The Apology of Chaos

One grows weary of the, “it’s a non-linear, chaotic system” apology for human failing to accurately model weather and climate. This idea seems to have stunted roots in the very different worlds of the Lorenz Butterfly, and quantum probability.

Never mind that the scales of these two worlds are wildly different, and what may be currently unknowable at quantum scale is definitely not unknowable at butterfly scale.

The logical extent of this conflation of concepts leads to the conclusion that by merely observing Lorenz’ butterfly, we disturb it and irretrievably alter the future.

In the real world the effect of a butterfly wing flap is as insignificant to weather and climate, as quantum wave function probability is to navigating to the moon.

Chaos becomes an apology, an excuse to throw up ones hands and declare the task impossible. The task of understanding weather and climate is not impossible. It is just very difficult.

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Alternate “Facts” and Science.

Every “fact” so far that we naked apes have ever believed we have known, has proven to be incorrect at some level.

The operative verb in the sentence above is “believe”.

Alternate facts discovered by inquisitive people challenge the existing system of belief.

Often, when there is no clear economic benefit from the challenging fact, the investigators are persecuted.

Human nature.

[metalogue]: Science is the business of discovering alternate facts.

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Einstein and Weight Loss

Weird idea of the day: why your excessive butt or belly is so unresponsive to exercise, but so responsive to how much you eat.

Of course it is the equation. He won the Nobel Prize for something else, but the famous three term equation rings like a common field through separate universes from weight loss to particle physics.

The equation is pretty intuitive compared to most mathematical hieroglyphics, but I’m going to write it in English anyway.

Energy (what you eat)=mass (your butt) times the speed of light squared. Needless to say, the speed of light is a really big number to start with. When you square it, it gets completely out of hand.

Pretty simple. When you eat more energy than your body consumes, your butt (or belly) grows at the speed of light squared.

We all have our little indiscretions. It tastes sooo good. To get that excess mass off your butt, you must set the treadmill to the speed of light squared.

Simple (wink).


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Emissivity of CO2

There is a lot of confusion about emissivity. Emissivity is the tendency to emit; particularly the tendency to emit light after absorbing it. In the infrared part of the spectrum, where the earth emits radiation towards the atmosphere, there is very little scattering. Incident light is either absorbed, or it passes through.


It can be seen that the light transmitted is equal to the light that passes unhindered plus the light that is emitted after absorption. The rectangle above can be thought of as a slab of atmosphere.  Transmission is the incident light minus the transmitted light. Transmissivity is the transmitted light divided by the incident light. Emissivity is the proportion of light emitted to the light absorbed.  For a perfect blackbody, emissivity is 1.

CO2 is far from a perfect blackbody.

Hottel and Leckner measured the column emissivity of CO2 in the 1940’s and 1970’s, respectively, at .14. Nasif Nahle has calculated a much smaller emissivity, .002, using two different methods. Staley and Jurica (1972) get .19. The range seems to be between poor and spectacularly poor emissivity, .002 to .2.

It turns out that in either case, the extremely high (98% in one meter) absorption of CO2 at 15 microns/wave number 667.4, and the poor transmission (2%), renders emissivity unimportant as regards upward radiation. The range of emissivity simply bounds the amount of light that passes straight through without being absorbed.


Above is the one meter absorption of CO2 at 400 ppm. The Q branch at 667 is 98% absorbed in one meter. This means that only 2% is transmitted. This transmission includes both absorbed and re-emitted light and unhindered light passing straight through.

Where emissivity becomes important is back radiation, as an equal amount to what is absorbed and re-radiated up, must be re-radiated downwards as well. The highest published full column emissivity is ~20%. This would seem to be the high limit of full column back radiation.

What about individual layers?

Staley and Jurica give a value of .08 for CO2 emissivity of a slab of one centimeter optical depth, a value of .14 for 10 centimeters, and a value of .19 for a meter. Optical depth is defined in several different ways by astronomers, chemists, and atmospheric scientists. Astronomers treat optical depth as the mean free path through a slab. In this treatment “mean free path”, “path length”, the distance a photon travels after entering before interaction, the average distance between interactions, the distance between the final interaction and escape, the distance you can “see” into the material, and optical depth; are essentially the same.

