Part III: How the feedback factor f was exaggerated
By Christopher Monckton of Brenchley
In this series (Part 1 and Part 2) I am exploring the cumulative errors, large and small, through which the climatological establishment has succeeded in greatly exaggerating climate sensitivity. Since the series concerns itself chiefly with equilibrium sensitivity, time-dependencies, including those arising from non-linear feedbacks, are irrelevant.
So far, it has been established that the models’ failure to determine the central estimate of equilibrium or final climate sensitivity ΔT from their central estimate of the unitless feedback factor f (see Part I of this series) combined with their erroneous official mixing of surface temperature and emission-altitude flux in the Stefan-Boltzmann equation to generate an excessive value for the climate-sensitivity parameter λ0 (see Part II) had led to a 40% exaggeration of the central estimates of the reference pre-feedback sensitivity ΔT0 and hence of final sensitivity ΔT in the CMIP5 ensemble of general-circulation climate models.
Part III will consider a further effect of the official exaggeration of λ0 on climate sensitivity –the overstatement of the temperature feedback factor f.
The official equation (1) of climate sensitivity as it now stands, which was well calibrated against the outputs of both the CMIP3 and CMIP5 model ensembles in Part I, is –
(1) 
where 
Fig. 1 Illumination of the official climate-sensitivity equation (1)
Fig. 1 illuminates the interrelation between the various terms in (1). We shall now determine equilibrium sensitivity stepwise, making corrections for the errors identified in Parts I and II along the way and, this time, also correcting the value of the feedback factor f.
The net incoming flux density F0, at the emission altitude about 5 km above ground level depends solely on the total solar irradiance S0 = 1361 W m–2 and on the mean albedo or reflectance α = 0.3, thus: F0 = S0(1 – α) / 4 = 238.175 W m–2. From the fundamental equation of radiative transfer, assuming emission-altitude emissivity ε0 = 1 and the Stefan-Boltzmann constant σ = 5.67 x 10–8 W m–2 K–4, emission temperature T0 = [F0 / (ε0 σ)]1/4 = 254.578 K.
Add the CO2 radiative forcing ΔF0 = 5.35 ln(2) = 3.708 W m–2 to obtain the pre-feedback or reference flux density Fμ = 241.883 W m–2, from which the Stefan-Boltzmann equation gives Tμ = 255.563 K, so that reference sensitivity ΔT0 = Tμ – T0 = 0.985 K.
With these preliminaries, we begin the consideration of temperature feedbacks, which are additional forcings ci, summing to c = Σi ci, expressed in Watts per square meter per Kelvin of the reference warming ΔT0 that triggered them. This time, we shall concentrate only on the central estimate of climate sensitivity. In the next article, we shall examine the upper and lower bounds, for the hitherto poorly-constrained breadth of the climate-sensitivity interval arises chiefly from variations in temperature feedbacks between models.
IPCC’s interval of climate sensitivities in AR5 is [1.5, 4.5] K, just as it was in the Charney report for the National Academy of Sciences in 1979. Where λ0 is the official reference-sensitivity parameter 3.2–1 K W–1 m2, the ratios G of these bounds to IPCC’s estimate ΔT0 = λ0ΔF0 = 1.159 K of reference sensitivity fall on [1.294, 3.883], implying [0.227, 0.742] as the bounds of the interval of feedback factors f = 1 – 1 / G. The central estimate of f is here taken simply as (0.227 + 0.742) / 2 = 0.485, implying a feedback sum c = f / λ0 = 1.550 W m–2 K–1.
At this stage we are not going to challenge IPCC’s implicit central estimate of the feedback sum. If we were to retain IPCC’s concept and estimate of λ0, and consequently its estimate of reference sensitivity ΔT0, then the central estimate of equilibrium sensitivity based on the feedback sum c = 1.550 W m–2 K–1 would be 2.2 K, as Fig. (1) shows.
Fig. 1 shows a stable, an unstable and a climate-unphysical region. The stable region, where the feedback factor is either negative or at most 0.1 (and preferably little more than 0.01), reflects the fact that process engineers designing electronic circuits designed to perform stably even where the reliability of componentry and the stability of ambient operating conditions cannot be guaranteed often use a rule-of-thumb maximum design value for feedbacks, since any value above the maximum may lead to unwanted instability.
Why is the operation of feedbacks in electronic circuits of interest when looking at the climate? The answer is that the mathematics of feedback amplification was originally developed for electronic circuits, typically amplifiers, and that the two papers that between them established the present mathematical approach to feedbacks in the climate – Hansen (1984) and Schlesinger (1985) – refer back specifically to the treatment of feedbacks in electronic circuits as the origin of and justification for the method they proposed.
