We had discussed the temperature of radiation to space in a “mark with chalk, cut with a chainsaw” kind of way based on MODTRAN graphics below.

We see that CO2 takes a gap toothed bite out of the Planck radiation curve and that the bottom of the deviation conforms to and matches the 220 K Plank curve in a general sort of way. The “P” and “R” rotations surrounding the fundamental bend conform generally to 220 K, but the fundamental “Q” branch itself jumps up to something like 240 K, and the other absorption lines fall unsatisfyingly between the 20 K Planck increments. We want a much more precise way to judge the radiative temperature.

Our first approach was to try to reproduce the MODTRAN graphic with 5 K temperature increments. When you push the “Show Raw Model Output” button you get **Radiance** in 2 wave number increments. We then sought to plot the Plank curves for the same increments according to the equation:

We got beautiful curves.

Unfortunately, our MODTRAN gap toothed bite was some 4 orders of magnitude less than the Plank units, and even when scaled on a secondary axis did not match the MODTRAN graphic in either implied temperature or units. Rats!

In trying to understand why, we noticed that the MODTRAN graphic is in units of **Intensity **W/m2/cm-1. These units are commonly used for **Irradiance **or incident radiation as opposed to **Radiance** or radiant exitance per solid angle. In fact. Planck curves in units of **Irradiance** match the units in the MODTRAN graphic, but this left the unwelcome task of converting the model output in **Radiance** to **Irradiance.** This is a can of worms in its own right, and involves transcendental numbers.

Even if we could reproduce the MODTRAN graphic with 5 degree Planck increments, we would still be interpolating. What we really want is a Planck temperature for each wave number’s total radiance. Brightness temperature seemed to fit the bill. It basically treats each wave number’s total radiance as the maximum intensity of a new Planck curve to derive the temperature.

This basically inverts the Planck equation to give temperature. The *I* is confusingly intensity. We plugged in MODTRAN total radiance for *I *and got:

Using the equation above we got brightness temperature in the ridiculously low units on the left, somewhat like of the microwave background radiation from space. We found another brightness equation that used base 10 log and used the physical constant “C1”, which is 2hc^2 in a way unfathomable to us. Remarkably, the two different equations scale closely, with Eq. 2 giving units to the right in the range of nuclear fusion in the sun.

Brightness and radiance scale closely, with the difference increasing in shorter wavelengths.

A fine kettle of fish! We still don’t have reasonable temperatures by wave number.

For now, we are blaming our travails on transcendental numbers. Numbers have no business being transcendental. They should unfailingly hold their values. Using the constant *e *implies a symmetry. Perhaps emergent properties of *e*^x in both the Planck and brightness equations are messing us up. (Wink)

“At the present rate of nitrogen build-up, it’s only a matter of time before light will be filtered out of the atmosphere and none of our land will be usable.”

The quote above from Kenneth Watt, a self-appointed and widely read guru, in 1970. He was worried about cooling. Nobody worries much about “nitrogen build-up” these days, nor cooling; we have new witches to burn.

Al Gore likened the CO2 greenhouse effect to a blanket, keeping the planet warm by reducing energy loss to space. This might work if the blanket was a “space blanket”with a coating to reflect the dim light radiated from the surface back down. If you have ever used a space blanket to stay warm, you will know that it helps, but you would much rather have a down blanket. This is because conduction and convection are very important as well as radiation, and the down blanket reduces conduction and convection better than a space blanket.

Somewhat more sophisticated came the argument that rather than a blanket, the greenhouse effect relies on pushing the “net radiative altitude” higher. Since the troposphere cools with altitude, and radiative intensity varies to the fourth power of temperature, pushing this radiative altitude higher would reduce radiation to space. We will show that significant CO2 radiation to space comes from the stratosphere where temperature increases and radiation to space increases to the fourth power of increasing temperature with increasing concentration.

According to MODTRAN, as seen from the altitude of polar orbiting satellites at 70 km, the fundamental bend of CO2 (and its rotations) radiates at a temperature of 220K. This temperature corresponds to an altitude of 12-13 km at the tropical lapse rate. The atmosphere continues to decline in temperature in the tropics to 17 km, where the lapse suffers a relapse and the atmosphere begins increasing in temperature with altitude.

