emergence (except those due to aerosols); differences in where the forcing is occurring on the planetary surface; increasing or decreasing water vapor; variations in cloud albedo; the oft-predicted weakening of the Atlantic Meridional Overturning Circulation … none of that makes any difference in the model.
We know that because if any of them did make a difference, we couldn’t reproduce the output so exactly using just the input forcings.
And of course, the corollary of that conclusion is that the current generation and type of climate models is completely unfit for the purposes to which they are put. If their output is just a simple scaled and lagged version of their input, they cannot forecast the climate in the year 2100. They cannot tell us if a given drought is made more or less likely by “climate change”. In short, they are of little use for any serious purpose.
Here’s the thing. Climate models can only do what the modelers tell them to do. In this case, all of the models are running under the central paradigm of mainstream climate science, which says that changes in temperature are a linear function of changes in forcing. Expressed mathematically, that is
∆T = λ ∆F
So of course the models’ output is a scaled, lagged version of their input. It’s what the models have been told to tell us … so that’s what they tell us.
So that’s Part 1 of my look at the GISS E2.1 model. When the madness next strikes me, I’ll write up Part 2, which will be about the success of the model in hindcasting the past … or the lack thereof.
Late night here, a foggy evening after an overcast day. Must be climate change …
My very best to everyone,
w.
I know, I know, you’ve heard it before: When you comment, please QUOTE the exact words you are discussing. I choose my words carefully and can defend them. I can’t defend what you might think they mean.
Mas handwavium.
63+63=63 is still wrong.
396 BB/333 “back”/63 duplicate GHE loop is still a theoretical calculated fantasy.
The concept of forcing is as bogus as caloric and luminiferous ether.
it’s sounds like rape- ergo, all who use fossil fuels are raping the Earth- that’s what they want us to believe
I was going to ask : Define ` Forcing .
I don’t know of its use in any other field .
The most fundamental non-optional computation for any quantitative analysis of Earth’s temperature is the gray body radiative equilibrium , ~ 278.7+-2.3 around our orbit . Then the correlation of our measured color , Schwarzschild , spectrum with that of the Sun’s power spectrum determines the planet’s equilibrium . I have never seen even that essential calculation .
Within that envelope the only force which can , and does , cause the internal temperature to be higher is the adiabatic tradeoff of gravitational potential , and thermal , kinetic+radiant , energy keeping total energy density constant .
All these arguments are about planetary means , which are like gas laws . The internal chaos is a separate question .
This is classical quantitative physics . I’m too simple minded to be interested in pixel or voxel models , altho CoSy is uniquely suited to implement them ( https://cosy.com/y26/NL202605.html https://cosy.com/4thCoSy/Physics/general.f ) , until these fundamental constrains are understood .
Thanks, Bob. I’ve added the IPCC definition of “forcing” to the head post.
w.
I may be wrong, but I have always thought of “forcings” as similar to the engineering terms of deltas. ΔT, ΔP, ΔH, ΔC, etc. all drivers of change in a system.
Forcing = ASR – OLR if + = warming.
Both are ToA not physical surface.
Basic heat transfer class.
Sun heats surface, surface heats atmosphere per Q = U A (Tsurf – Ttoa)
If albedo decreases Q increases as does dT and Tsurf.
Eerth be cooler w atmos/water vapor/30% albedo not warmer.
Ubiquitous GHE balance graphics don’t + violate GAAP & LoT.
Kinetic heat transfer processes of contiguous atmos molecules render “extra” GHE energy from BB surface impossible.
396 BB & 333 “back” & 63 duplicate based on theoretical, imaginary. calculation for denominator of emissivity ratio: 63/396=0.16.
GHE = bogus & CAGW = scam.
You are using an incorrect definition of forcing.
What you are using is a hijacked, repurposed, social/common language context derived definition that is always presented with not clearly defined context.
I really wish you would stop using GAAP, which is Generally Accepted Accounting Practices when Kirchhoff’s Law is better.
I really wish you would stop promoting that bogus energy graphic. All you are doing is fueling AI to accept it as credible.
Nor is “back” radiation anywhere but GHE.
Forcing:
In physics and dynamical systems, forcing is an external or driving influence—such as a time-dependent force, energy input, or boundary condition—applied to a system from the outside to change or alter its behavior.
Mechanics and Oscillations
External driver: An imposed time-varying input like an oscillating mechanical pull or electrical voltage.
Driven systems: Causes phenomena like driven harmonic motion, resonance, or specific wave patterns.
Mathematical term: Appears as an inhomogeneous non-zero term F(t) in a differential equation describing the system.
Forcing Function:
A forcing function is an external input or force applied to a physical system over time. It acts as the non-homogeneous term in a differential equation, driving the system to change its natural behavior.
Core Role in Physics
External Drive: Represents an outside influence like a push, wind, or voltage acting on a system.
Math Term: Appears on the right side of a differential equation (such as (F(t)) in: F(t) = m x y” + c x y’ + k x y.
System Response: Forces the system to move past its normal resting or free vibration state into a forced response
Common Types
Step Function: Turns on suddenly and stays at a steady level.
Sinusoidal Function: Repeats in a smooth wave pattern, like an alternating electrical current or vibrating motor.
Impulse Function: A very large force applied for a tiny fraction of time, like hitting a ball with a bat.
Nothing has changed in these terms since I first learned them in high school in the late 1960s.
So they do not include clouds in the IPCC definition as an example of a “driver of climate change?”
Of course not! Cloud formation is a complex and chaotic operation that involves many forms of energy in supporting the isothermal process of condensation. It can’t be measured by a thermometer, therefore the thermometer-readers feel justified in ignoring the whole mess!