Optical depth is also defined as the path length times the absorption coefficient. The one meter absorption coefficient for the CO2 fundamental bend is .98.  Path length is defined as partial pressure times the layer thickness. If the layer thickness is one meter, and the partial pressure is .04, the path length also becomes .04. We multiply this by the absorption coefficient to get an optical depth .04*.98*100(centimeters of CoE to a meter)=3.9

Beer’s Law defines optical depth as the negative natural log of transmittance. This is one of those mysterious empirical fits that work with surprising frequency. Transmittance is one minus whatever doesn’t get through (absorptance), so the negative natural log for a meter layer thickness and therefore the optical depth is -Ln(.02)=3.9. What luck.

The difference between the astronomical approach to optical depth and the chemistry and physics approach is that for astronomers optical depth is an actual distance, while the physical/chemical optical depth is dimensionless.



We can do a sanity check above where we plot linear values through a one meter slab by projecting centimeter scale values from measurements and calculations (.02 transmission, .08 emissivity/cm optical depth) We know perfectly well that none of this is linear, but transmission, and transmission minus emission are virtually indistinguishable. What this crude exercise can tell us is that it is very unlikely that any 667.4 photons pass through a one meter slab of atmosphere at current CO2 concentration. What we see transmitted through the slab is absorbed and re-emitted.

We can therefore assume that transmission, emission, and back radiation are all equal.

Back radiation must run the gauntlet of CO2 molecules on its way down as well. What we see radiated back down has the same mean free path to escape as what is transmitted up.


While the optical depth and free path remains the same up and down, the number of photons absorbed coming up, compared with those re-emitted in either direction, is reduced by the factor of emissivity. Back (downwelling) radiation can be no more than 2% of upwelling radiation.

The Schwarzschild equation used in radiative transfer models gets this completely backwards. The logic of this equation includes a “sink function” for absorption and a “source function” for re-radiation.  This “source function” is given as the absorption coefficient, .98 in this case.

This would be true according to Kirchhoff’s Law  if CO2 was a good blackbody with an emissivity of 1. We have seen that CO2 is a lousy blackbody with full column emissivity somewhere between .002 and .2. The value we have developed above, .02, falls in this range.

The “source function” in the Schwarzschild equation must be very significantly adjusted downward to reflect the real world emissivity of CO2.









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Berkeley Earth, The Arbitrary Use of Parameters to Create a Spurious Correlation


Temperature, CO2, and volcano data | More recent data | High-resolution image The annual and decadal land surface temperature from the BerkeleyEarth average, compared to a linear combination of volcanic sulfate emissions and THE NATURAL LOGARITHM of CO2. It is observed that the large negative excursions in the early temperature records are likely to be explained by exceptional volcanic activity at this time. Similarly, the upward trend is likely to be an indication of anthropogenic changes. The grey area is the 95% confidence interval. (Capitals mine)

The graphic above has been widely used to establish a relationship between human CO2 and temperature. I was unable to replicate it. Stephen Mosher very kindly steered me towards a link on the Berkeley Earth site with an excel sheet.

Using their spreadsheet I first analyzed the components.


The volcanic correlation is interesting, but a very large negative temperature excursion in 1758 seems unsupported by major volcanism, and many temperature drops seem to have begun before the corresponding volcano.

There is certainly no important correlation (.46) between CO2 and the Land Only data. So I wanted to see how the correlation in the Berkeley Earth graphic was achieved. We are given an equation for the “fit”: Fit = alpha + beta * log( CO2 / 277.3 ) + gamma * Volcanic. The values of the parameters are given as:

alpha:  8.342105
beta:  4.466369
gamma:  -0.01515

It is well known that there is an approximately logarithmic diminution in transmission as CO2 in a jar is increased, so a logarithm is a reasonable place to look for getting CO2 in line with temperature.


You can see that the log of CO2 is no fun at all. It is way too flat. We are given the common log, not the natural log, so we take this as a error in the caption of the Berkeley graphic.


I know, let’s add a parameter. We hear a lot about parameterization, but rarely get to see it in action. Here we step through the equation one term at a time.


Hot damn! That’s a much better fit at the end, but there are some jagged toes on that foot.


And with the volcanic parameter the shoemakers have finished their fit.

