geosciencebigpicture

We left off the last post having accommodated MODTRAN radiance to the Planck curves, and having promised to derive the temperatures of the individual lines from an inversion of the Planck radiance formula to give the Planck brightness temperature.

Unfortunately, we were unable to make this work. Several different formulae for brightness can be found that yield somewhat different results, but they all show conformance to the general shape of the CO2 radiance deviation.

We decided to use the following two because despite their different approaches, when scaled they become identical.

Below is what we get using the first equation above not scaled:

We really don’t know what to make of this. The systematic decrease in brightness from lower to higher wave numbers across the CO2 deviation in relation to radiance seemingly gives little hope that useful temperatures can be gained this way.

The good news is that a fresh look at the Stephan-Boltzmann approach to temperature using an emissivity slightly above .2 gives a far more satisfying result.

Staley and Jurica (1970) derived a full column emissivity for CO2 of .2. We are frankly astonished that by tweaking this slightly higher we were able to get such good agreement across the ~250 wave numbers of the CO2 deviation. Perhaps .225 is the correct column emissivity. At any rate, we will use this approach to get our temperatures henceforth.

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 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)

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.

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.

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?

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.