Fig. 1 The rectangular-hyperbolic curve of equilibrium climate sensitivity ΔT in response to the feedback factor f = λ0Σici, based on the official method of determining climate sensitivity, showing that implicit official final sensitivity in response to the central estimate f = 0.485 is 2.2 K.
Now, the mere fact that process engineers often try to impose an upper bound on feedback where it might lead to instability does not prove that climate feedbacks in the region shown in Fig. 1 as unstable are impossible. However, it suggests that they are unlikely; and, in the next article, we shall demonstrate that, in the climate, feedbacks do not occur in that region, and that they only appear to do so owing to a substantial error in climate feedback analysis.
For now, we shall take IPCC’s implicit central estimate of the feedback sum c = 1.550 W m–2 K–1 and use it as the basis for determining the central estimate of climate sensitivity, but without using the defective official quantity λ0.
Instead, we shall redetermine the unitless feedback factor f as the product of c and the first derivative of the Stefan-Boltzmann equation at the emission altitude after taking into account the pre-feedback increase in radiative flux density at that altitude, thus:
(2) 
1.550 
0.409.
From this value, the final gain factor G = 1 / (1 – f ) = 1.693. The product of G and ΔF0 gives the final flux change ΔF, so that the final flux density F = F0 + ΔF = 244.454 W m–2, whereupon the final temperature T is 256.239 K, and the final sensitivity ΔT = T – T0 < 1.7 K.
Charney (1979) gave the central estimate of ΔT as 3.0 K. The CMIP5 models’ value is 3.2 K, which is a 92.5% exaggeration compared with the value 1.661 K found here. As we shall see later in the series, even this corrected central estimate is substantially too high.
Table 1 summarizes the calculations in this article.
| Determination of the central estimate of final climate sensitivity | |||
| Variable | Derivation | Value | Units |
| 2 x CO2 forcing ΔF0 | 5.35 ln (2) | 3.708 | W m–2 |
| Emission flux density F0 | S0 (1 – α) / 4 | 238.175 | W m–2 |
| Amplified flux density Fμ | F0 + ΔF0 | 241.883 | W m–2 |
| Amplified temperature Tμ | (Fμ / σ)1/4 | 255.563 | K |
| Emission temperature T0 | (F0 / σ)1/4 | 254.578 | K |
| Reference sensitivity ΔT0 | Tμ – T0 | 0.985 | K |
| Official feedback factor foff | (0.227 + 0.742) / 2 | 0.485 | Unitless |
| Implicit feedback sum c | foff / λ0 | λ0 = 3.2–1 | 1.550 | W m–2 K–1 |
| Corrected feedback factor f | c Tμ / (4Fμ) | 0.409 | Unitless |
| Final gain factor G | (1 – f )–1 | 1.693 | Unitless |
| Final flux change ΔF | G ΔF0 | 6.279 | W m–2 |
| Final flux density F | F0 + ΔF | 244.454 | W m–2 |
| Final temperature T | (F / σ)1/4 | 256.239 | K |
| Final sensitivity ΔT | T – T0 | 1.661 | K |
Ø Next: How the breadth of the climate-sensitivity interval was exaggerated.
References
Charney J (1979) Carbon Dioxide and Climate: A Scientific Assessment: Report of an Ad-Hoc Study Group on Carbon Dioxide and Climate, Climate Research Board, Assembly of Mathematical and Physical Sciences, National Research Council, Nat. Acad. Sci., Washington DC, July, pp. 22
Hansen J, Lacis A, Rind D, Russell G, Stone P, Fung I, Ruedy R, Lerner J (1984) Climate sensitivity: analysis of feedback mechanisms. Meteorol. Monographs 29:130–163
IPCC (1990-2013) Assessment Reports AR1-5 are available from www.ipcc.ch
Schlesinger ME (1985) Quantitative analysis of feedbacks in climate models simulations of CO2-induced warming. In: Physically-Based Modelling and Simulation of Climate and Climatic Change – Part II (Schlesinger ME, ed.), Kluwer Acad. Pubrs. Dordrecht, Netherlands, 1988, 653-735.
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Christopher Monckton of Brenchley, thank you again.
A few relevant quotes from Bode:
The first sentence in chapter 1 of Bode’s book states,
“The networks to be considered consist of ordinary lumped
inductances, resistances and capacities, together with vacuum tubes.”
It goes on to say,
“For purposes of discussion the tubes will be replaced by equivalent
structures consisting of ordinary circuit elements connected between
the accessible terminals, together with a source of current or voltage
to represent the amplification of the tube.”
The first sentence of the second paragraph states,
“It will be assumed throughout that all the elements are linear.”