I am not at all impressed with this process. The author states the rationale:

Because the RCP estimates that most forcing time series are highly correlated to CO2,

it follows that 98.5% of the variance in their total forcing curve can be mapped by just

CO2 and volcanism.

He started with a preconception based on Potsdam forcing data that the correlation must be close, and felt no qualms about using tunable parameters to create a spurious correlation.

True scientists don’t run in those shoes.

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Twitter is a Sack

If you can’t figure out what kind of sack…

The real problem is that it is a confirmation bias, groupthink, censored, scored by perceived approval, no dissent allowed, we’re all grooving’, in crowd, don’t rock the boat, typical human doom scenario.

It’s the way we roll. Pick a shaman so you can quit thinking. Watch TV.  Tweet something, knowing it cannot be challenged. Even if you are full of the same thing the sack is.

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One often hears it said that the “heat absorbing” properties of CO2 in the atmosphere are “well known” or “well understood”. This is definitely not the case. The properties of CO2 in the atmosphere are presupposed  to be simple, and held up as doctrine, but the investigations outlined below reveal many surprising complexities and ambiguities.

MODTRAN is a very mature model derived from thousands of measurements by very serious folks in the US Air Force with no agenda but to get it right. The public release in a form that allows comparison of both up and down perspectives from any altitude, in one meter increments, and to isolate individual gasses; allows an insight into the role of CO2 in the atmosphere that to the best of my knowledge has not been previously explored.

MODTRAN is a model. It is a good one, but it is not perfect. It is possible that some of the results found here are model artifacts. MODTRAN is about light, not heat. Even though the modern custodians of the program have labeled a principal value “Downward IR Heat Flux”, the currency is photons.

Although interpretations of the surprising results of this investigation are presented, no claim is made that these are the only possible interpretations. In fact, my own thinking has changed as this series progressed.

The first surprising discovery in this series was that in the first 300 meters of the atmosphere MODTRAN looking down sees no deviation from a blackbody spectrum in the CO2 bands; but looking up it sees strong deviation. The upwelling radiation (seen looking down) seems to be so completely thermalized that the atmosphere radiates as a unit, a brick if you will. CO2 is not individually distinguishable in reducing or increasing the radiation.The down welling radiation seen looking up from the same elevation is strongly increased in intensity, and the CO2 bands are clearly distinguishable.

1Modtran up and down 1m

In all of these images the “looking down” is the red background run or “Model Alt” and the blue is looking up.

It is fundamental in physics that radiation takes place equally in all directions (Isotropy). If from any altitude  looking down there is no distinct signature of absorption coming up, we expect that when we turn around from the same place and look up we will also see no distinct signature of absorption  coming down…but we do, and we see the same thing consistently from one meter to 300 meters in elevation.

CO2 molecules should not be able absorb or radiate in one direction only. The same molecules should not appear to be a brick looking down and a powerful absorber looking up. It is interesting to consider the blue curve from which the huge “spike” of downwelling intensity emerges. One should really try to imagine the graphic upside down because the warmest temperatures and greatest intensity are usually at the surface.

The Planck curves are the magic of this general approach to radiation because they unify temperature and intensity. When the blue downwelling curve from which the blue spike emerges clearly is not a Planck function we have no way to know if it harks from temperature (which can be equated to altitude by the lapse rate) or intensity. The most intense and warmest part of the blue spike clearly is a Planck function. The part that conforms corresponds to the fundamental wavenumber 667.4 “bend” and its rotational sidekicks.


In the illustration above the relationship between the wavenumber ranges of the fundamental vibration (bend) of the CO2 molecule and the sloped Planck (blackbody) parts of the up and down CO2 signatures is shown for 5 kilometers elevation. At this elevation the downwelling blue spike emerges from a zero flatline showing only blips of weak third order transitions. The second order transitions (618, 667.8, and 720.8) are also shown. The 667.8 second order bend is so close to, and so much weaker than the 667.4 zero to 1 transition, it is essentially subsumed by it. The relationship of the ranges is what is important at this point. Details of CO2 transitions will be explored a little later.

3Modtran up and down 100m

At 100 meters in the image above we see much the same thing, except that the fundamental CO2 bands looking up are no longer warmer than the brick.

