
By Christopher Monckton of Brenchley
In the closing minutes of the final plenary of the U.N.’s Doha climate summit, when no one else had anything further to add, I spent a few seconds telling the delegates something that the bad scientists and the malicious media have done their level best to conceal. There has been no global warming for 16 years.
In the real world, this surely welcome news would have been greeted with cheers of relief and delight. Since the beginning of 1997, despite the wailing and gnashing of dentures among the classe politique, despite the regulations, the taxations, the carbon trades, the windmills, the interminable, earnestly flatulent U.N. conferences, the CO2 concentration that they had declared to be Public Enemy No. 1 has not stabilized. It has grown by one-twelfth.
Yet this startling growth has not produced so much as a twentieth of a Celsius degree of global warming. Any warming below the measurement uncertainty of 0.05 Cº in the global-temperature datasets is statistically indistinguishable from zero.
The much-vaunted “consensus” of the much-touted “ensembles” of the much-heralded “models” has been proven wrong. The much-feted “modelers” had written in 2008 that their much-cited “simulations” ruled out, to 95% confidence, intervals of 15 years or more without global warming. To them, 16 years without warming were as near impossible as makes no difference.
Yet those impossible years happened. However, you would never have known that surely not uninteresting piece of good news from reading the newspapers or watching ABC, BBC, CBC, NBC, et hoc genus omne. The media are not in the business of giving the facts or telling the truth any more.
Precisely because journalists no longer bother to provide the inconvenient truth to their audiences, and because they are no longer willing even to provide the people with the straightforward facts without which democracy itself cannot function, the depressingly ill-informed and scientifically-illiterate delegates in Doha can be forgiven for not having known that global warming stopped a long while back.
That is why they should have been excited and delighted when they heard the news – nearly all of them for the very first time.
But this was the alternative reality that is the corrupt, self-serving U.N. Howls, hoots and hollers of dismay and fury greeted my short, polite announcement. This absurdly inappropriate reaction raises a fascinating question.
How are we to dig a rat-hole wide enough to allow the useful idiots and true-believers to escape as each passing year makes it more and more obvious that their fatuous credo has all the plausibility of the now somewhat discredited notion that the world was to be snuffed out at this year’s winter solstice?
Every student of the arts of diplomacy in the civil-service and staff colleges of the U.K. hears much about the rat-hole problem. How does one let the other side off some hook on which they have imprudently impaled themselves, while minimizing their loss of face?
A cornered rat will fight savagely, even against overwhelming odds, because it has no alternative. Give the rat a way out and it will instinctively take it.
The first step in digging a diplomatic rat-hole is to show that one understands how one’s opponents came to make their mistake. One might make a point of agreeing with their premise – in the present instance, the long-proven fact that adding a greenhouse gas to an atmosphere such as ours can be expected, ceteris paribus, to cause some warming.
Then one tries to find justifications for their standpoint. There are five good reasons why the global warming that they – and we – might have expected has not occurred for 16 years: natural variability in general; the appreciable decline in solar activity since the Grand Maximum that peaked in 1960; the current 30-year cooling phase of the ocean oscillations, which began late in 2001 with the transition from the warming phase that had begun in 1976; the recent double-dip la Niña; and the frequency with which supra-decadal periods without warming have occurred in the instrumental record since 1850.
The next trick is to help them, sympathetically, to focus the blame for their error on as few of their number as possible. Here, the target is obvious. The models are to blame for the mess the true-believers are in.
We must help them to understand why the models got it so very wrong. This will not be easy, because nearly all of our opponents have no science or math at all.
We can start our deconstruction of the models by pointing out that – given the five good reasons why global warming might not occur for 15 years or more at a time – the modelers’ ruling out periods of 15 years or more without warming shows they have given insufficient weight to the influence of natural variability. We can poke gentle fun at their description of CO2 as “ the tuning-knob of the climate”, and help them to put things into perspective by reminding them that Man has so far altered only 1/10,000 of the atmosphere, and may alter 1/3000 of it by 2100.
We cannot altogether avoid the math. But we can put it all in plain English, and we can use logic, which is more accessible to the layman than climatological physics. Here goes.
The fundamental equation of climate sensitivity says temperature change is the product of a forcing and a climate-sensitivity parameter.
The modellers’ definition of forcing is illogical; their assumptions about the value of the climate-sensitivity parameter are not Popper-falsifiable; and their claims of reliability for their long-term predictions are empirically disproven and theoretically insupportable. Let us explain.