If you set MODTRAN to 410 ppm CO2, and set all the other greenhouse gasses to zero; you can vary the altitude to see where the CO2 radiance is coming from.

Above you can see that the total IR upward flux looking down from the tropopause at 17 km is 397.21 W/m2. You can also see that without the other absorbing gasses, the planet radiates at the Planck curve for surface temperature except for the deviations caused by CO2. The lowest part of the CO2 deviation is seen radiating at the temperature of the 17 km altitude, about 195K, as seen from the blue lapse curve to the right.

When we jump up to 70 km we see that the upward IR flux ** increases** to 400.35 W/m2. This means that 3.14 W/m2 of the CO2 radiance polar orbiting satellites at 70 km see is coming from the

When we drop down to 24km the upward flux is 398.15, meaning that 2.2 W/m2 of the CO2 radiance seen from 70km takes place *above*** **24 km.

We decided to continue the exercise above and record the MODTRAN upward radiance at one meter and thereafter at 1 km increments to 70 km. It can be seen that CO2 radiance increases in the lower stratosphere and then levels out beginning at 43 km. Seemingly, 43 km is the last inflection point in CO2 radiation to space.

We began this post with examples of muddled thinking about the impacts of human absorptive gasses in the atmosphere. We end the post having presented data indicating that we need a far more nuanced approach.

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]]>We had never seen an analysis like this, and what struck us was the how close the lowest seasonal maximum temperatures were to the highest seasonal minimum temperatures. The closeness of these values in absolute terms suggests a conserved quantity or thermostatic mechanism.

Above it can be seen that while the seasonal fluctuations in average maximum and minimum temperatures are on the order of 15C, the difference between the lowest highs and the highest lows averages only .25C.

As the earth rotates, each day the sun warms every location on earth, creating each location’s daily high temperature or TMAX. Locations then rotate into darkness, creating the daily low temperature or TMIN. The graphic above is averaging these lows and highs, essentially treating the entire planet as a single location.

The simple explanation for the closeness of the TMAX lows and the TMIN highs would be that the temperature is “handed off” from the top of the low to the bottom of the high. This is not the case because the TMAX lows vary between November and February, and the TMIN highs are nearly always in July.

Above we got rid of the seasonal amplitude to focus on the difference between the high lows and the low highs. It can be seen that as the planet has warmed, the difference has diminished. This reduction in the difference is a clue, but it is not really surprising since we know that the planet has warmed more at the poles, and more in TMIN than TMAX. Essentially, the planet is warming more at the cold extremes.

We thought it would be interesting to compare Berkeley absolute TMAX with CERES net flux. It can be seen in the unsurprising result above that the highest temperatures correspond to periods of negative net flux to space; and the reverse. The TMAX highs typically lag the net flux lows by a month. It can also be seen that the TMAX warming over the CERES period was *not* caused by a reduction in radiation to space. Radiation to space actually increased slightly over the period.

The apparent thermostatic control that limits the TMAX lows and the TMIN highs to such a narrow range is probably the elastic nature of the specific heat of the ocean. The capacity of water to store energy increases as the water warms, and decreases as the water cools. It works like a rubber band.

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In physics, this is called a symmetry, a conserved quantity. The reason the quantity is conserved is unclear and taken on faith as a fundamental property of the universe. For fun, we go out on a limb here and propose that physics can take a lesson from biology.

The basic idea of conserved quantities is that they remain constant amid the mayhem around them. When we say energy or baryon number is conserved, we mean only that at the end of a long day of collisions, phase transitions, etc., we get the same quantity we started out with. If we wish to take the lesson from biology, the continuous symmetry of aerobic respiration and photosynthesis means that all the powerful disruptive forces continue cancel out, leaving the relationship the same within the bounds of inevitable fluctuation.

Less well appreciated than the symmetry of aerobic respiration and photosynthesis, is the symmetry of aerobic and anaerobic metabolism. In the guts of hungry multicellular respirers from ourselves to termites, beneath the water surfaces, and as deep in the crust as we have yet drilled; lives a vast anaerobic world.

The vast anaerobic world is ancient and mostly microscopic Archaea, the first living things. The anaerobic world developed some internal symmetries of its own during a billion years or more of evolution before cyanobacteria got around to producing much Oxygen.