Excellent look at forcings in the climate models. I always contend it’s forcings that determine the output and they are determined by the whim of the modeler.
7 negative forcing, 1 positive forcing.
Forcings based on 396 are equally junk.
“Forcing” is exactly the right word to describe most climate models.
Yuh, forcing us or trying to force us to believe nonsense.
I was thinking forced results but yours are just as applicable.
Anyone else of that generation, can’t here the words ‘black box’ without having to primal scream..
Wa-ho, wa-ho, wa-wa-ho, wa-wa-wa-wa-ho
You’re such a, you’re such a, you’re such a
You’re such a hot temptation, you just walk right in, walk-walk-walk right in and…..
“In 1993, Black Box added American singer Charvoni Woodson to the lineup. They released the single “Rockin’ to the Music”, which performed poorly on the charts. In 1995, Black Box released their second album, Positive Vibration, which failed to chart or rise to the same level of success as their previous record. The album spawned the singles “Not Anyone” and “A Positive Vibration”, both of which fared well on the charts. In 1997, the album was re-issued with three additional singles: “I Got the Vibration”, “Native New Yorker”, and “Fall into My Love”.”
Excellent reminder WE that your old simplistic model finding is still true for CMIP6.
There are two additional reasons the whole CMIP thing is a waste of time and money.
First, an observation first made by Judith Curry some years ago concerning CMIP4. The first required CMIP submission is a 30 year model hindcast. CMIP always shows the various model hindcast comparisons as anomalies, and that way they always look ‘close’ to each other. In reality, as real temperature hindcasts they vary by about +/-3.7C—awful. They disagree over the past 30 years by over twice the supposed rise to 2100.
Second, to best hindcast, model parameters are ‘tuned’ (there are two basic methods, both the subject of a previous post here). Tuning automatically drags in the attribution problem—how much change is ‘forcings’ and how much is just natural variation—of the many sorts you note here CMIP6 manifestly ignores. Even AR4 WG1 SPM figure 4 said the GAST rise from ~1920-1945 was mostly natural, as there was only a small rise in forcings (CO2).
It is simple. Hindcasting with control knob tuning is simply curve fitting.
From the article: “When the madness next strikes me, I’ll write up Part 2, which will be about the success of the model in hindcasting the past … or the lack thereof.”
I’ll look forward to that.
So the complex coupled earth energy systems is:
CO2 is input
IR is transfer function
Temperature is output.
Got it.
If the science is settled, then why are we spending so much on that settled science?
Climate scientists gotta eat too. So settled but not too settled.
You are right and that unsettles me.
Thanks for this update, Willis! Your insights and analysis on this topic are gem-standard work.
“My conclusion is that despite the many hundreds of thousands of lines of code in the GISS E2.1 model, in reality, nothing makes a difference in the outcome except for the claimed forcings.”
Readers here will remember that, similarly, Pat Frank used simplified computation from the forcing inputs to successfully emulate GCM air temperature projections. This finding was then used to propagate uncertainty on an annual-step basis in his 2019 work here.
https://www.frontiersin.org/journals/earth-science/articles/10.3389/feart.2019.00223/full
So my point is that once a simplified emulation from the time-scheduled forcings is demonstrated, as both you and he have done, the circular cartoon-caricature nature of the entire modeling exercise using complex GCMs is apparent.
Well done.
‘Readers here will remember that, similarly, Pat Frank used simplified computation from the forcing inputs to successfully emulate GCM air temperature projections.’
Yes, and if there are uncertainties in the ‘forcings’, these compound over ‘time’, hence the results tell us absolutely nothing about future temperatures.
Hmmm. Seems to me another poster here has also shown how the model outputs degenerate into a linear projection also. Sooner or later you’d think the modelers and their supporters would wake up and say “what are we doing here”!
They cannot ‘wake up’, because then they would all be out of work. CMIP modeling is an annual multibillion dollar global business. NCAR alone has an annual budget of about $250 million, which is why Trump wants to shut it down.
A question and a comment.
Question: Is the “All” quantity in Fig. 10 what you use for F?
Comment: I suspect that your model was intended to be a step in the discrete-time calculation of the solution to dF/dT = λF – T/τ. It doesn’t look quite right, though, so some readers might appreciate your providing a derivation.
Hey, Joe, always good to hear from you. Yes, I’ve used the “All” quantity.
As to the derivation, no clue. It was based on the comment by Paul_K here.
Hang on, let me ask my research assistant … perplexity.ai.
OK, here you go. Is this the correct derivation? Dunno … but it sure works.
w.
===
1. Start from a simple ODE
Assume global mean temperature T(t) responds to net forcing F(t) according to a one‑box energy balance model with a single relaxation timescale τ and sensitivity λ: dt/dT =τλF(t)−T(t)
This says T relaxes exponentially toward the instantaneous equilibrium value λF(t) with e‑folding time τ.
Define ΔT(t)=T(t)−T(t−1) and ΔF(t)=F(t)−F(t−1) when we move to discrete time steps of length 1 (e.g. years).
2. Solve over one time step
Consider the interval from t=n to t=n+1. Over this step, approximate the forcing as changing from F(n) to F(n+1), and write the solution of the linear ODE with a time‑varying equilibrium. A convenient way is to treat the system as:
A homogeneous part (decay of existing anomaly) with solution ∝ exp(−Δt/τ).A particular part driven by the change in forcing over the step, which will add an increment toward the new equilibrium.For a unit time step Δt=1, the homogeneous solution gives
Thom (n+1)=T(n)exp(−τ1 )+λF(n)(1−exp(−τ1 ))
But we want the increment due specifically to the change in forcing between n and n+1, so we separate the effect of F(n) (already embodied in T(n)) from the new forcing F(n+1).