The IPCC defines a forcing as the net down-minus-up flux of radiation at the tropopause, holding surface temperature fixed. Yet forcings change that temperature. A proposition and its converse cannot simultaneously be true. That is the fundamental postulate of logic, and the models’ definition of forcing manifestly offends against it.
No surprise, then, that since 1995 the IPCC has had to cut its estimate of the CO2 forcing by 15%. The “consensus” disagrees with itself. Note in passing that the CO2 forcing function is logarithmic: each further molecule causes less warming than those before it. Diminishing returns apply.
We can remind our opponents that direct warming is little more than 1 Cº per doubling of CO2 concentration, well within natural variability. It is not a crisis. We can explain that the modelers have imaginatively introduced amplifying or “positive” temperature feedbacks, which, they hope, will triple the direct warming from CO2.
Yet this dubious hypothesis, not being Popper-falsifiable, is not logic and, therefore, not science. If a hypothesis cannot be checked by any empirical or theoretical method, it is not – stricto sensu – a hypothesis at all. It is of no interest to science.
Not one of the imagined feedbacks is empirically measurable or theoretically determinable to a sufficient precision by any method. As an expert reviewer for the IPCC’s Fifth Assessment Report, I have described its strongly net-positive feedback interval as guesswork – and that, in logic and therefore in science, is exactly what it is.
There is a powerful theoretical reason for suspecting that the modellers’ guess that feedbacks triple direct warming is erroneous. The climatic closed-loop feedback gain implicit in the IPCC’s climate-sensitivity estimate of 3.3[2.0, 4.5] Cº per CO2 doubling falls on the interval 0.62[0.42, 0.74], though you will find no mention of the crucial concept of loop gain either in the IPCC’s documents or – as far as I can discover – in any of the few papers that discuss the mathematics of temperature feedbacks in the climate object.
Process engineers building electronic circuits, who invented feedback mathematics, tell us any loop gain much above zero is too near the singularity – at a loop gain of 1 – in the feedback-amplification equation. At a gain as high as is implicit in the models’ climate-sensitivity estimates, the geological record would show violent oscillations between extremes of warming and cooling.
Yet for 64 million years the Earth’s surface temperature has fluctuated by only 3%, or 8 Cº, either side of the long-run mean. These fluctuations can give us an ice-planet at one moment and a hothouse Earth the next, but they are altogether too small to be consistent with a feedback loop gain anywhere near as close to the singularity as official estimates imply, for homeostatic conditions prevail.
The atmosphere’s lower bound, the ocean, is a vast heat-sink 1100 times denser than the air. Since 3000 bathythermographs were deployed in 2006 no significant ocean warming has been found.
The upper bound of the atmosphere is outer space, to which any excess heat radiates harmlessly away.
Homeostasis, then, is what we should expect, and it is what we get. Accordingly, the climatic loop gain – far from being as impossibly high as the IPCC’s central estimate of 0.62 – cannot much exceed zero, so the warming at CO2 doubling will scarcely exceed 1 Cº.
It is also worth explaining to our opponents the fundamental reason why models cannot do what the modelers claim for them. The overriding difficulty in attempting to model the climate is that it behaves as a chaotic object. We can never know the values of its millions of defining parameters at any chosen moment to a sufficient precision to permit reliable projection of the bifurcations, or Sandy-like departures from an apparently steady state, that are inherent in all objects that behave chaotically. Therefore, reliable, very-long-term prediction of future climate states is known a priori to be unavailable by any method.
The modelers have tried to overcome this constraint by saying that the models are all we have, so we must make the best of them. But it is self-evidently illogical to use models when reliable, very-long-term weather forecasting is not available by any method.
This fundamental limitation on the reliability of long-term predictions by the models – known as the Lorenz constraint, after the father of computerized or “numerical” weather forecasting, whose 1963 paper Deterministic Non-Periodic Flow founded chaos theory by examining the behavior of a five-variable mini-model of the climate constructed as a heuristic – tells us something more, and very important, about the climate.
Bifurcations (or, in our opponents’ intellectual baby-talk, “tipping-points”) in the evolution of the climate object over time are not a whit more likely to occur in a rapidly-warming climate than in a climate which – like our own – is not warming at all.
Sandy and Bopha, and the hot summer in the U.S., could not have been caused by global warming, for the blindingly obvious reason that for 16 years there has not been any.