On page 108, it states as an assumption,
“1. A passive circuit is always stable”
It goes on to say
“[because] … the fact that a passive system cannot contain a source of power”
Clearly, Hansen and Schlesinger were unaware of these conditions.
Bottom line the climate is absolutely stable as long it is linear. The IPCC is throwing in positive feedback to make it look like we are closer to a tipping point where something breaks over.
“Bottom line the climate is absolutely stable as long it is linear”
The climate is absolutely stable as long as it is passive. Other than that, you are right that the IPCC is using otherwise scary positive feedback to make the climate seem far more fragile than it really is.
“Clearly, Hansen and Schlesinger were unaware of these conditions.”
Can you quote what Hansen and Schlesinger said that indicates that?
“Can you quote what Hansen and Schlesinger said that indicates that?”
What makes you think that their framing of the climate system into Bode even remotely correct in light of the obvious ignorance about Bode’s prerequisites for the systems his analysis applies to? Adequate peer review was never applied to the mapping Hansen established, otherwise, it wouldn’t be as broken as it is.
They omitted recognizing that Bode assumes linearity, they omitted recognizing the assumption of unit open loop gain, they ignorantly assume active, powered gain that can amplify 1 W/m^2 of incremental forcing into 4.3 W/m^2 of surface emissions.
If you want a quote, look at either of their papers and pick almost any sentence.
“What makes you think that their framing of the climate system into Bode”
You could start with just quoting where they frame the climate system into Bode. In the Schlesinger version I read, he only once mentioned Bode in passing. He said he was doing a linear sensitivity analysis.
You can look for yourself. The Hansen paper (1984), Schlesinger paper (1985), Roe (2010) and Bode (1945) are relatively easy to find with Google. All directly reference Bode as the foundation analysis and make specific correspondences between Bode’s terms and climate system attributes. If climate related feedback is not based on Bode, then don’t call it feedback and don’t try and infer runaway conditions.
More precise references are in this link: http://www.palisad.com/co2/fb1/fb1.pdf
“You can look for yourself.”
I did. What you say just isn’t true. Schlesinger makes one passing reference to Bode. None “reference Bode as the foundation analysis”. Roe notes that
They managed that without the “foundation analysis”.
Your link is basically to the text of your recent post.
Nick,
If you don’t think that the consensus climate feedback model is based on Bode’s feedback system analysis, what is it based on? Is it just BS pulled out of thin air like the rest of the warmist pseudo science? If you won’t accept that it is based on Bode, then what gives you the basis to invoke feedback, much less runaway feedback where understanding the stability of feedback networks is the primary topic of Bode’s book? Bode’s book is also the primary non climate science related reference having to do with feedback that each of those papers cited. Where do you think the concept of feedback comes from?
And BTW, while climate system feedback is ostensibly based on Bode, there are so many mistakes in the mapping you simply can not make a legitimate correspondence. So on the one hand you are right that the climate feedback model doesn’t conform to Bode, although it’s certainly claimed to do so and this contradiction is why climate science is so obstinately wrong.
Lets take a step back and examine reality for a moment. If your answer to any of these questions is no, you better offer a solid explanation other than deferring to authority, otherwise, I see no point in continuing this discussion.
Do you agree that if the surface temperature increases by 3C, its radiant emissions increase by about 16.3 W/m^2 and that this increase is claimed to arise from only 3.7 W/m^2 of equivalent solar input after reflection (forcing per the IPCC definition)?
Do you agree that the planet has no other source of input power other than the stimulus arriving from the Sun, thus is a passive system according to Bode’s definition of a passive system?
Do you agree the only possible sources of the more than 12 W/m^2 of excess input required to sustain a 3C surface temperature rise is either feedback or comes from an internal energy source that is not the incoming forcing?
Do you understand that feedback is tangible energy and is the physical return of joules of emitted surface energy back to the surface?
Do you agree that COE must apply to the climate system?
Do you understand that the T^4 relationship between temperature and emissions quantified by the Stefan-Boltzmann Law is immutable, derivable from first principles quantum mechanics, rigorously tested and is actually a validated theory and not just a hypothesis?
Do you understand that the sensitivity claimed by the IPCC is only a hypothesis and is as far from a validated theory as any hypothesis can be?
Please try and answer these questions, rather than filibuster.
co2,
Sorry I missed these for a while. OK:
Do you agree that if the surface temperature increases by 3C, its radiant emissions increase by about 16.3 W/m^2 and that this increase is claimed to arise from only 3.7 W/m^2 of equivalent solar input after reflection (forcing per the IPCC definition)?
Yes
Do you agree that the planet has no other source of input power other than the stimulus arriving from the Sun, thus is a passive system according to Bode’s definition of a passive system?