3. Express in terms of changes ΔT and ΔF
Write the equilibrium temperature at step n as Teq (n)=λF(n) and at step n+1 as Teq (n+1)=λF(n+1). The standard discrete solution for a first‑order lag toward a moving equilibrium over a unit step is:
T(n+1)=T(n)exp(−τ1 )+Teq (n+1)(1−exp(−τ1 ))
Substitute Teq (n+1)=λF(n+1):
T(n+1)=T(n)exp(−τ1 )+λF(n+1)(1−exp(−τ1 ))
Now write T(n)=T(n−1)+ΔT(n), and similarly F(n+1)=F(n)+ΔF(n+1). The piece involving T(n−1) can be grouped with the previous step; what matters for the update is:
A term proportional to the latest change in forcing ΔF(n+1).A term carrying forward the previous change in temperature ΔT(n) with an exponential decay.If you expand and regroup terms, you can write the update in “increment” form:
T(n+1)=T(n)+λΔF(n+1)(1−exp(−1/τ))+ΔT(n)exp(−1/τ)
which is exactly your discrete‑time formula.
Thanks a lot; that clears things up. (My question arose because I had thought I remembered that years ago you had rolled your own weighting of the forcing constituents.)
Although I haven’t yet taken the time to verify the last step, the AI derivation otherwise makes sense if you start with τdt/dT =λF(t)−T(t) instead of dt/dT = τλF(t)−T(t) and read “τ1” as simply “τ.” (Incidentally, in my comment I flubbed the initial equation myself.)
As you say, this is a black-box exercise, so what matters is the output, not whether the model’s innards seem to make sense to the casual observer. And your output couldn’t get much better, so it makes sense not to be to concerned with the derivation.
But permit me to make a couple of observations anyway.
First, it would probably appeal to more readers’ intuition if instead of the form you used in the head post you presented the model as the derivation’s penultimate result, i.e., T(n+1)=T(n)exp(−τ1 )+λF(n+1)(1−exp(−τ1 )); my experience with guys who know this kind of stuff suggests that they’d more readily recognize that form as a simple superposition of the equation’s homogeneous and driven solutions.
Second, note that the derivation treats T(n) as a temperature sample taken at the end of the nth time interval and F(n) as a value that’s constant throughout that time interval. That is, one might consider the time measurements as being delayed by half a time interval from the forcing measurements.
Again, this is a black-box exercise, so you don’t really care whether the parameters have good physical interpretations. If you wanted to attempt a more-physical τ value, though, I’d suggest using a derivation that instead treats the temperature and forcing values as coincident samples, maybe assuming that the forcing changes linearly between samples. That would lend itself readily to a Runge-Kutta calculation. Such a calculation would be more involved, but for guys who regularly do numerical differential-equations solutions I think it would be readily understood.
I agree Joe, “T(n+1)=T(n)exp(−τ1 )+λF(n+1)(1−exp(−τ1 ))” is a better form.
With the original form:
T(n+1) = T(n)+λ ∆F(n+1) * (1- exp( -1 / τ )) + ΔT(n) exp( -1 / τ )
where “∆T(n) is the change in temperature since the previous period, or T(n) minus the previous temperature T(n-1).”
The equation involves three samples: n+1, n, and n-1.
Some people might recognize the similarity of these equations to an exponential moving average, S(n) = αX(n)+(1-α)S(n-1) where X(n) is current observation and S(n) is the smoothed statistic. The smoothing constant α is related to the time constant τ by α=1-exp(-ΔT/τ) where ΔT in this instance is the time step, not the change in temperature.
Willis, you wrote that “All that the equation does is to resize and lag the input forcing”. This depends on what you meant by resize. Scaling is frequency dependent. You can test this by injecting first a high-frequency sinusoid (e.g. 2 year period) and then a low-frequency sinusoid (e.g. 20 year period) of the same amplitude.
The forcing function is, at best incomplete. For all practical purposes the oceans integrate solar activity and the time constant is very long (centuries at least). An integrator has a -90° phase response so the delay is variable, e.g the nominal delay for 60-year cycle would be 15 years. The oceanic response appears to apply for periods longer than 10 years. The atmospheric and sea-surface response is different and obviously faster with less heat capacity.
According to Google, model CCSM3 will produce at low resolution about 35 simulated years of result for each day of run time on a supercomputer. At high resolution each day of runtime will produce about 4 years of simulated results. For your chart covering 250 years at high resolution it would take about 63 days (2 months) to get a result.
How long did it take whatever computer you have to get the result shown on your chart? Perhaps you could offer a deal to NOAA?
For the record, NCAR says that the average run time for a single CMIP6 model run was indeed about 60 days of continuous supercomputer operation, with a grid spacing median of about 200 km by 200 km at the equator. CMIP6 finest was 100km, coarsest was 280km.
Sure, things aren’t complicated, but rather trivial. One only needs to dare taking them on.
One such simple thing is featured in the chart above. Just take a closer look at that relation between neg. aerosol forcing and anth. GHG forcings. In the year 2000 it were like -1.2 / +2.4, or 0.5, roughly. Now think what this might have look like for the northern hemisphere, where 90% of worlds population lives and over 90% of aerosols are emitted.
We should not have had warming in the NH up to the year 2000..
Thanks for revisiting this Willis. I like it that your heuristic approach gives nearly identical results as GISS model E2.1, which for reasons I won’t go into now, is also a heuristic representation of reality.
Note, there is nothing wrong with heuristic methods, as all of us use them subconsciously almost every day in order to avoid becoming potential ‘Darwin Award’ recipients. But they should always be challenged when they serve as the basis for formulating major public policies, which is currently the only role of climate models.
