ITS QUIRKS AND USES. • Watts Up With That?

Guest post by KEVIN KILTY

An Atmospheric Transmission Model

Introduction

This post is not one I planned to do. It has grown from work I was doing with MODTRAN (moderate resolution transmission program) in support of my post on June 14, 2019. As I mentioned in that thread I had gotten odd results from MODTRAN, and needed to ponder them further. As in so many things on WUWT what I have learned is directly connected with a number of Willis Eschenbach’s posts (especially 08/11/2011, and 04/12/2014); but also is related to dozens of others, especially to a recent post by Nick Stokes (06/06/2019) and something Dr. Spencer said on a thread so long ago I can’t even recall the year, let alone a date. What I have to say in this post is pretty fundamental to interpreting the output of model runs from MODTRAN, as well as to understanding its relationship to models of the greenhouse effect including feedback.

1. Radiation measures

There are four quantities used to quantify radiation which are related to one another and which use standard symbols in most texts: radiant intensity (I), irradiance (G), emitted power (E or W), and radiosity (J). I used I_s for solar irradiance in my previous post, but I will stick to the conventional symbols from now on. Older versions of MODTRAN use I_out as the symbol for irradiance. The definitions of these are:

1. Intensity is the power flowing along a pencil of rays from or toward a unit area on a surface and delimited by a unit solid angle in some direction. It is what we would think of as a beam of radiation.
2. Irradiance is intensity integrated over the entire view that a unit surface area has of incoming radiation. It is the power flux landing on a surface.
3. Emitted power is the irradiance produced by emitting sources on a surface. The Stefan-Boltzmann law is emitted power.
4. Radiosity is the combination of emitted power plus irradiance derived from reflection at a surface. I won’t use radiosity at all in this post, but radiation is reflected frequently on the surface or in the atmosphere. Radiosity is what one needs to handle such instances.

2. MODTRAN Input/Output

MODTRAN, like so many legacy programs written in FORTRAN, reads a formatted file, a virtual card deck if you will, that specifies the current job. It calculates radiant intensity over a specified path, and also transmission coefficients at particular wavelengths on this path. Through the input file one can define a path, what sort of surface this path reaches, and the composition, pressure and temperature structure of the atmosphere along the path. While this sounds like a great deal of flexibility, most of this isn’t available in the portal that most of us have access to.

3. The University of Chicago Wrapper

The public copy of MODTRAN that Willis uses is provided by the University of Chicago. Access to the underlying program is through a graphic interface wrapper written in some other language. This wrapper does not provide full access to MODTRAN. It allows only vertical paths in the atmosphere, it seems to have no provision for input of an arbitrary model atmosphere, but rather lets the user choose from among a number of models–U.S. standard atmosphere (1976), Tropical, Midlatitude summer, etc. and then make limited adjustments thereto, including a surface temperature offset. Its output is irradiance at a specified height, and at a view of either the upper or lower hemisphere; and makes available raw model output with a pushbutton on the wrapper.

Figure 1. Screen output from the University of Chicago wrapper for MODTRAN. Note the range of potential input values, and the raw model output button.

How the University of Chicago wrapper turns an intensity value into an irradiance is by assuming that the intensity is the same in all directions (isotropic radiation) at which point irradiance is just π times the intensity; G = πI.

Having an unfettered input to MODTRAN would allow a person to calculate radiant intensity at a number of view angles, and integrate over a hemisphere. The restricted operation of the wrapper using a fixed value of π leads to an obvious bias.

Imagine being high in the atmosphere, 70 km above the surface, and looking down. The view is not of Earth covering the entire hemisphere, but rather includes cold, dark space at grazing angles. I figure the actual view is only 98% of an illuminated hemisphere, which produces a bias. This is a moot point, however, because we do not actually know if intensity is the same in all directions and so the factor of π is likely not correct in any case. The accuracy of irradiance values provided by the wrapper might be as much as 26W/m² different from true values on top of uncertainty of another 10W/m² contributed by MODTRAN itself. Despite this, difference between models can be much more accurate as long as one takes care in specifying the model.

A final point about the wrapper concerns the temperature offset it allows. Each of the model atmospheres has a default surface temperature; 299.7K for the tropical model, 294.2K for the midlatitude summer model, and so forth. By specifying an offset, though, one actually adjusts the entire atmospheric path by this offset value, and is not what one intends to do in most circumstances. It would be great if one could just adjust temperature of the boundary layer, or just the surface, but this is not possible in any easy manner.

Finally, there are two assumptions regarding water vapor that a user can choose–constant relative humidity or constant mixing ratio. The constant relative humidity choice has an interesting interaction with a negative temperature offset. This is likely to produce a relative humidity exceeding 100%, and will produce a long list of error messages in the MODTRAN raw output, which never reach the graphic output of the wrapper. Look at the raw model file. Caveat emptor!

4. The MODTRAN Oddities

In his post about the MODTRAN oddities, Willis noted that his calculation of greenhouse forcing from a doubling of CO₂ using various model choices would never meet values stated by James Hansen, and he wondered why this is so.