No. The Sun is a power supply in the circuit sense, and an active device is one that modulates the supply. A transistor and battery has no other source of power, but the transistor is an active device. An ipad has no external source, but is active.
Do you agree the only possible sources of the more than 12 W/m^2 of excess input required to sustain a 3C surface temperature rise is either feedback or comes from an internal energy source that is not the incoming forcing?
No. All that is required is to increase the impedance on the flow of the 240 W/m2 from sun. That is the source. Feedback is a way of modifying the impedance.
Do you understand that feedback is tangible energy and is the physical return of joules of emitted surface energy back to the surface?
Feedback as conventionally defined returns energy (in finite gain system) to the input, which here is the forcing (at TOA). Temperature at surface is the output. I say finite gain, because you can’t return energy to the input of an ideal op-amp. It has infinite input impedance.
Do you agree that COE must apply to the climate system?
Yes
Do you understand that the T^4 relationship between temperature and emissions quantified by the Stefan-Boltzmann Law is immutable, derivable from first principles quantum mechanics, rigorously tested and is actually a validated theory and not just a hypothesis?
There is a great deal more to radiative transfer than S-B (which yes is accepted science). T^4 relates to emission from a surface. From a gas it’s harder; the gas has a distributed emissivity expressed as m^-1, but there is mix of emission and absorption.
Do you understand that the sensitivity claimed by the IPCC is only a hypothesis and is as far from a validated theory as any hypothesis can be?
There is surface temperature, which is affected by flux, and there is no doubt that all else being equal, a sustained change in flux will determine a change in equilibrium temp. The relation might not be linear, in which case we are describing a derivative, which describes finite changes to first order. It may be that we are not keeping all else equal over periods of time, even though we think we are. There are the normal uncertainties of science.
Nick,
“The Sun is a power supply in the circuit sense”
Absolutely not. The Sun is the stimulus and not the implicit INFINITE power supply assumed by Bode that supplies the output power, despite the fact that the Sun is the ONLY source of power entering the system. You still don’t understand that Bode’s amplifiers provide power gain where the extra power comes from the implied infinite source and that this disconnects the input and output from the requirements of COE. The climate does not exhibit power gain because it has no source of input power to draw from above and beyond the stimulus.
“All that is required is to increase the impedance”
On the one hand, you agreed that the 12 W/m^2 of extra surface emissions must be replenished or else the surface will cool, but then you fail to account for where these joules are coming from. Impedance doesn’t create joules. You are right that this does warm the surface and this is the origin of the extra 0.6 W/m^2 per W/m^2 of forcing received by the surface making it warmer than it would be if limited to 239 W/m^2 of input.
If each of the 239 W/m^2 of incident energy results in only 1.6 W/m^2 of surface emissions (0.6 more per W/m^2 of forcing), how does this jump up to 4.3 W/m^2 for the 240’th W/m^2 of input?
George
“not the implicit INFINITE power supply assumed by Bode”
If Bode only worked for infinite power supplies, it would be of no use. In fact every power supply that tries to be a voltage source has output impedance, and can only supply a finite amount of power. And while that impedance is normally ignored, there is no problem including in the circuit analysis – it makes very little difference. And a voltage source with impedance in series is equivalent to a current source with the same impedance in parallel. That’s how the Sun functions here.
“Impedance doesn’t create joules. “
Watts is more relevant. And we’re talking about a temperature (voltage) rise, which doesn’t equate to power. If you have a through current from a source, and put a resistance in the way, the voltage on the source side rises. The resistor doesn’t create joules, but it raises voltage.
Nick,
“If Bode only worked for infinite power supplies, it would be of no use.”
You have this backwards.
When the power supply runs out of joules, the system goes non linear and Bode’s analysis no longer applies.
Bode’s analysis is of an idealized system where power supply limitations and the like are not an issue.
George
“When the power supply runs out of joules, the system goes non linear and Bode’s analysis no longer applies.”
I don’t think the Sun will go non-linear. And a power supply with output impedance is not non-linear.
I don’t think Bode’s analysis is helpless here. But in any case that’s irrelevant. They aren’t doing Bode’s analysis. They are doing their own. Quite competently.
NIck,
“I don’t think Bode’s analysis is helpless here. But in any case that’s irrelevant. They aren’t doing Bode’s analysis. They are doing their own. Quite competently.”
Really? I would say quite incompetently. You agree that they’re not conforming to Bode’s analysis, which is the definitive authority on feedback systems and can’t cite what their analysis is based on other than it being something Hansen made up. What happened to the deference to authority your side is so keen on?