The whole problem with climate models is that they are all fundamentally flawed mathematically.
As someone who received a MS and PhD in mechanical engineering with an emphasis in the thermal sciences, (thermodynamics and heat transfer) and who has developed and validated thermal models of solar thermal receivers, (Earth is a solar receiver.) I can assure you that the climate models upon which most of the hysteria is based are fundamentally flawed. (I was hired by Sandia Labs to lead solar thermal receiver development, including the development and validation of thermal models, which I did.)
The mathematical treatment of the greenhouse effect was described in a National Academy of Science (NAS) report entitled Carbon Dioxide and Climate: A Scientific Assessment published in 1979 and chaired by Jule G. Charney. Note 2 under Climate Sensitivity in Wikipedia, confirms the Charney equation, and indicates that CO2 addition increases the top of atmosphere temperature. According to Charney and the mathematical models used to predict significant global warming, doubling CO2 concentrations in the atmosphere causes a global average temperature increase
(Delta T) of Q/4sigmaT^3.
DeltaT = Q/4sigmaT^3
Where Q is supposed to be the reduction in radiative heat loss from the top of the atmosphere caused by doubling CO2 concentrations in the atmosphere (radiative forcing); sigma is the Stefan-Boltzmann constant (5.67×10^-8W/m2/T^4; and T is the apparent average temperature of the earth as viewed from space, i.e., top of atmosphere (TOA) temperature.
But this equation is merely the first derivative of the energy balance equation of the earth expressed in incremental form.
The energy balance equation between sun, earth and space, where back radiation from space is assumed to be negligible and earth’s emissivity is assumed to be 1.0 (both good assumptions) is:
Q=sigmaT^4
The derivative of the above equation is:
dQ/dT = 4sigmaT^3
Rearranging the above equation to solve for dT and expressing it in incremental form (DeltaT) yields the Charney equation for the direct effect of doubling CO2 concentrations in the atmosphere (see above). Mathematically, the Charney equation is an approximation of simply adding radiative forcing from CO2 doubling to the energy absorbed by earth. In other words, Charney and the climate modeling community mathematically treat a reduction in heat loss at the top of the atmosphere as an increase in earth absorbed solar energy. This is a clear-cut violation of the first law of thermodynamics. Mathematically, Charney is creating energy from nothing. Adding CO2 to the atmosphere does not increase the amount of solar energy absorbed by earth. In addition, substituting a high temperature heat source for a low temperature heat loss is implicitly a violation of the second law of thermodynamics.
Proper analysis of heat transfer from earth’s surface to the TOA where heat absorbed by earth is radiatively rejected to space yields a significantly lower temperature increase than calculated by climate scientists.
Heat is transferred from earth’s surface to the top of the atmosphere by convection (99W/m2) and radiation (64W/m2). The TOA temperature is about 255K and is unaffected by additional CO2. With an average earth surface temperature of 288K, the temperature difference between earth’s surface and the TOA is 33K. This implies a global average convective heat transfer coefficient (Hc) of 3W/m2/K.
99W/m2 = HcW/m2/K(288-255)K
Surface emission from an average surface temperate of 288K is 390W/m2 and implies an average global sky back radiation of 326W/m2. (390-326) = 64W/m2
Assuming a radiative forcing of 4W/m2 for doubling CO2 concentrations in the atmosphere leads to an average global sky back radiation of 326+4 = 330W/m2
(Note that this is an actual reduction in radiative heat loss, unlike Charney, caused by doubling CO2 concentrations in the atmosphere. Note also that this leads to a slightly higher average sky temperature, which is in line with physical expectations. In other words, unlike Charney the math reflects physical reality.)
Thus, the mixed heat transfer equation for doubling CO2 concentration in the atmosphere is:
163 = 3(T2 – 255) + sigmaT2^4 – (326+4)
Solving for T2, the average earth surface temperature after doubling CO2 concentrations in the atmosphere yields 288.47 an increase of only 0.47 degrees, a factor 2.3 less than the 1.06 calculated by Charney.
This simple review of the math behind global warming theory shows that more CO2 in the atmosphere will have a much smaller effect than calculated by climate models, which simply add radiative forcing to earth-absorbed solar flux.
Complex models will not result is significantly different results. For example, Charney predicted 1.5 to 4.5 degrees of warming while the detailed climate models that followed generally fall in the same range. This should not be surprising since both the global and detailed models ultimately average the parameters that affect temperature increases.
This was something that was somewhat surprising to me at first when evaluating the performance of solar thermal receivers. I could get accurate estimates of thermal performance with simple thermal models based on the key design parameters. Detailed finite element analysis results did not differ significantly from the simple analysis.
In fact, the simple global models will tend to be conservative (predict slightly high) because of the non-linear effects of the T^4 response. Radiative heat transfer from the warm areas of the earth more than compensate for low temperature regions.
Because feedback from increased water vapor and reduced ice coverage are also moderated by convection and driven by a much lower CO2 induced temperature increase, feedback temperature increases will be insignificant.
This brief review of global warming math suggests that climate scientists have vastly overestimated the global warming potential of increasing CO2 concentrations in the atmosphere; should stay in their lane, far away from thermal analysis. They obviously do not know what they are doing. No one should listen to them.
My only problem with your solution is that CO2 only radiates in a small band of frequencies. You cannot just add its intensity to the earths intensity because the earths intensity is based on a black body’s full Planck curve. The total energy in each are vastly different.
In essence the 326 W/m² is much, much less across the entire spectrum of radiative energy from the earth. My pure guess without doing the math, is that an average emissivity of 0.1 across the full Planck curve is more realistic. That would make the 326 closer to 32.6.