4Modtran up and down 1km

At a kilometer we see deviation in the red from the blackbody curve as evidence of specific absorption (and radiation) in the CO2 bands looking down. CO2 is individually beginning to  reduce the intensity of upwelling radiation.

4Modtran up and down 10km

At ten kilometers the CO2 spectral signature in upwelling radiation is very well developed and we can see a second surprising feature, the kissing conundrum. Up and down welling CO2 signatures always meet (kiss) at the wave number 667.4 spectral spike above 300 meters in the troposphere. We will get back to this this.

In the bottom left hand corner of the MODTRAN screens there are some numbers whose meaning is not immediately clear. “Downward flux” seems easy, but is it for just the selected gas or the entire atmosphere? “Background” seems general. To help understand them they are plotted for the first kilometer of the atmosphere below.


The two series to the left correspond to the “Downward IR Heat Flux” numbers on the MODTRAN screens and the two right series the “IR Heat Loss (Background)” for CO2 only and water only respectively. The background upwelling for CO2 only and water only is very similar.

I have not found a satisfactory explanation of what these numbers represent in either the raw model output or the MODTRAN guidance, but this could be a personal problem. In any event the graphic above constrains the possibilities considerably and the common terms upwelling for “background” and downwelling for “downward IR flux” seem reasonable.

Incoming solar radiation to the planet is thought to be 340 W/m2, so the sheer magnitude of the “background”, always over 400, tells us that it must be the composite radiance of the entire atmosphere. The MODTRAN intention seems to be to subtract the downwelling from the background to get a planetary radiance for each altitude, given your selection of gasses. Below is what this looks like.


Both CO2 only and water only “net” radiation increase in intensity with altitude. in a linear way up to 5 kilometers and then begin catenary regressions to the values that remain unchanged to 60 kilometers. There is clearly something very important about the 5 kilometer level. (this figure and comment were changed 11/27 due to my spreadsheet error) If we assume this net radiance is isotropic, CO2 only would appear to cool the planet as its net radiance exceeds 340 W/m2 for most of the curve.


MODTRAN addresses the IR part of the spectrum from wavenumbers 100-1500. The sun, at 5800K does not radiate significantly in these spectra. All the energy in the MODTRAN part of the spectrum harks from the surface of the planet. The atmosphere is initially radiated and otherwise heated from the bottom up.

With this in mind it seems interesting to subtract the values of “background” upselling radiation for progressively higher altitudes from the lowest altitude in MODTRAN (half a meter). We might expect that background upwelling would diminish with altitude, leading to a steady increase in the differences. This exercise was performed for CO2 only and water only. Below are the results for the first kilometer.


CO2 was done first and the very surprising result was that CO2 differences were “quantized”. The data points were chosen to parse out the “quanta”. The stepwise progression in energy is very regular and appears to be a model increment. The stepwise progression in altitude is very irregular. The same data points were used for water. There could be quanta for water too if its data were systematically parsed at meter scale, but they would not match those of CO2. This effort will focus on CO2 and use water only for a frame of reference.

The other surprising result is that the background radiance is higher than the surface leading to negative differences for both CO2 only and water only. The background does not drop back to surface (half meter) levels until maybe 370 meters for water and 400 meters for CO2.

How can this be? How can “background” radiation that emanates from the “ground” increase with altitude?

It is very interesting in this regard that the 400 meter level  where background upwelling values higher than the surface end for CO2 onlyis the level we begin to see a signature of specific CO2 absorption.


Four hundred meters is the top of the “brick” for CO2 only.

Energy can  be neither created nor destroyed. Neither can it be amplified without borrowing. Greenhouse gasses do not create energy. They can absorb it and transform it, but there is always a processing loss to other forms of energy. The increase in radiation above the surface could rely on energy “borrowed” in the form of downwelling from higher levels.

We can evaluate this possibility with the downwelling difference from the surface shown below.


It can be seen that downwelling differences from the surface are very linear and decline in energy with altitude. This is not promising. One would expect an increase in downwelling in the same 0-400 meter zone if it were the cause of upwelling values higher than the surface.

How can CO2 only radiation be stepwise?

The stepwise altitude progression of CO2 shows very equal increments in energy (.31 W/m2) but altitude increments ranging from 14 to 386 meters. These could represent layers with different properties, but the layers would apply to CO2 and not water.