Why does everyone who has written a paper about climate feedback invoke Bode as the primary (and often only) non climate related reference about feedback in their attempts to justify the ‘consensus’ feedback model that presumes massive positive feedback amplifies 3.7 W/m^2 of forcing into more than 16 W/m^2 of surface emissions?
“I don’t think the Sun will go non-linear. And a power supply with output impedance is not non-linear.”
What happens when the system demands more than 240 W/m^2 from the Sun to maintain the temperature? Does it draw more from the Sun or does it start to cool? If the Sun was the power supply, it would draw more from the Sun, but of course, it can’t and will cool. If you can figure out a way to do this as you way, patent it and you will become rich as the inventor of free energy from perpetual motion.
This is a trivial concept and shouldn’t be that hard to grasp, unless understanding is blocked because of a fear that once the reality of a passive climate system is accepted, the massive amplification claimed from positive feedback becomes impossible, the skeptics have been right all along and the political consequences of this are too harsh to accept so reality is denied. Never before in our history has such a powerful political party latched on to the wrong side of science is such an obstinate manner. We are in uncharted political territory as the collapse of CAGW is inevitable and could just drag the Democratic party down with it.
“What happened to the deference to authority your side is so keen on?”
They are doing maths. No deference to authority is required. You just have to get it right, as they do.
“amplifies 3.7 W/m^2 of forcing into more than 16 W/m^2 of surface emissions”
There’s something you could work on. There is no real difficulty about analysing transimpedance amplifiers. But you can also reconsider what you regard as input or output. Now you’re framing it as a current amplifier, where the gain is dimensionless. That should lead to fewer mistakes.
“Does it draw more from the Sun”
The Sun is a current source, and doesn’t vary when drawn, It is detemined by TSI and albedo. It would be the same 240 W/m² if there were no GHG and the temperature of surface was 255K. The existing GHG has raised it to 288, without varying the 240. More GHG can raise it firther without any sort of supply shortage.
In fact, there is a limit on the sun’s source, discussed here. The apparent temperature of the Sun is 5700°K, but the maximum a solar furnace can get to is 2310K. This is in effect an impedance matching issue. The implied impedance of the solar source, on average, is 5700/240=23.75 K/(W/m²). With linearity, at max power transfer, the temp would be 5700/2=2650K. It is less because of the T⁴ effect. But these are extreme temperatures; the impedance is negligible for the perturbations induced by climate.
ps the comments are getting out of order. This is a response to
co2isnotevil September 10, 2016 at 4:05 pm
If you want to model the Sun as a current source, then the stimulus is a current souce, not a voltage source. This doesn’t change the fact that we are still talking about the stimulus and not the implicit power supply of Bode.
BTW, transimpedance amplifiers are built from OP amps, which are voltage gain devices. It just happens that voltage, current and impedance are linearly related to each other through Ohms Law, moreover; op amps have implicit power supplies and provide power gain.
And they are not even doing math, but bungling arithmetic by considering the open loop gain has 2 different values, depending on where it appears in the equations.
Apologies, the solar furnace max is 2310°C, not K.
It seems that Lord Monckton and Clarence Darrow have a few things in common— they both took on “religious” zealots in a courtroom presided by deaf and blind justices who have no intention to be swayed by inconvenient facts or alternative hypotheses. “The Science is Settled”, and the trolls have been unleashed… Good luck, Lord M!
“in a courtroom presided by deaf and blind justices who have no intention to be swayed by inconvenient facts or alternative hypotheses”
Is this WUWT?
Who here is a troll and who is not: that’s the question indeed. Maybe you could review the thread and its preceedings, comment by comment.
You then might discover that nearly none of these “alternative hypotheses” happened to be defended to the end against counterarguments going somewhat farer in the depth.
Your lordship is a great communicator. I followed your advice by contacting Thomas F. Stocker who works for the IPCC. Before that he worked on the EPICA ice cores along with Luethi et al. Thomas knows the truth but prefers to lie so I call him “The Prince of Darkness”:
https://diggingintheclay.wordpress.com/2013/05/04/the-dog-that-did-not-bark/
I am a little exasperated with several on this thread expressing opinions about feedbacks and applying Bode to try to prove this or that, and understand instability.
First things: We are talking about LINEAR systems that are realizable in a circuit with real components or in a physical system found in nature or on a plant floor, or robot for example, a subject I have some experience with.
Instability is defined as a system running away to infinity, either by ramping positive or negative or by an increasing sine wave. Real systems with no feedback will not do that but can come close. Practical systems with feedback can be unstable only until something saturates. The simplest example is a stream of water flowing into a bathtub. This is integrating to infinity until it overflows then it is a different system with a feedback representing the water going over the side in proportion to the level. In electronics see: oscillators.