The spectral response of CO2 is baked into the 4W/m2 radiative forcing. In other words doubling CO2 results in a sky temperature that feels 4W/m2 warmer. The 326W/m2 accounts for the integrated absorption characteristics of the atmosphere, including water vapor, CO2, etc. Doubling CO2 concentrations in the atmosphere simply adds 4W/m2 more.
It may be baked for the frequencies that it radiates. However, it is not equivalent to a black body radiation of a Planck distribution.
You may add the intensity at the wavelengths that CO2 radiates to the wavelengths contained in a black body Planck curve but that is all. The emissivity at those wavelengths may be close enough to do so. You cannot add CO2 intensity to those wavelengths where its emissivity is close to zero. Basically, anywhere close to 15 um can be added but no other wavelengths will receive any added intensity. You end up with a bump in the Planck curve at CO2 primary emission frequency and nowhere else.
In other words if you integrate the Planck law over all the wavelengths emitted by a black body at a given temperature, you will have the total radiant energy emitted per unit time per unit surface area.
CO2 does not radiate as a black body. The emissivity is zero for most of the black body radiation. You can not treat CO2 equal to the surface radiation. Converting temperature to intensity or vice versa technically requires black body radiation. It can be reduced by using emissivity, but the assumption is that the emissivity applies to all wavelengths not just a few.
Here is a picture I created in CoPilot that shows the little bump from CO2 added to the surface Planck curve.
Here is another problem. The next instant after absorbing, the surface will spread that energy our over all the wavelengths it radiates out. It won’t amount to much of an increase at any given wavelength.
The TOA radiates to space like a blackbody at 255K. Doubling CO2 concentrations in the atmosphere reduces heat loss from the earth’s surface to the TOA by 4W/m2. Because of continuity, heat loss from the TOA to space is also 4W/m2. The details of wavelength dependent characteristics of CO2 are immaterial.
Here is what you said.
“”Assuming a radiative forcing of 4W/m2 for doubling CO2 concentrations in the atmosphere leads to an average global sky back radiation of 326+4 = 330W/m2″”
326 is a black body temperature intensity. It is based on radiation across a full integration of the Planck equation. It is an incorrect value for back radiation from CO2. The 4 W/m² is also a black body estimate of total energy based on integrating the Planck equation. The energy given by these are not based on the transmittance of CO2.
I have yet to be shown where the 4 W/m^2 decrease actually comes from.
R_out ∝ nR_co2, where n is the number of radiating molecules and R_co2 is the emitted radiation per molecule toward space, and R_out is total radiation toward space.
If CO2 concentration goes up then expectation would normally be that “n” goes up.
In order for ΔR_out to go down, either “n” has to go down or R_co2 has to go down, or perhaps a combination of the two. It is doubtful that “n” will go down with increased concentration of CO2. Leaving the problem that R_co2 has to go down.
Continuity would seem to imply that R_co2 would at least equal R_-co2 (i.e. the radiation away from space) if not be greater when integrated over the height of the atmosphere. The corollary to this is that if the earth gets warmer and emits more LWIR then R_co2 and R_-co2 would both go up as well because of increased absorption.
If R_co2 + R_-co2 both go up then how does R_total change? You would expect it to go up. ΔR_co2 > 0 and ΔR_-co2 > 0.
One possibility for a different result would be to add some kind of an additional term, R_retained. R_retained + R_out = nR_co2 –> R_out = nR_co2 – R_retained.
Over the long term, R_retained has been approximately zero. Anything else would have left the Earth a molten ball over the past millenia. It would also imply that the ice ages could not have happened.
There are other possibilities but none seem to make any physical sense.
Convection cannot remove energy from the Earth system. The single strongest challenge to the 0.47°C:
The calculation treats the 255 K emission level as fixed, whereas radiative-transfer theory predicts that increasing CO₂ changes the altitude and temperature of the effective emission layer, reducing outgoing radiation until warming restores balance. That is the central mechanism behind the conventional ~1.1°C no-feedback sensitivity estimate.
The 255K TOA temperature is fixed. Look at the energy balance equation between earth, sun, and space that establishes TOA temperature. There is nothing that additional CO2 can do to effect TOA temperature. The TOA temperature of 255K is a boundary condition for the heat transfer problem between earth’s surface and TOA. Doubling CO2 can only add to atmospheric thermal resistance, resulting in 4W/m2 less radiative transfer from earth’s surface to the TOA and then to space.
Convection certainly transfers heat from earth’s surface to space. Not including convective transfer in what is obviously a mixed heat transfer problem is the main reason climate models overestimate warming.
“Doubling CO2 can only add to atmospheric thermal resistance, resulting in 4W/m2 less radiative transfer from earth’s surface to the TOA and then to space.”
This would be true for instantaneous radiative balance, i.e. between black bodies. But the assumption that “thermal resistance” exists means that instantaneous black body balance doesn’t apply.
Since the earth radiates out for 24 hours but only receives incoming over 12 hours it doesn’t have to radiate at a difference of 4W/m^2 in order to maintain balance between heat-in and heat-out in terms of joules of heat.
You are trying to confuse the alarmists with facts and data.
Shame on you.
Sorry. Not sorry.
It really plisses me off when so-called “climate scientists” delve into a subject which they obviously have no comprehension of (heat transfer), and then tell me that I am unqualified to comment on it because I’m not a climate scientist.
Might “retained” at the TOA be more descriptive of the consensus community position than an “increase” in earth absorbed solar energy?
What they appear to have done is equate a reduction in heat loss at the top of the atmosphere from doubling CO2 concentrations to an increase in absorbed solar flux, which, I guess, sounds reasonable if you don’t know any better. But that is not the way it works. Your math has to line up with what is physically happening.