The primary modes of anaerobic metabolism are methanogenesis and fermentation. Fermentation produces CO2 and alcohol. Alcohol itself is valuable low entropy carbon (carbohydrate) used by other critters. Methanogenesis can occur from carbohydrate, producing methane and CO2; or it can combine CO2 and Hydrogen, producing methane and water.

A few arcane critters use methane anaerobically, but methane from CO2 is essentially the final step in anaerobic metabolism. The resulting methane is waste to the Archaea, but an important gift of low entropy Carbon to the aerobic world.

]]>A lot of the particles in wildfire smoke are about the size of the wavelength of visible light, .4 to .7 microns, so they block sunlight effectively, but are relatively transparent to the long wavelength radiation to space from the surface that fosters frost on otherwise dry, clear nights.

Dust masks are designed to filter out particles larger than .3 microns, but this is the *average* size of the particles in smoke, so the only filter about half of the particles, and the smaller particles they do not filter are considered more dangerous. Most people I have seen have not even bothered to pinch the masks around their noses, which lets *all *of the particles in. The air would much rather go around than through the mask.

They closed the schools. More of the foolishness that led to the closure of schools for weeks after the fires here last year. Kids live in houses subject to the same conditions as the schools, and schools are better places to reduce activity levels for many kids with working parents. These kids are just turned loose when schools close.

Like cold smoke, foolishness and superstition seems to drive much human behavior.

]]>By flat, we do not mean that you could walk to the edge and fall off, like pre Columbian notions of the earth. Rather, we mean that despite all of Mr. Einstein’s distortions of spacetime, at the largest scales we can manage, the universe follows the geometry of Euclid drawn flat in the sand.

If the old Greek guys had done their geometry on a sphere or a saddle (ignoring the impracticality of sand on these shapes for the sake of argument), the sum of angles in a triangle would not be 180 degrees. The same is true for any surface not perfectly flat.

Astronomers measure the distances to stars and such and can calculate the angles of triangles connecting them. Always 180 degrees.

This seems as peculiar as energy being a function of the speed of light *squared*.

Nevertheless, the flatness of the universe has become as embedded in our cosmology as pre Columbian notions of the earth. The critical density to achieve flatness is the cornerstone of many equations.

If you could fall off the edge of the universe, would you have found the multiverse?

]]>Nevertheless, the human concept of squareness derives from carefully drawn figures in the sand, subdivided and counted. A square divided in two each way yields four smaller squares, so it is natural to say that four is two “squared”. In this frame of reference, when you don’t subdivide your square at all, each side remains one, and you have but one square. That one squared equals one is unassailable.

We have noted before that equating energy to matter times the speed of light *squared *seems peculiar. Here we explore an alternate notion that the huge asymmetry between energy and mass in units smaller than the speed of light, disappearing to unity at the speed of light, is precisely why the equation works.

Mr. Einstein did not know about fermions. Fermions are subatomic particles that take up space, have mass, and constitute the matter in the universe. Electrons, quarks, and the triplets of quarks we call protons and neutrons are fermions.

Mr. Einstein knew about photons. He was instrumental in their discovery. Photons have no mass, take up no space, represent force rather than matter, and travel at the speed of light. In our standard model photons are bosons, and we see the difference between fermions and bosons as a fundamental division of the universe.

Complicating the picture, the divide between fermions and bosons is not about mass. The divide is about the ability (even propensity) to condense into or occupy the same space. Many bosons have mass derived from Mr. Einstein’s equation because they do not travel at the speed of light. Of the bosons, only photons, gluons (carriers of the force holding quarks into protons and neutrons), and the (as yet) hypothetical gravitons have no mass and travel at the speed of light.

This brings us to an important distinction between mass and matter, often loosely interchanged as the “m” in E=mc^2. All matter has mass, but not all mass is matter. While we can write E=f(ermions)c^2 and the expression is true, we cannot write E=mb(osons)^2.

If we use units of the speed of light, in some sense we pose our subject energy, mass, and matter to be travelling at the speed of light. This is not possible, as mass and matter cannot travel at the speed of light. There is still something the *matter *with our conception.