Instability in this context is NOT small changes in component values that change over time. That is a completely different sort of engineering problem that mostly gets solved by avoiding differencing large values. Its not changes in in parameters like cloud cover that are a function of temperature or time. If you want that answer put it in the system equation.
A linear system is just that, linear. If its stable its output will vary linearly in proportion to its input. Changes to feedback parameters do not respond linearly. Again, running into a limit, clipping or distortion, is a change to the system and the stability analysis must be done around the new operating conditions. All I ask is that inputs not be conflated with parameters, this means you LCM.
Now Bode starts off in a discussion in chapter 2 that EEs start seeing in their sophomore year. To do a circuit analysis a resistor is represented by its value, an inductor is represented by SL, and a capacitor by 1/SC. By writing the mesh equations (EE stuff) you can get the transfer function of a network or physical system in terms of the ratio of two polynomials in S. If you started out doing differential equations you can get a differential equation with the same coefficients. Now the EEs get tricky here, math majors and scientists know this but the EEs are good at it. The inverse Laplace transform will get you the transient response of the transfer function. By substituting iWt for S you get something close to a Fourier transform that will get the magnitude and phase of the system at any given frequency, W. It may seem a bit of magic to non-EEs but it has a solid foundation in differential calculus.
Bode and Nyqvist realized that the frequency response can be expressed as a complex value and that by plotting the roots with the real part on the x axis and the imaginary on the Y axis you can learn a lot about the system. Roots of the denominator are called poles and roots of the numerator are called zeros. Complex poles and zeros come in conjugate pairs.
Real systems will have none of their poles to the to the right of the Y axis.
Bode goes on in Chapter 3 to write the feedback equation 𝜇/(1-𝜇𝜝). I expressed it earlier as G/(1+GH). The sign in the denominator is the sign of the summing node. Negative feedback decreases the output and positive feedback increases the output. Positive feedback is not a certain recipe for unstable operation, just an enhanced output which is the subject of the original post. EEs are not accustomed to going there, the benefits from feedback usually come from large negative values. This was the source of my initial confusion about Lord Monckton’s point.
The denominator of the system (feedback) equation is all that needs to be examined to determine stability.
If a pole exists to the right of the Y axis or a real pole is sitting on zero, it unstable. Biggest problem is solving for the poles in some systems.
A Bode plot does it slightly differently. Bode plots both the magnitude and phase of the open loop transfer function, 𝜇𝜝. Stability is when the magnitude of the forward transfer function falls below 1 as the frequency increases before the phase reaches 180 degrees. This is the engineering way to get answers to the intractable in that you don’t need to find the roots. If you do know the roots thats better because other shortcuts open up.
Pure delays in the system are particularly nasty because there’s phase shift in proportion to frequency with no decrease in magnitude. Pure delays are possibly responsible for the decadal oscillations seen in the climate, not really needing much gain.
Now, the simple linear climate feedback equation that was presented has no imaginary values. It has to have its one root>=0 to be unstable. You could put in heat storage and get transient responses and stability if so inclined and skilled. Integrators are 1/S and differentiators are just S.
As to whether the climate system has really represents feedback or whether gain violates COE, I will leave that to the scientists.
ristvan: Your squealing PA system, has poles all over the place and delays in the feedback. If the loop gain drives at least one pair of the poles to the right of the Y axis it squeals. Look up Nyqvist. An equalizer can sometimes cancel troublesome poles.
” By substituting iWt for S”
Should this be iω?
“Now, the simple linear climate feedback equation that was presented has no imaginary values.”
This relates to a point that I have been trying repeatedly to make here to restrain EE enthusiasm. All of Lord M’s posts just relate equilibrium states. DC analysis. There is no energy storage, no reactance, nothing to determine a frequency. So Bode plots etc reduce to a single point, or at least only one that we know about.
I think I now get why the EEs are running off in all directions. Lord Monckton has conflated the input with the feedback factor, f. So when we say linear he’s confused. This is understandable since, for him or rather the climatologists who have promulgated this mess, f IS the input. The radiative forcing is for them a constant. For us the response is linear to radiative forcing.
Now to essentially prove none of this make sense. The only place the simulated system is unstable is when f=1, f can be arbitrarily close to 1. In such a case, as in fig 1, the output, T, does go to infinity as f approaches 1. We know this can’t be. Either this is an invalid analysis or f isn’t an independent variable. If f depends on something else then start over.
Frankly I think a bad model is good only for thought experiments.