Heat loss and gain has to be calculated in terms of joules, not joules/sec-m^2, since the heat loss and gain occur over different time intervals. I’ve never seen climate science actually try to equate joules-in versus joules-out over different time intervals. The excuse of “we are using normalized values” just means that they had to calculate joules-in and joules-out over some time period in order to normalize to a common time interval. If you know joules-in and joules-out then why normalize them over a common time interval? Either joules-in and joules-out balance or they don’t.
From post:”…earth’s emissivity is assumed to be 1.0 (both good assumptions) is:”
What emissivity is assigned to the CO2? Or when doubled?
CO2 does not emit thermal radiation I.e. emission based on temperature.
Earth’s emissivity cannot be 1.
That violates LoT 1. 396 out vs 342 in
Emissivity 1 = 63/160 = .39
Emissivity 2 = 63/396 = .16
This is the third time in one article.
????
If you draw a Planck curve for a black body at a given temperature, say 300K, CO2 does not emit at any of the wavelengths except 15 um. the emissivity at absorptance/transmittance is about 0.95 at 15 um. At all others it is near zero. You can’t even use the SB equation of I = εσT⁴ because it doesn’t emit at all Planck curve wavelengths.
I agree.
My question to Grok and the response.
Emissivity for CO2 at <30°C and <1 atm: Given these conditions, the emissivity would likely be:
– Very close to zero for practical purposes. Let’s say, for an extremely rough estimate, somewhere around 0.001 or even less, depending on the exact conditions. This is because at such low temperatures and pressures, CO2’s interaction with thermal radiation is minimal.
While the emissivity of CO2 molecules are small, it is not zero, and because of the the “cavity like” effect of the cloud of molecules it can and does approach a black body. Eventually, IR radiation in the appropriate wavelength will encounter a CO2 molecule that absorbs it. The net effect is absorptance/emissivity approaching 1.0.
The emissivity of CO2 is only near 1.0 at the frequencies that it absorbs, primarily 15 um. At every other wavelength it is near zero. It does not act like a black body with radiation at all Planck curve wavelengths that would be expected for a black body at a given temperature.
At the other frequencies, water vapor, ozone, earth’s surface, etc. fill in the rest of the blackbody spectrum. They are mostly unaffected by adding more CO2. The 4W/m2 forcing from doubling CO2 concentrations in the atmosphere is caused by the “extra radiative insulation” both in increased thickness (height) and increased concentration in the approximately 26% part of the IR spectrum covered by CO2.
“At the other frequencies, water vapor, ozone, earth’s surface, etc. fill in the rest of the blackbody spectrum.”
But the 4 W/m^2 value is attributed solely to CO2 in the atmosphere, not to all the other components. And the earth’s surface would not figure into the atmosphere emission spectrum.
>> Simulations (1850–2014) With GISS-E2.1“. The E2.1 model is described as “an updated and more skillful version of the GISS-E2 model used in CMIP5”.
Modellers test for skillfullness is typically a comparison of the calculated trend with some past real world measurements.
That is a necessity, but not enough to prove skillfullness. For example the drawing of a child could in principle match the measurements. Only a full uncertainty analysis using the full range of possible parameters without restrictions by the modellers believe can show skillfullness, anything less is meaningless.
In particular high-co2 sensitive scenarios show unrealistic trends in CMIP6 after the models resolution was increased and things like aerosol cloud physics was improved. The knowledge that these improvements affect calculated trends impacts the results of all older models! Older analysis have not considered these facts and are therefore incomplete/unreliable until updated.
The I reflected usage of CMIP5 results (or older) without considering the potential effect of the lower resolution and errors in physics is unscientific (and that I am afraid does include your post here as well!).
“If their output is just a simple scaled and lagged version of their input, they cannot forecast the climate in the year 2100. “
Willis, we have been through this before. Your fallacy is that the forcings you cite are not the input to the models. Those inputs are just gas concentrations and TSI etc. Your analysis is circular. The forcings are derived from the output, or at least some advanced part of the modelling. All your simple arithmtic does is reproduce backwards the simple arithmetic by which the forcings were derived from the model outputs. Your black box is not the functionality of the models.
Your criticism does not persuade me. If a linear equation can reproduce a models output with using reasonable inputs, especially over a long time period, then it must have some correlation.
You need to show that Willis’s inputs are unreasonable, or that the output is spurious and not statistically significant.
“then it must have some correlation”
Of course it does. The forcing is derived, by simple linear calculation, from the GCM output. Willis’ black box has the GCM output as input, and forcings as output. And there is no mystery about the mechanism. It’s a simple calc which he runs backward. Nothing to do with the actual operation of a GCM.
You haven’t shown that the inputs are unreasonable. Where they come from is irrelevant.
Not irrelevant. Willis says
“This is where you have both the input and the output of some unknown process, and the challenge is to figure out what is happening functionally inside the black box.”
If your black box has an input that is not the input of the process (GCM), that is a fundamental defect.
Never mind the bollocks Nick, let’s cut to the chase and explain why Nobel Prize “winners” such as Michael Mann and Al Gore’s predictions haven’t stood up to observed realities.
Or why haven’t any of the ClimateGate Emails crew answered Jimmy Hansen’s question to them all –
“what if we’re wrong? . . . “
I think you are trying to change the subject…
My point still stands — there is a simple lagged linear relationship between the forcings and the temperature. Whether the temperature is calculated from the forcings or the forcings are calculated from the temperature is immaterial.
This means that the model is ignoring everything but the forcings, which is assuredly not how the world works.
w.
You still haven’t shown what is wrong with the inputs.
Here is the equation.