However, there are many variables in the climate object other than CO2 concentration and surface temperature. Even the tiniest perturbation in any one of these millions of parameters is enough, in an object that behaves chaotically, to induce a bifurcation.
Nothing in the mathematics of chaos leads one to conclude that “tipping-points” are any more likely to occur in response to a large change in the value of one of the parameters (such as surface temperature) that describe an object than in response to an infinitesimal change.
The clincher, in most diplomatic discussions, is money. Once we have led our opponents to understand that there is simply no reason to place any credence whatsoever in the exaggerations that are now painfully self-evident in the models, we can turn their attention to climate economics.
Pretend, ad argumentum, that the IPCC’s central estimate of 2.8 Cº warming by 2100 is true, and that Stern was right to say that the GDP cost of failing to prevent 3 Cº warming this century will be around 1.5% of GDP. Then, at the minimum 5% market inter-temporal discount rate, the cost of trying to abate this decade’s predicted warming of 0.15 Cº by topical, typical CO2-mitigation measures as cost-ineffective as, say, Australia’s carbon tax would be 48 times greater than the cost of later adaptation. At a zero discount rate, the cost of action will exceed the cost arising from inaction 36 times over.
How so? Australia emits just 1.2% of Man’s CO2, of which Ms. Gillard aims to cut 5% this decade. So Australia’s scheme, even if it worked, would cutting just 0.06% of global emissions by 2020. In turn, that would cut CO2 concentration from a predicted 410 μatm to 409.988 μatm. It is this infinitesimal change in CO2 concentration, characteristic of all measures intended – however piously – to mitigate future warming that is the chief reason why there is no economic case for spending any money at all on mitigation today.
The tiny drop in CO2 concentration would cut predicted temperature by 0.00006 Cº. This pathetic result would be achieved at a cost of $130 billion, which works out at $2 quadrillion/Cº. Abating the 0.15 Cº warming predicted for this decade would thus cost $317 trillion, or $45,000/head worldwide, or 59% of global GDP.
Mitigation measures inexpensive enough to be affordable will be ineffective: measures expensive enough to be effective will be unaffordable. Since the premium exceeds the cost of the risk, don’t insure. That is a precautionary principle worthy of the name.
When the child born in Bethlehem ~2012 years ago grew up, He told His audience the parable of the prodigal son, who had squandered his inheritance but was nevertheless welcomed by his father with a fatted calf when he returned and said he was sorry.
However vicious and cruel the true-believers in the global-warming fantasy have been to those few of us who have dared publicly to question their credo that has now been so thoroughly discredited by events, we should make sure that the rat-hole we dig for their escape from their lavish folly is as commodious as possible.
If all else fails, we can pray for them as He prayed looking down from the Cross on the world He had created.
Father, forgive them, for they know not what they do.
Yep, that’s the fly in the ointment – they’re rats, not just people driven by ego. Not that they don’t have overinflated egos to begin with but schmoozing them to admit defeat does nothing to keep food on their table, rats have to eat.
Government is practically the only employer of alarmist climate scientists, so I think they are more worried about where their next meal comes from when this farce is over than how to swallow their pride, (and I doubt many of them have much pride left by now, assuming they had some to begin with).
Given that and the certainty of economic disaster in pursuing CAGW mitigation, perhaps it would be a real bargain to buy-out the cabal’s ring leaders? Say ‘we’ had $20 million and offer the first of their ilk $10M to publicly admit the fraud and go away; the second gets $5; the third $2 and $500K to next six rats to jump ship. I wonder who would be mann enough to be first in line?
“The overriding difficulty in attempting to model the climate is that it behaves as a chaotic object.”
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Precisely. If those who model the climate are so confident in their ability to model chaotic systems, why haven’t they modeled the stock market and gotten filthy rich by now?
My guess is that if they tried to model the stock market, they would be outed as frauds within days or hours of the attempt and their careers would be over. However, with climate it takes years or decades to debunk a forecast. By then the modelers will have revised their algorithms and moved on to “new and improved” models. They will then offer an updated forecast that will require another decade or two to discredit. It may remind some of a prediction from ancient times about people who are “ever learning, and never able to come to the knowledge of the truth.”
JoBrighton says:
December 25, 2012 at 4:01 pm
We’re now 333 months where every single month globally has recorded a temperature above the global 20th Century average. Every single month.
And your point is?