First, Willis measured the instantaneous forcing by using the difference in upwelling irradiance of paired MODTRAN runs at 70km in the atmosphere. The only difference is CO₂ concentrations–375ppm versus 750ppm. I have verified Willis’s values ranging from 3.2W/m² in the tropics to 1.6W/m² in the subarctic. These are much smaller than Hansen‘s stated value of 4.5W/m². I have tried to figure out what Hansen was thinking, but I can‘t. In various places he states different values, and he applies different assumptions, and considers different end points. I don‘t think chasing Hansen around in print is very useful, but staying with the story Willis has begun is worth pursuing.

5. Calculation of the Greenhouse Forcing

CO2	Observation height	View	Offset T	Irradiance
ppm	km		˚C	W/m2
280		70 km	Down	0	269.35
280		Surface	Down	0	382.14
280		Surface	Up	0	267.18
560		70 km	Down	0	266.4
560		Surface	Down	0	382.14
560		Surface	Up	0	270.73
560		70 km	Down	0.5	268.41
560		Surface	Down	0.5	384.65
560		Surface	Up	0.5	272.43

Table 1. Irradiance values calculated using the public portal for MODTRAN at the University of Chicago website.

As Willis states, the discrepancy can‘t be the result of long term feedbacks. Indeed, something a bit surprising comes from considering two necessarily nearly instantaneous feedbacks which take the disturbance of a doubling of CO₂ back toward equilibrium. Table 1 shows the pertinent details of a series of experiments done with MODTRAN, which tell this interesting story.

First, let‘s use a 1976 Standard Atmosphere model with a surface temperature of 288.2K and 280ppm of CO₂. MODTRAN calculates an upward irradiance at the top of atmosphere (70 km) of 269.35W/m². Let’s call this an equilibrium baseline state. This output irradiance is exactly what is needed to balance energy considering solar irradiance of 1370W/m² and some assumed albedo.

If we now disturb this by suddenly doubling CO₂, MODTRAN keeps the surface temperature and the atmospheric temperature distribution constant, and calculates a new irradiance at the top of atmosphere of 266.4W/m². The difference of 3W/m² is the new forcing–or almost. MODTRAN keeps all sorts of things constant, but in a real atmosphere and surface several of these things cannot remain constant. Neglecting long term feedbacks, we still have to consider an almost instantaneous effect. Within a week or two, the atmosphere will warm slightly because of the new absorptivity the increased CO₂ provides, and the surface will warm slightly because of new downwelling LW radiation.

To model this with what freedom the wrapper around MODTRAN allows, I will offset the surface temperature by 0.5^◦C. This is not a perfect rendition of what happens, but because such a large fraction of water vapor is close to the surface, it isn’t a terrible approximation either, and it illustrates what will happen.

As Table 1 shows, this reduces the 3W/m² difference at the top of atmosphere to 1W/m². In the long term other feedbacks will increase the top of atmosphere value to 269.35W/m² once again because this is what is needed to restore energy balance. At equilibrium the enhanced forcing from the doubling of CO₂, in fact from any disturbance involving CO₂ isn’t apparent at all at the top of atmosphere. This is an example of what Dr. Spencer meant when he stated that one can learn nothing about the feedbacks involved in a regulator through its output at equilibrium. It was also the point, partially of Nick Stokes‘s post.

Consider what Table 1 shows about goings on at the surface. The difference in downwelling between the two constant surface temperature runs is 3.5W/m², approximately equal to what we observe at the top of atmosphere. However, the model run at slightly higher surface temperature shows the difference in downwelling radiation is enhanced by immediate feedback and now different by 5.4 W/m². As longer term feedback kicks in the surface values will continue to adjust to eventually indicate the full enhanced greenhouse forcing. The new forcings are fully observed at the surface, not at top of atmosphere, which seems reasonable to me.

6. Conclusions

Whether this fully explains the departure between Willis‘s and Hansen‘s instantaneous values I cannot say. One can argue that the quick reaction of the atmosphere to a doubling of CO₂ is a feedback that should be excluded from consideration, but I would respond that it is so quick as to be completely different from something like water vapor feedback or melting glaciers. Also, people may argue about the equilibrium values of surface temperature or top of atmosphere radiation I apply. However, one has to begin with some assumptions and recognize that MODTRAN cannot fully handle conservation of energy except at the top of atmosphere because it is not equipped to account for the full range of heat transfer mechanisms at the surface. It is not a full-fledged heat transfer code.

In some future installment I plan to return to these points in connection with feedback because what one can determine about the internal workings of a system, at equilibrium or otherwise, depends to great degree on what one measures. The different response of upwelling and downwelling radiation at the surface versus at the top of atmosphere to a doubling of CO₂ illustrates this.

7. Notes:

The definitions and standard symbols for radiation can be found in the text Fundamentals of Heat and Mass Transfer, by Incropera and DeWitt, John Wiley and Sons, in any of its eight editions. In his well known text Physics of Atmospheres, Houghton uses the symbol F for irradiance rather than G, and appends an up or down arrow to indicate upward or downward flux. He also uses a radiant intensity for the blackbody function, calling it B. This is therefore the Stefan-Boltzmann law divided by π.

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