We do know that a watermelon is usually over 90% water. We can weigh the watermelon and calculate the number of water molecules with 90% accuracy. We can then take the ideal gas constant and divide it by Avogadro’s number (this is the Boltzmann constant), multiply this by the log of the enormous number of ways we could exchange water molecules throughout the watermelon without changing its appearance, and derive a number for the entropy of 90% of the watermelon. This would be foolishness, because the reason a watermelon can swim upstream against the tide of entropy to exist at all is a result of information; the information in its DNA. Information can create singularities of negative entropy.

A watermelon is not a gas. Most gasses are invisible to us even at the macroscopic levels we chunk their molecules up, like pressure, temperature, and wind. Our conceptions of thermodynamics were really developed for steam engines, machines designed to extract work from the disequilibrium of pressurized gas. The molecules in a gas want to spread out. The molecules in a watermelon do not, at least not as quickly. The watermelon seems content, as if it has reached some (temporary) quantum of negative entropy equilibrium.

Steam is the gas phase of 90% of a watermelon. Not the steam we think we can see, that is actually liquid water that has condensed from the invisible gas phase. Most of the water in a watermelon is liquid or bound up in solid organic molecules. Solids, by definition, do now want to spread out as much as liquids or gasses. The very property of being solid constrains the spread. The information coding for particularly the solid rind of the watermelon temporarily defeats the tendency to spread out.

Boltzmann entropy was the intellectual beginning of the notion that a probability field extends through the universe from the point human perception fails. The existence of the watermelon in negative probability phase space makes the value of W (the number of ways the invisible components can be rearranged without perceptible difference) negative. The fundamental reason for this negative entropy is information. Information implies purpose.

Entropy is perplexing because while we easily understand why it increases toward the future, since all the fundamental laws of physics seem reversible, entropy should also increase toward the past. Boltzmann wrote this off to probability, simply maintaining as a brute fact that probability flows to the future. Yet probability fails to explain why an entire universe should evolve with billions of galaxies each containing billions of stars, just to get to a human brain. It is far more likely that a human brain, whose perceptions reputedly define the line between the classical and the probabilistic, would randomly fluctuate into existence.

So while we can derive a number for the entropy of a watermelon based on the statistical mechanical properties of the high proportion of water it contains, we really have no way to evaluate the probability of the information it contains. In some sense all living things are watermelons. In some sense our planet is a watermelon.

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1/r^2 Newton’s inverse square diminution of gravity.

F=ma Newton’s second law. Describes the relations between force, mass, and acceleration.

S=klogW Boltzmann’s formula for entropy.

E=hv Planck-Einstein energy of a photon, Planck’s Constant x *frequency.*

E=mc^2 Einstein’s General Relativity. In units of the speed of light, energy=mass.

These are a few famous examples, and except Boltzmann, they are classical mechanics. In classical mechanics there is no need for probability, and there is no margin of error (except in measurement). The rules are absolute. Space is a field, or reference frame with fixed Cartesian coordinates, x,y, and z. Even today, you need nothing but classical mechanics to send a rocket to the moon. Except for a small deviation in the orbit of Mercury, classical mechanics describes the motions of the planets in our solar system.

The flamboyant Monsieur Laplace seems to have been the first to observe that using Newton’s laws, if you knew the position and momentum of every particle in the universe in the present moment, you could extend this information indefinitely into both the past and the future. You would know everything there was to know. (Momentum is a vector quantity that includes direction). Asked by Napoleon why his work contained no mention of God, he replied that he had no need.

In Special Relativity Einstein argued the concept of spacetime, where every xyz coordinate becomes a clock that is inseparable from the point. He showed that in a curved trajectory different observers would record different amounts of time passing. Space and time are relative.

In General Relativity Einstein argued that spacetime itself is warped by gravity, and that not only space and time, but energy and mass are relative. The only constant is the speed of light.

Newton’s and Einstein’s theories are classical, there is no probability, no statistics, and no dice. The title of this post refers to a quote from Albert Einstein made in a letter discussing statistical mechanics with Max Born. He said (in German), that he believed God does not play dice with the universe. We will argue in favor of Einstein in this regard. We will argue that Boltzmann’s introduction of probability in his study of entropy set the stage for probabilistic notions in quantum mechanics. We will argue that both probabilistic approaches founder on an anthropocentric philosophical error.

