Trouble with Trancendental Numbers

The title of this post was supposed to be The Temperature and Altitude of Radiation to Space. With a bit of luck and perseverance that post will happen, but we’ve had so much trouble, the trouble itself seems to warrant a post.

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 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)

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A Brief History of Muddled Thinking about the Greenhouse Effect

We humans kick up a lot of dust and burn a lot of stuff. A concept developed in the 1970’s was the “human volcano”. There is a lot of truth in this analogy. Back then, despite dead wrong rants, the sober scientific consensus was that it was unclear whether the net effect of the human volcano was to warm or cool the planet.

“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 stratosphere. As you can see from the blue lapse curve to the right, temperature increases with altitude in the stratosphere. The bottom of the CO2 deviation conforms to and radiates at the 220K Planck curve, a temperature 25 degrees higher than the bottom of the deviation at 17 km. In the tropical atmosphere, 220K matches an altitude of  24 km in the stratosphere.

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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Berkeley Earth Absolute Temperature and the Remarkable Closeness of TMAX Lows and TMIN Highs

Bob Tisdale recently did a series of posts exploring the differences between using absolute temperature and temperature anomaly to understand the recent temperature evolution of our planet. Bob was interested in the extracted TMAX highs and TMIN lows, and showed that the TMIN lows have warmed about twice as fast as the TMAX highs.

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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Aerobic and Anaerobic Metabolism: A Symmetry

The symmetry between aerobic carbohydrate respiration and photosynthesis is widely appreciated. In the presence of Oxygen, many living things (including photosynthesizers) burn carbohydrates, producing useful energy and CO2. Photosynthesizing creatures, using the energy of sunlight, manufacture carbohydrates from CO2 for their own use; and Oxygen. By outrageous good fortune, photosynthesis has been so wildly successful that has produced a surplus of carbohydrate and Oxygen sufficient to balance the evolution of hungry multicellular respirers; like ourselves.

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.

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Cold Smoke

Santa Rosa is comfortably away from the fires this year. Last year’s baptism cleaned out the tinder around here. Yet the Camp Fire has given us the peculiar combination of dense smoke and hard frost.

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.

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The Flat Universe Society

While it is clear that the earth is sort of round, rather like a baseball that has absorbed too many home runs, to the best of our understanding, the universe is flat.

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?

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Energy and Matter

An interesting upshot of Mr. Einstein’s famous equation is that in units of the speed of light, energy and mass are equal. Mr. Poincare once observed that mathematics is the exercise of making different things the same (equal), and sophistry above with the square of the speed of light might be just the sort of trick he was referring to.

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.

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