David,
Yes, you are correct. I’ve been struggling trying to get Monckton to understand the ramifications of Bode and his understanding has certainly improved in the last month or so and I suspect his next posting will show even more understanding. Nonetheless, he hasn’t gotten past the broken idea that feedback amplifies sensitivity, while the feedback model is explicitly amplifying input forcing to produce an output temperature and the gain of this model is the sensitivity.
To be fair, this is not his error, but can be traced all the way back to Hansen’s 1984 paper and is still present today in all the climate feedback related literature and even most skeptics fail to see this error. I believe that even Lindzen thinks this to be the case as well since I believe that Roe was a student of his who has it wrong in his paper and I think this is why Monckton is having a hard time letting go of this broken concept.
As I have said before, nobody on either side of the science has a solid grip on feedback analysis.
‘As I have said before, nobody on either side of the science has a solid grip on feedback analysis.”
And that is the only reason I tried to jump in. I don’t comment here much and only when I think I can contribute positively. Most of this is just over my head, no pun. I have been confused since part 1 and tried to write down the model in some way and just couldn’t follow it.
“This is understandable since, for him or rather the climatologists who have promulgated this mess, f IS the input.”
No, I think that is uniquely Monckton (as here). One thing about this whole confused discussion is that people talk about what climatologists do or say, but never quote or reference, and it is often way off. I don’t know of any climatologists who share that confusion.
‘It ought to be entirely plain from the form of the final transmission characteristic in the Bode equation that the output of an amplifier circuit in the presence of positive feedback is not linear but rectangular-hyperbolic.’
That’s the comment that convinced me that serious confusion/conflation was afoot.
David,
Yes. He was actually referring to the fact that the relationship between feedback and the output is rectangular-hyperbolic because he incorrectly believes that the input to the model is the feedback coefficient and not the stimulus.
“Take a close look at lambda 0, it’s delta Ts/delta F0. Sticking to the transistor analogy the delta anything wouldn’t be what you would use to calculate the operating point.”
lambda is not used to find the operating point. The operating point is found by solving S-B with the input flux density which equals the solar flux density after albedo reflection + the 2xCO2 forcing (3.7 W/m^2). Evaluate dT/dF at that point to get lambda-zero. dT/dF*DeltaF gives Delta T (no feedback). With feedback, Delta T is scaled by (1- f)^-1 with f= c*lambda-zero (unitless).
“Instability is defined as a system running away to infinity, either by ramping positive or negative or by an increasing sine wave.”
The term thermal runaway to which is what I think you refer is a misnomer. The singularity exists only in the linearized model. As Lambda*f approaches unity in the small signal model, the output grows to the point where the small signal assumption of the model no longer is valid. What happens is as the output temperature grows, the small signal gain (derivative at the operating temperature) decreases as we move up the S-B solution curve whose slope decreases with T. The slope decreases faster than lambda*f approaches unity such that the singularity is never hit. Another way of saying this is that positive feedback can never overcome the negative feedback provided by S-B which will always find a new stable equilibrium. But just because the process can’t “run away”, doesn’t mean that the new equilibrium will be to our liking.
Not that I think we’re in any danger of that occurring given the best evidence points to a lower climate sensitivity that the models predict. This happy fact gives us time for a smooth and orderly transition to alternative energy sources. I’m much more worried about the other non-linear coupled system melting down, namely the global economy.
Nothing about this makes sense! We are all just talking around the main problem here.
It just looks like the feedback gain equation but it just does not make sense. It gets worse as you look at it. We are arguing about how much temperature changes as a function of radiative forcing from CO2. Take a close look at lambda 0, it’s delta Ts/delta F0. Sticking to the transistor analogy the delta anything wouldn’t be what you would use to calculate the operating point.
Look long enough and likely we will find a hidden divide by zero.
David,
It’s not exactly a divide by zero, but is a lambda0 divided by lambda0 cancelling it out of the feedback loop.
David, it actually is coherent. It’s just not formulated in the way EE are use to because we’re used to thinking in unitless gain blocks while they think in terms of the output’s of those blocks.
In the equation, first set c to zero. Next divide through by Delta F. This give Delta T/Delta F = lambda-zero. So lambda-zero is the zero-feedback “gain” but pay careful attention to it’s units which are K/(W/m^2). When you multiply by the input delta F, the denominator units cancel and you get delta T in K. delta T is the change in temperature for a doubling of CO2 if there was no feedback. The increase in flux, delta F, from doubling C02 is 3.7 W/m^2 (or so they say, I think this is off a bit due to saturation effects but that’s a different can of worms).