It is ironic that the probabilistic approaches to entropy and quantum mechanics both begin with a wildly improbable assumption: that what humans can perceive somehow matters at a fundamental level in the universe. The basic problem is that the nanoscale world of molecules, atoms, and sub-atomic particles is largely invisible to us.

For Boltzmann, the solution was to count the ways the invisible components can be arranged and remain visibly the same. Entropy becomes the log of this number of ways the invisible components can be arranged multiplied by Boltzmann’s Constant (k); which is the ideal gas constant divided by Avogadro’s number. Boltzmann’s insight was that the behavior of gasses well understood at a practical level by engineers of the industrial revolution, was dictated by the interactions of the invisible molecules and atoms.

We can see above that it works very well for the density of the earth’s atmosphere. The molar concentrations of the invisible (to us) gasses in the atmosphere are well-known. If we suddenly developed the ability to see the different gasses, and they became distinguishable to us, nothing would change. Entropy here is the separation between the molecules in the air, density–regardless if we can see the molecules or not. Is it simply more probable that the spaces between molecules seek a higher entropy equilibrium as pressure is relieved, or is it certain they will do so?

Probability is a tool for the blind. If we can see it, it is classical; if not, we would have God play dice. We are blind at the scale of particles, atoms, and most molecules. Quantum mechanics is fundamentally the notion that the energy of electrons is not a smooth linear progression, but rather a stairway with discrete intervals or steps; quanta. Our efforts to locate electrons and other subatomic particles are hindered by the unfortunate circumstance that when we ping them we unavoidably alter them. Werner Heisenberg formulated this as his uncertainty principle. We can know either the positions or momenta of invisible particles, but not both. This quandary evolved into the probabilistic notion of a wave function, where waves of probability are our only information until we collapse the wave to a particle by pinging it. It is even maintained that *there is no such thing* as where the particle actually is until it is observed.

Can you imagine anything more anthropocentric? If a tree falls in a forest and nobody hears it, did it really fall? Of course it did. It is not a matter of phase space between probability waves. It is easy to go and see many fallen trees that very likely nobody heard. Trees fall and particles have positions and momenta regardless the perceptions of naked apes, or even – aliens capable of somehow garnering information on both position and momentum.

Take a macroscopic object, a watermelon. They differ somewhat in color, size, and skin pattern. These differences arise from somewhat different arrangements of components, but people will generally agree that a watermelon is a watermelon. A watermelon is composed of all manner of quarks held together in Protons and Neutrons by the strong nuclear force. These may be surrounded by electrons in quantized shells held to the protons by the electromagnetic force (and to both neutrons and protons weakly by gravity). These atoms may be bound to others by ionic or covalent bonds in complex molecules. Add these layers up and you sometimes get a watermelon.

When we observe a watermelon, we change the universe. Photons of visible light reflect from the surface of the watermelon and our retinas absorb these photons, which otherwise would have travelled to a different fate. Did the watermelon exist before we observed it? Did the photons exist before our retinas absorbed them, or were both the watermelon and the photons mere probability waves before we intervened? When we turn our back to the watermelon, our eyes no longer absorb the photons. Some may be reflected from our clothing, and others may be absorbed by our skin without sending information about the watermelon to our brain. We change the universe even when we don’t observe the watermelon, but the watermelon is still there.

Imagine that our vision deteriorates and we can no longer see individual watermelons, but can barely discern a large pile of watermelons. We will have lost a lot of information. Our degraded retinas absorb a different set of photons.We could then argue that the positions of the watermelons in the stack is a probability function, and *there is no such thing* as exactly where a watermelon is in the pile until we reach in blindly and grab one. This would be foolishness, but it is exactly what the probabilistic Copenhagen Interpretation of quantum mechanics would have us do. If the pile of watermelons is subject to gravity and forms a cone, we could even develop formulas that determine the probability of grabbing a watermelon as being much higher when we reach blindly toward the bottom of the pile rather than the top.

By having God play dice, we elevate probability to the status of a force of nature, joining the strong, weak, electromagnetic, and gravitational forces. The field and waves are established in the form of the wave function. A probability particle can’t be far behind.