T(n+1) = T(n)+λ ∆F(n+1) * (1- exp( -1 / τ )) + ΔT(n) exp( -1 / τ )
Are the inputs of temperature, forcing, and tau incorrect? Are they reasonable? If you can use a linear equation to match the output of a GCM closely while using inputs that are reasonable, then the question remains.
Nick, you say:
“The forcing is derived, by simple linear calculation, from the GCM output. ”
This is not true. From the Miller paper linked to above:
“We characterize perturbations to the pre-industrial climate using the effective radiative forcing (ERF), defined as the difference in TOA net radiation between two AMIP-style simulations with pre-industrial SST and sea ice, where one simulation contains the forcing agent: for example, an increase in greenhouse gas (GHG) concentration (Hansen et al., 2005). The difference is constructed from 30-year averages after a 1-year adjustment period, following the RFMIP-ERF protocol (Forster et al., 2016; Pincus et al., 2016; Smith et al., 2020).”
There’s nothing in there about a “simple linear calculation” … and despite that, the forcings can be emulated by a simple linear calculation using only the temperature.
w.
Willis,
Miller sets out exactly how they are derived:
“the difference in TOA net radiation between two AMIP-style simulations with pre-industrial SST and sea ice”
Firstly, they are from the output of GCM calculations – that is explicit. You can’t use them as input. And difference is a simple linear calculation.
But the main fact is that if you set up a black box with an input that is not the GCM input, your BB is not telling you about GCMs. maybe some other calculation process.
Of course it is telling you something about the GCM’s. Even if the inputs are different from the GCM input, when you get the same output, it tells you that the complexity and cost of the GCM is not worth the effort. You need to deal with the inputs being incorrect in order to dismiss the work.
Thanks, Nick. You are correct that the forcings are “the difference in TOA net radiation between two AMIP-style simulations with pre-industrial SST and sea ice”.
However, that is based on the input concentrations of CO2, aerosols, and the like. And all that calculation is doing is converting the inputs from units of concentration to units of forcings.
That calculation is done in separate experiments by each model, not by the model runs shown above. The calculation tells us the change in forcing that a particular model gets from the changes in concentrations.
Given that the GCM is driven by concentrations/emissions, and the modelers have separately diagnosed the forcings corresponding to those concentrations/emissions from fixed‑SST experiments, I can treat those forcings as effective inputs to a black‑box emulator of the GCM’s global‑mean temperature response.
They’re a different‑unit representation of the same underlying input perturbations, for the purposes of my emulation, not for the GCM’s internal integration.
My point remains. The fact that those forcings alone are sufficient to calculate the output temperature means that the model is basically ignoring modulation of the North Atlantic Oscillation; changes in the frequency and strength of the La Nina Pump; variations in the Gulf Stream; alterations in cloud type, cover, and time of emergence (except those due to aerosols); differences in where the forcing is occurring on the planetary surface; increasing or decreasing water vapor; variations in cloud albedo; the oft-predicted weakening of the Atlantic Meridional Overturning Circulation; and all other climate phenomena.
Short version: My black‑box emulation shows that, for global‑mean temperature, the GISS model behaves as if forcing were the only driver. Forcing is not literally the model’s input, but it is a compressed representation of its physical inputs that suffices by itself to reproduce its global‑mean temperature response to an extremely high degree of accuracy.
Best regards,
w.
Willis,
“My black‑box emulation shows that, for global‑mean temperature, the GISS model behaves as if forcing were the only driver.”
You said of black box analysis that
“The only requirement is that it takes the same input and produces the same output.”
and yet that fails here. The input is not the same as for the GCM. So what is the BB emulating? A notional process in which forcing is the input and T the output. But therer isn’t such a process. There is a process in reverse, where T is the input and F the output. It is how F was deduced. The BB models it in reverse.
But this is just a closed calculation at the end. It tells you nothing about GCM operation. It tells you about how you deduced F from T, after the GCM had done its work. Your key formula is very similar to
ΔTₙ = λΔFₙ
They could very well have calculated F from the GCM results using
ΔFₙ =(1/ λ)ΔTₙ (as they did, almost, here)
If they had, then your “emulation” would be almost exact. But it is trivial. It is just telling you about how F is defined and derived from end results – not about how the GCM calculated T.
“The input is not the same as for the GCM.”
Converting miles/hour to meters/sec does nothing but change the units. the inputs remain the same – velocity. Willis tried to explain to you how changing the units through conversion, be it by constant relationships or equation calculations, doesn’t change the forcings into being different forcings.
You are arguing that 60mph ≠ 1mile/minute.
‘The forcings are derived from the output, or at least some advanced part of the modeling.’
Nick, are there any details on this you wish to share, or is this like the old cartoon where the guy in the white coat points at the spot on the blackboard where it says ‘a miracle occurs’?
Also, I seem to recall that the various CMIPs use various canned scenarios (RCPs) designated by the amount of ‘forcing’, e.g., RCP 8.5 (RIP). Did they know the amount of ‘forcing’ before they ran the models?
“Did they know the amount of ‘forcing’ before they ran the models?”
The designations are just labels, and come from knowledge of previous runs. It is the content of the RCPs that is the GCM input, and that is not the forcing, but gas concentrations, or at last emissions, etc. That is why they are called Representative Concentration Pathways. Here’s Wiki:
“These pathways (or trajectories) describe future greenhouse gas concentrations (not emissions) and have been formally adopted by the IPCC. The pathways describe different climate change scenarios, all of which were considered possible depending on the amount of greenhouse gases (GHG) emitted in the years to come. The four RCPs – originally RCP2.6, RCP4.5, RCP6, and RCP8.5 – are labelled after the expected changes in radiative forcing values from the year 1750[1][2] to the year 2100″
‘The designations are just labels, and come from knowledge of previous runs.’