I am 63 years old. We’re now 516 months where every single month my height was above the average for the last half of the last century. Every single month.
Lord Monckton is correct in sensing that something is wrong with the IPCC’s argument. However, he is incorrect in his diagnosis of what it is that is wrong. What’s wrong is that no statistical population underlies the IPCC-referenced climate models. In lieu of this statistical population the IPCC-referenced study of global warming is scientific and logical nonsense.
Isaiah 24:20 NIV
The earth reels like a drunkard, it sways like a hut in the wind; so heavy upon it is the guilt of its rebellion that it falls–never to rise again.
Process engineers building electronic circuits, who invented feedback mathematics, tell us any loop gain much above zero is too near the singularity – at a loop gain of 1 – in the feedback-amplification equation. At a gain as high as is implicit in the models’ climate-sensitivity estimates, the geological record would show violent oscillations between extremes of warming and cooling.
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I seem to remember this story started at a dinner party discussion between Christopher and a lecturer at monash university.
Seems this particular lecturer is under the impression that the climate is a linear system. It’s definitely not.
I have previously pointed out that real control systems, even if linear, have to deal with saturation effects. Linearity is a idealization after all.
Willis also pointed out the obvious. Most temperature dependent effects are themselves intrinsically non-linear. This applies to the vapour pressure of water, chemical reaction rates and a whole lot of stuff affecting climate. Many of these depend on an exponential relationship.
All this has been explained to Christopher before. But he still repeats the same dubious idea as if it’s a proven fact.
Homeostasis, then, is what we should expect, and it is what we get.
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If there was homeostasis there would be no ice ages. There are ice ages so homeostasis is disproved.
Consider the sheer subltety of the effect that causes the oscillation between glaciers and interglacial. It’s a barely discernible change in solar insolation. This causes a change that would freeze our civilisations. It will take an equally subtle effect to fry it.
As long as CO2 continues to climb they’ll shriek how harmful it is in some way or another. If GW falls apart then Ocean Acidification will gain prominence. If we then dismantle Ocean Acidification they’ll come up with some other disastrous outcome from increasing CO2. Personally, I hope Beck was right but I’m not holding my breath. If only we could somehow familiarize the general public with carbon to the extent that when someone said carbon pollution it meant there’s some contaminant in the carbon the same way water pollution and air pollution mean there’s a contaminant in the water or air.
The much-feted “modelers” had written in 2008 that their much-cited “simulations” ruled out, to 95% confidence, intervals of 15 years or more without global warming.
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That’s odd. Because I recollect model paper which predicted temporary changes in the warming rate, similar to what is observed. Apparently changes in ocean circulation, all those swirling currents you see, can change the rate at which warm surface waters are replaced with cooler water from the deeps.
The modelers have tried to overcome this constraint by saying that the models are all we have, so we must make the best of them. But it is self-evidently illogical to use models when reliable, very-long-term weather forecasting is not available by any method.
——
You would think after all this time Christopher would understand that the model runs are used to produce averages over time and space that allow the chaotic behavior of individual runs to be ignored. The grand constraint of energy conservation limits the chaotic trajectories to a range of accessible states.
“””””…..Icarus62 says:
December 25, 2012 at 1:15 am
In reality, every measure we have of global temperature shows that the warming trend continues unabated. Every study of climate sensitivity shows that our no-feedback warming will be amplified by at least a factor of 2 in the short term, and much more than that if we allow slow climate feedbacks to kick in before trying to arrest the warming. It’s time to accept the evidence and look for solutions……”””””
Well for once Icarus62, it is possible you could be right; or correct; your choice. But then we’ll never know, since we have NO measure of the global Temperature, to which we can apply your criterion. We might have meaures of local Temperature; VERY local Temperature; say within mm of actual Thermometers. From there, it is largely guesswork, as to how much larger a local area, may be validly represented Temperature wise, by the declaration of that thermometer. Dr Hansen seems to believe it is the same Temperature 1200 km away from that thermometer.
Then there’s the slight inconvenience of the lack of simultaneity. A simultaneous difference in Temperature, between adjacent points, must result in a flow of “heat” between those locations, and lacking simultaneous recording of Temperatures, prohibits the computation of those heating and cooling flows, so there is no way to account for the simple fact that the earth rotates under the solar blow torch of 1362 W/m^2 power level, so that input energy has to spread out from the point of application, and that spreading determines the Temperature map around the present input region. Lacking that spreading function, prevents calculation of the thermal radiative cooling, and also the “heat” flow cooling by conduction, convection, and phase change (melting and evaporation).