Lambda-zero is just the “small-signal gain”, i.e. slope of the line tangent to the S-B curve at the equilibrium temperature, hence my transistor bias operating point analogy. S-B has: F = sigma T^4 so T= (F/sigma)^1/4 so dT/dF = 1/(4*F^3/4 * sigma^1/4). We evaluate this at the equilibrium flux F= 238.175 + 3.7 W/m^2 to get lambda-zero = .264. To get the equilibrium sensitivity (another term which appears unorthodox to EEs because we think of sensitivities as a type of gain, not an output but oh well) we multiply lambda-zero (.264 K *m^2/W) by delta F(3.7 W/m^2) which gives delta T = .988 K.
Now we connect the feedback. Note that f = lafmdba-zero*c is unitless. Most EEs would call this value the open loop gain (normally G*H). Evidently climateers lump the whole denominator together and call it the gain factor. But in any case the closed-loop gain is just the familiar G/(1- G H) with G= lambda-zero and H = c. Since (1-GH) is unitless, the units of the CLG is the units of lambda-zero (K m^2/W). The post-feedback sensitivity (again an output, not a gain) is delta F-zero * G/(1-G H).
David,
“It just looks like the feedback gain equation but it just does not make sense.”
I think it does make sense, and I have tried to explain why here and here.Firstly, the operating point is not found from the equation; it is just the state of the atmosphere at a reference point in time. ΔT_S and ΔF are the increments. What it looks like, most simply, is Ohm’s law. If you write:
ΔT_S (1/λ₀-c) = ΔF
and think of negative feedback c, ΔT_S~V and ΔF~I, then (1/λ₀-c) is just a sum of conductances. Or you could put that conductance as the feedback of an inverting op-amp amplifier, and it would express output V from input I.
But sum of the c’s may be positive feedback, and so appear as negative conductances. So add another stage of unit gain inversion, and use those conductances to feed back from that output. Then you have this implementation (borrowed from Bernie Hutchins):
R1 is the parallel combination of negative c’s and R2 is the combination of positives.
But in the second link, I’ve tried to explain why a better way of seeing it is just as the expression of chain rule differentiation.
Incidentally, the transimpedance amplifier has caused much angst. You could just consider the variables to be ΔT_S and (λ₀ ΔF), which have the same units (kelvin). The latter can be interpreted as the temperature produced by ΔF under reference conditions (just a rescaling). Then its is just a unit gain voltage amplifier.
“If a pole exists to the right of the Y axis..”
There are no poles or zeros or delays here. The equations assume instantaneous response so w=0. Think of biasing a transistor. We’re setting the DC operating point of the IV curve. The small signal gain is then the derivative of the IV curve at that op point. Same thing here. The S-B solution determines the operating point (equilibrium temperature) for a given fixed flux density and the derivative of the S-B curve at that point then determines that gain that the loop sees when feedback is applied.
Excuse me if I’m wrong on this, but isn’t it a huge mistake to even attempt to model Earth’s radiative balance and related “forcings” based on the use of the Stephan-Boltzman equation (ref. second paragraph in the article)?
Obviously, the Earth cannot be represented as a blackbody radiator (average emissivity = 1.0). The Earth is not even close to being a gray body radiator (average emissivity < 1 and constant).
Surely, a realistic approximation of the equivalent of an emissivity factor applied to Earth must account for at least the following:
— dependence on radiation wavelength (e.g., spatial and temporal variations in reflection and absorption bands),
— dependence on season (e.g., dependence on presence/absence of both vegetation/leaf and ice/snow areal coverages),
— dependence on view angle relative to Earth coordinates (e.g., polar field-of-view will be significantly different that equatorial field of view; actual albedo is a function of angle-of-incidence).
Independent of the problems with deriving an approximate non-temporal value for Earth's "average" emissivity, is there any basis for assuming a T^4 dependence, given that both the geosphere and biosphere are highly responsive to (and thus moderate) even slight changes in Earth's "average" temperature.
I realize that the various factors discussed in this article may cover or smooth over some of the above (e.g., factors applied to some of the individual forcings) but I still have concern that the basis of the fundamental mathematical equations presented in the article are inherently wrong, despite being convenient to use in modeling.
The basic Stephan-Boltzman equation (with applied fudge factors) may be the best physics-based approximation that we have, but is this a case of garbage in = garbage out?
“Excuse me if I’m wrong on this, but isn’t it a huge mistake to even attempt to model Earth’s radiative balance and related “forcings” based on the use of the Stephan-Boltzman equation (ref. second paragraph in the article)?”
You’re right.
Monckton is showing some simple introductory level equations that typically appear in first year textbooks. Generally they’re used to illustrate physical principles or in a few cases for some very simple approximations to estimate global response.
There are more grown-up calculations that look nothing like this but Monckton doesn’t seem to understand them. Read, for example, the methods section here:
https://www.gfdl.noaa.gov/bibliography/related_files/bjs0601.pdf