Oh. So what’s the rationale for re-running the models? Repeatability? Making sure the oceans don’t boil off…?
“RCPs”???
In CMIP1 through CMIP3, the actual forcings were specified as inputs. This gradually transitioned to CMIP5, where “RCP”s, or “Representative Concentration Pathways” were the inputs.
Contrary to your comment, RCPs are no longer used. In CMIP6, the inputs are now called Shared Socioeconomic Pathways (SSPs), expressed as either emissions or concentrations.
w.
Willis,
“the actual forcings were specified as inputs”
That just isn’t true, and can’t be true. First, without a GCM output you have no numbers for forcing. And second, they would be a useless input for a PDE solver. Those operate locally, cell by cell. They can’t handle global averages like forcing. They can handle gas concentrations, since these are assumed uniform in space, or at least with only small corrections. And they can handle TSI etc, with known geometry.
The CMIP3 experiments are here. All in terms of gas inputs. Forcings are listed as output, table A5.
For Experiment 8: “Hold CO2 fixed after reaching doubled concentration.”
This is a cell-by-cell value?
“And second, they would be a useless input for a PDE solver. Those operate locally, cell by cell. They can’t handle global averages like forcing.”
Really? The assumption that CO2 is well-mixed is not part of the CGM’s? Is CO2 concentration parameterized cell-by-cell? Region by region? Globally?
The assumption that cloud cover can be parameterized is not a part of the CGM’s? Is cloud cover parameterized cell-by-cell? Region by region? Globally?
What other pieces in the CGM’s are parameterized cell-by-cell?
Thanks, Nick. However, you have just proven my point—since the forcings are simply a rescaled, de-lagged version of the temperature output, then the temperature output is a rescaled, lagged version of the forcings.
Which is what I said … the point is the simple relationship between the two, not the direction of the calculation.
GISS‑E2.1’s diagnosed forcings and its global temperature can be related by a simple linear, lagged transformation; therefore, for global‑mean behavior, the model is effectively emulable by a low‑order black‑box driven by the forcings.
w.
Willis,
“ the point is the simple relationship between the two, not the direction of the calculation.”
No, the point is whether your black box actually represents the GCM. And it doesn’t, because the forcings are not an input to the GCM’s.
What your black box emulates is a trivial calculation at the end, whereby the forcings are deduced from the GCM output. Your formula shows that you can reverse that to get the output surface temperatures from the forcings. But it does not cover what GCMs actually do.
Kamala Harris would giggle.
First error: gas is not a forcing.
Second error: incident solar EM energy is the only forcing that is external to the planet.
This error: The black box model uses the Incident solar EM energy as the forcing input.
“When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast”
White Queen to Alice
I remember that original post and discussion.
My thought at the time was to relate your equation to Newton’s F=MA.
We now know it wasn’t “precise”, but it is darn good enough for down here on planet earth.
(Bohrs model is similar.)
Single digit forcings are lost in the noise when compared to elliptical orbit, tilt and albedo.
Plus all these ubiquitous GHE balance numbers are just pulled out of some PhD’s heavily papered butt.
Hello Willis:
From your emulation equation, it looks like your time steps are yearly.
How was tau determined? Was that the value given by climate models you replicated?
And finally, the scaling factor Lamda? You give it as .504. Is that a “climate sensitivity” derived by something? In order to make that work against a forcing in Wm-2, I’m surmising it is from the derivative of the Stefan Boltzman equation with a certain emissivity calculated by factors not disclosed at the mean earth temperature.
I’ve always said, “climate models” in their present form are nothing but overrated heaps of junk. The modelers have always had an attitude that they could pull a temperature trick out of the rabbit hat by treating the output as a mathematical BVP where weather runs random in the system. Knowing that, they always knew they could not get absolute humidity and cloud fractions correct in space time coordinates, but because the earth is a sphere, somehow, they claim the radiation exchange from TOA to surface over all wavelengths will somehow balance out and you could get a correct answer. Sheer nonsense.
Thanks
Tau and lambda were iteratively fitted to give the best answer.
Lambda is a climate sensitivity of some sort. It works out to ~1.8°C/doubling of CO2 or so. Is it transient climate response or equilibrium sensitivity? Given the short nature of tau, only about six years, I’d say its tending towards equilibrium sensitivity.
But that all assumes that changes in temperature are a linear function of the forcing, which I don’t think is true.
w.
“ but because the earth is a sphere, somehow, they claim the radiation exchange from TOA to surface over all wavelengths will somehow balance out and you could get a correct answer. Sheer nonsense.”
Since the temperatures of the bodies are different and joules-in/joules-out happen over different cyclical time intervals the radiation exchanges will never balance. Joules-in vs joules-out is the only balance that applies.
Forcing = ASR-OLR net+=warmer
Brief search of forcing seems to set upwelling temp at surface not top of troposphere.
Copernicus site.
So what is it?
The ubiquitous graphics are surface not ToA.
Again, you used the Trans-Reality Alarmist lexicon definition of forcing, which is bogus.
Why do they omit water from the models, are they saying it has no effect?
They don’t.
“are they saying it has no effect?”
No. The issue is, they do not know how to quantify and average water.
The climate models run heavily on averages.
Average reflectivity for a 25km^2 or 100 km^2 grid?
Average solar incident EM energy when there are trees and grass and water and asphalt all responding differently due to their own specific material properties.
There seems to be a notion that Earth exists in some kind of balanced thermal equilibrium which mankind’s miniscule CO2 disrupts, i.e. “forces” out of balance to the detriment of all life.
That notion is an article of faith not science.
The elliptical orbit, tilted axis and albedo have Earth in a constant ebb & flow of imbalance.