Earth weather and climate is determined by hundreds or thousands of physical, chemical, or bilogical processes (at least), and a good fraction, if not all of those processes are subject the le Chatalier principle of inate resistance to change. To argue that we can model all of that accurately, is simply absurd. And even if we could, how would we calibrate that model, when we have no valid two variable (time and location) Temperature map of the planet; let alone a continuous record of that over a climatically significant length of time.
He told His audience the parable of the prodigal son, who had squandered his inheritance but was nevertheless welcomed by his father with a fatted calf when he returned and said he was sorry.
———-
Rather apt since Christopher is encouraging people to squander their inheritance.
In the real world if you squander your inheritance that’s it, there is no more money. Father is dead and buried and has no more money to give.
Interesting:
“On an objection about Luke, Quirinius, and Herods:”
http://christianthinktank.com/quirinius.html
Could I choose if so compelled between language being a creation of man or product of ongoing evolution?
No, I don’t think I could nor should. Creation and evolution are not mutually exclusive.
Should I proclaim all those that came before me that didn’t speak a modern language to be unintelligent for creating useless words and evolutionary dead ends?
No, I don’t think I should or would. All the extinct words and languages had a purpose at one time.
“””””…..AlecM says:
December 25, 2012 at 2:06 am
………………………………
The basic principle of the Maxwellian electromagnetic treatment of optical waves is that only the net vector from opposing wave fronts can do thermodynamic work. Because the CO2 GHG IR band thermal emission is the same amplitude as that emission from the surface, they mutually annihilate at the surface. This means there can be no CO2-AGW and no interaction via the water cycle with the moist part of lapse rate warming, the basis of the climate models……””””
Better hit the books again Alec, or google wikigoodluck, or however you get youe EM theory.
You are endowing totally incoherent sources of EM radiation with properties that are available ONLY to coherent sources of EM radiation. You can only get cancellation by adding amplitudes of coherent wave trains. Incoherent source interract only in the summation of the sqare of the amplitude, and since real world squares are positive, no cancellation can occur. You don’t even have congruence of frequency; let alone phase coherence.
LazyTeenager @ur momisugly 12:31 pm
With apologies to Hemingway, “ask not for whom the Lord drills rat-holes, he drills them for thee.”
Richard Verney/DirkH: I do not deny there is a temperature signal in the electromagnetic continuum for every object above absolute zero and that for the Earth’s surface it peaks at about 10 microns.
However, when it comes to doing thermodynamic work on the atmosphere with which the surface is in equilibrium you go to different physics, that of radiative equilibrium. A sensor signal is a different beast. this is because sensors work by blocking off radiation for the other direction.
All pyrometers work by having a radiation shield behind the detector so it only detects the Poynting vectors from the required view angle. Put two back to back in zero temperature gradient in air and they measure net zero signal!
To get radiative equilibrium, you look at the difference between the S-B equations, then at the differential of those, the Planck Irradiance Function for a wavelength by wavelength net sum. For a collimated beam, the amplitude of the PIF at any wavelength/waveband is the Poynting Vector.
This means there is no net 15 micron CO2 band IR emission from the surface to or from the atmosphere in radiative equilibrium because it cancels out. Climate Science claims there is an up and down radiation flow which does work. That cannot be the case because only net energy flow can do thermodynamic work. A radiometer detects those individual flows by blocking off the other, a subtlety that has been completely missed by Houghton et. al.
“””””…..Bill H says:
December 25, 2012 at 9:00 am
Pamela Gray says:
December 25, 2012 at 7:54 am
Got everything right except the Sun part. The direct solar influence on temperature waxes and wanes with the solar cycle producing only a smidgen of temperature difference here on Earth that cannot be seen or deducted from the temperature record. It can only be mathematically construed it is so small. The indirect variable components of the sun’s interaction with Earth (cosmic rays, magnetic influence etc) produce far less temperature-forcing capability and cannot even be remotely considered as an agent in the past century’s warming record.
The sun can be considered, in comparison with Earth’s significant intrinsic variability, to be a constant. It is Earth itself which produces warming trends, cooling trends, and nada trends in the temperature record we debate.
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The total output of the solar fusion reaction is indeed fairly constant. However, that is only 1/2 the story. It is the Infrared bands of light/heat hitting the earth which have the warming effect on fm earth. Depending on which band happens to be passing from the suns reaction in strength as measured on the surface of the earth it can indeed have major control over earths systems.