We have painted a picture of CO2 acting strangely in the atmosphere. It appears to radiate in only one direction below 300 meters; its Planck intensity up and down always matches; it shows a pronounced stepwise altitude variance with the surface; and its upward radiance is higher than the surface up to 400 meters.

What could account for this strange behavior?

Possibly unique spectral properties.

Orders of CO2 Transitions

Carbon dioxide is unique among the major greenhouse gasses in having what is called a “Q branch” in the arcane. The HITRAN  image next below (thanks to Barrett Bellamy) shows what this means. The central spike is the “Q” branch corresponding to the 667.4 fundamental bend. The “P”  are rotational transitions gained along with the bend that reduce the energy of the molecule. The “R” are rotational transitions also gained with the bend that increase the energy. It is a package deal. You do not get the P and R without the Q, and the P and R substantially cancel out.

Similarly, in the image above it can be seen that all of the CO2 transitions are dependent on the fundamental bend or Q branch. They all spring from the first transition rather than the ground state. Another package deal. Without the fundamental bend the other transitions do not exist.

The image above also illustrates the preponderance of total energy represented by the fundamental bend. This indicates very powerful absorption. It also shows how the second order transitions are much weaker, and that all of these transitions are saturated at 280 ppm pre industrial levels of atmospheric CO2.

No light in either the first or second order bands has made it from the surface to the tropopause since the industrial revolution.

The third order transitions are in turn entirely dependent on the second order, and they are orders of magnitude weaker. Only these third order bands are unsaturated, and only these can further warm the “planet”. The saturated bands are already completely absorbed, and increasing CO2 in the atmosphere merely moves the complete absorption level closer to the surface .


The image above also shows the absorption fraction in the fundamental bend at 400 ppm CO2.It can be seen that the Q branch, on which the P and R depend, is very nearly completely absorbed in one meter.

The resolution of MODTRAN is one meter.

This may help explain the kissing conundrum and how there could be no distinct signature (the brick) looking down and a clear signal of powerful absorption looking up. The Q branch spikes “kiss” looking up and down because this CO2 band is an extremely powerful absorber in both directions; and, the entire CO2 absorption spectrum rests on its shoulders. With one meter resolution, MODTRAN can be seeing different molecules looking down and looking up, and nearly the complete absorption section for the fundamental 667.4 bend is contained in the same meter. We could be seeing downwelling energy warming the brick in the upper part of the one meter section.

The stepwise altitude progression of CO2 upwelling differences from the surface appear to describe atmospheric layers with no change in CO2 only upward background radiance. The deepest such layer extends from 14 meters to 400 in altitude. The following layer is also 14 meters to 414. Above this the steps are more regular, ranging from 69 to 88 meters. I have no explanation to offer.

The increase in upward background radiance above the surface for both CO2 only and water only is also difficult to explain. The top of this excess radiance seems to match the top of the “brick” for CO2 only, but the top of the brick for water only is much higher than its top of excess radiance, closer to the level at 5 kilometers. where both CO2 only and water only reach their maximum net radiance. CO2 and water differ in many aspects of their greenhouse behavior. Water never “kisses” up and down, for example. This excess energy phenomenon may have something to do with the intense radiative exchange between the surface and the lowest atmosphere described as the photon food fight. Kevin Trenberth’s energy budget gives this as 398 W/m2 upwelling and 340 W/m2  (same as TSI) downwelling taking place simultaneously. These numbers  square fairly well with MODTRAN, which gives a background upwelling value of 417 W/m2 and composite downwelling of 348 W/m2 at one meter. Perhaps the photon food fight acts like a reflector oven.

It is widely known that global temperature is not currently following the model script. Temperature controls the variation around the trend of increasing atmospheric CO2 even today.It is  widely acknowledged that CO2 follows temperature in the ice and benthic cores extending back at least several million years. CO2 does not appear to control temperature at any time scale.

The obvious and unaddressed question is, “Why not?” Hopefully this exercise provides the partial answer that we still have a lot to learn about how CO2 works in the atmosphere.

However one choses to interpret these investigations, it should be clear that the “heat absorbing” properties of CO2 (and water) in the atmosphere are not “well known” or  “well understood.”




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