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In the 78 miles we have navigated 16 rapids with holes big enough to turn our 18′ rafts upside down. These holes result from the water’s defiance of gravity and momentum to curl back upstream to fill voids in the flow when the water is forced to pass around rocks. It curls back upstream to reduce the gradient in potential energy from the disturbance of the water’s cohesive force, and increase the entropy.

Fortunately, age and experience can influence luck, and this was our first trip where nobody was even out of the boats in a rapid.

The Vishnu metamorphic complex brings us to a really staggering unconformity, called the Great Unconformity. We may recall that the Tapeats sandstone began to be deposited about 545 mya, just slightly before the Cambrian period, the first period of the Paleozoic era 540 mya.

The Tapeats was deposited variously on top of late Proterozoic Super Group members where they occur, and on top of the early Proterozoic Vishnu complex where the Supergroup is absent in much of the canyon. We may also recall that the Vishnu complex includes both highly metamorphosed sedimentary rocks and magmatic plutons which injected and overprinted any age information in the sediments. Our best overall dates for last crystallization are 1.7 billion years ago.

Where the Tapeats sits on the Vishnu, a little arithmetic reveals that the Great Unconformity spans 1.16 billion years. This is almost exactly 1/4 of the time since our planet is thought to have coalesced from stardust 4.6 billion years ago.

Above is the unconformity in Blacktail Canyon. The bedding within the Tapeats above the unconformity is highlighted by the different lime content and erosion resistance. If you look carefully, you will see some very thin bedding just above the contact, that based on color, seems to be erosion from the Vishnu. These redder bands alternate with lighter material from a different source, possibly the same source as the lighter bedding seen generally in the Tapeats above.

Really though, the Great Unconformity between the Tapeats and Vishnu overstates the case a bit. The Supergroup is absent here, but it exists elsewhere in the Grand Canyon. The Vishnu was folded nearly vertical at some point. The Supergroup was tilted against the Vishnu at some point, and both were eroded to a nearly flat surface during the Great Unconformity. This flat surface eventually received the Tapeats, and the Tapeats therefore overlies the Vishnu and each of the tilted layers of the Supergroup in turn.

Above we show a more fair representation of the Grand Canyon unconformities. These were assembled using date ranges for members from Wikipedia and Macrostrat. The two sources do not always agree, so a measure of our own judgement was required. This cannot be considered the last word, but the general trend for longer unconformities the further we go back in time is clear.

It may be that as you go back in time, the probability of having had a long unconformity that erases a lot of little ones increases. Or it may be that the breathing of the great beast has sped up.

Ludwig Boltzmann was the pioneer of statistical mechanics, a notion that although some important things are unknowable to us, like the position and velocity of every atom in a watermelon; we are not entirely powerless. He showed us that in situations where we can count the possible ways the atoms could be rearranged and still appear to be a watermelon, we can calculate the probability of a watermelon. Unfortunately, neither watermelons nor great breathing beasts are among these situations.

Unconformities are the most fundamental and largest scale changes evident in the Grand Canyon, but they are by no means alone. Every layer shows some level of fluctuation between more marine, calcerous, well sorted periods; and more terrestrial, silicious, and poorly sorted periods.

We saw the fluctuations in the Redwall above in the last post.

Above is some detail of the Cambrian Muav from National Canyon. If you measure the average distance between the bands and factor the section thickness and duration of the Muav you can calculate the intervals. This is a crude exercise. It can tell you it is probably not 100,000 years, and probably not 10 years.

Above is some context of the finely bedded Muav.

If you measure the spacing of these fluctuations within the Muav, or the Redwall or any other large scale stratum, you discover that despite the appearance of regularity, the spacing is *never* exactly the same.

One could take a section of aside creek like the one above that cuts through only the Muav, and imagine it as a miniature Grand Canyon. Sequences could be identified , grouped, and dated, and possibly unconformities could be found. This has been done for the more wildly varying members like the Supai.

Ultimately our groupings, and how sharply we focus our scope becomes somewhat arbitrary. It is our nature to rationalize and categorize, but at some point we are always left like Ludwig Boltzmann; trying to count the possible ways this all could have happened to assign a probability, when all we can see is the apparent result a great breathing beast.

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