Lower UV bands carry less energy/heat and are reflected by sea water tension or absorbed by the air and particulate matter. Higher UV bands pass these barriers (much like the difference between FM-108mhz line of sight and 2ghz or higher frequencies which will pass through solid objects) which then pass through surface tension of earth oceans and warm the upper layers.
While the suns total radiance and total output changes little, its that subtle change in strength within certain bands which have major effects on the earths surface……”””””
Where do you get such “information” Bill?
Pamela makes a perfectly reasonable observation of a position often cited by Dr Svaalgard, that cyclic variations in TSI cause very little change in earth Temperature (maybe 70mdeg C) from SB considerations. TSI is a measure of total incident power density, including frequencies which don’t survive a trip through the atmosphere. That still results in atmospheric heating, if not surface heating. Surface tension of water has nothing whatsoever to do with either reflectance or transmittance.
Your response to Pamela, is best described as “gobbledegook”. All solar EM radiation energy contributes to warming the earth; not just infrared, and any similarity between UV and FM radio, is limited to the fact that both are EM radiations. But one is a coherent source, while the other is not.
OOoops! I got my “a”s all scrambled. For “Svaalgard” read “Svalgaard”. Sorry Leif, or please fix Chasmod.
JoBrighton, you say:
“Actually, the NOAA makes a point of noting the last 3 decades on record are the warmest on record (with each subsequent one warmer than the previous one), because that’s indisputable evidence of a warming trend. If the next 3 decades on record are increasingly cooler, I’d say that was clear evidence of a cooling trend, but we both know that’s not going to happen”
We don’t need a cooling trend to falsify the IPCC prognosis. And I get tired of people claiming “the hottest decade” etc. Even if in the next three decades the tempratures stay level those alarmists will claim that the last four decades were the warmest on record, forgetting that the issue at hand is the trend rather than actual temperatures.
JoBrighton says:
December 25, 2012 at 4:01 pm
Larry, I suggest you familiarise yourself with the work of the NOAA, we’ve got pretty good global temperature records going back the 1850′s and detailed country records well before that.
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No, you don’t. Anyone believing that is demonstrating a very tenuous grasp of history. In 1850 most of the northern hemisphere had no temp readings. A very small percentage of the southern hemisphere had temp readings. That’s even allowing for an obscene radius for each thermometer. Not that a global temp average means anything, it doesn’t, but even if we were to pretend that it does, we shouldn’t pretend thermometers preceded western civilization in much of the world. Then, of course, calibration, standardization, siting, etc….. using thermometers, one can not reasonably get even a good land only global coverage until about 1930s or 1940s.
Sorry if I sound curt, but, it’s a sore spot for me.
highflight56433 refers to “a Requiem nam Rattus”.
A Requiem, on the other hand, the rat? Latin does not work by finding a word for “for” and affixing a word for “rat” thereafter—and, anyway, nam is a conjunction, not a preposition; Latin is an inflected language, and “for a rat” is ratto.
L. (December 26, 2012 at 12:37 am), your apologies ought to go to John Donne, the author of “Meditation 17” of Devotions upon Emergent Occasions, and not to Hemingway, who merely appropriated some of Donne’s words for a title.
LazyTeenager writes (December 25, 2012 at 11:46 pm):
“Christopher is encouraging people to squander their inheritance.”
Really? Where exactly in his words above, and how, is Lord Monckton encouraging people to squander that combined inheritance?
I had the impression that, contrariwise, Lord Monckton prefers that people utilise, preserve and perpetuate our shared heritage of civilisation, science and reason.
The circuit feedback analogy reminds me of dampening in harmonics, as well as airplane stability, both of which I have studied, the first from a multi-year daily calculus study and the second by taking flying lessons.
An example of the math for Critically Damped Simple Harmonic Motion:
http://mathworld.wolfram.com/CriticallyDampedSimpleHarmonicMotion.html
george e smith: you have a good point and I have anticipated it and am following it up.
Conservation of energy trumps all. When you predict net energy transfer by the difference between two S-B equations, you are also implying the by wavelength or wavenumber interaction of the derivative of S-B, the Planck Irradiance Function. Therefore you calculate the difference of energy between UP and DOWN powers in wavelength intervals. For this not to occur would be a breach of the 2nd Law of Thermodynamics.