Wednesday, February 14, 2018


Top-of-atmosphere (TOA) graphs of radiation flux from earth show ‘notches’ (‘ditches’?) centered at the nominal wavenumbers of long wavelength infrared (LWIR) active gas molecules, mostly CO2, which do not condense in the atmosphere. Figure 1 appears to be a typical mid-latitude TOA graph. Radiation flux intensity is plotted vs. wavenumber at any one geometric location. Wavenumber is simply the number of radiation wavelengths per centimeter. Wavenumber, cm-1 and wavelength in microns are easily converted: Divide 10,000 by either to get the other.


Figure 1: Typical mid-latitude TOA emission of radiation from earth. (original graph is from NASA [1])


Areas on these graphs represent power which for any elapsed time is energy. If there was no CO2, the average height of the black curve between 600 and 740/cm would be approximately 330 mW/m2 which results in an area of about 46 W/m2. The notches represent, for any time increment, energy that is ‘missing’. The first law of thermodynamics mandates that energy cannot simply disappear so where (and when) did that energy go?

On average, radiation from the liquid and solid surfaces of the earth is very closely described by the Planck spectrum for a black body at 288 K (15° C, 59° F) (The colors you see are reflected sunlight). The red trace on Figure 1 shows the Planck spectrum for a black body at 294 K. Part of this radiated energy goes directly to space through an atmospheric ‘window’ in the approximate wavenumber range 770-1230 (13-8.13 microns) where no gas molecules (except O3) absorb radiation. Over the rest of the range of significant terrestrial radiation (6.5-200 microns, wavenumber 1538-50 cm-1) the radiation is absorbed by LWIR active gases (which, including water vapor, are misleadingly called greenhouse gases (ghg)).

Logic mandates that the elapsed time between when a molecule absorbs a photon and when it emits one must be more than zero or there would be no indication the photon had been absorbed. This elapsed time is called the relaxation time. Experiments [2, 3] have determined the relaxation time is shorter at higher temperatures. It is about 5 microseconds for CO2 in the atmosphere where people are.

At the scale of atoms, the atmosphere can be visualized as molecules bouncing elastically (no energy loss) off each other in empty space. At sea level conditions, the time between collisions for the air molecules (molecule diameter 4E-10 m) is extremely short, less than 0.0002 microseconds [4]. Among other properties, these collisions are the basis for thermal conduction in the gas. Therefore, electromagnetic radiation (EMR) energy absorbed by ghg molecules is immediately shared with surrounding molecules both ghg and non-ghg. The sharing is thermal conduction in the gas. The process of absorbing radiation and sharing it with surrounding molecules is called thermalization.

A common observation of thermalization by way of water vapor is cloudless nights cool faster and farther when absolute water vapor content of the atmosphere is lower. Clear nights cool faster and farther in the desert than where it is humid.

At high altitude (above about 20 km), energy is conducted from non-ghg molecules to ghg molecules for radiation towards space. For lack of a better term, call it reverse-thermalization.

Ghg in the warmed air can emit photons only at a limited number of wavelengths (or wavenumbers) characteristic for each molecule species. Furthermore, all theoretically possible wavenumbers are not equally likely.

Radiance calculated by MODTRAN6
MODTRAN6 [5] is a computer program developed for the Airforce Research Laboratory which (besides other things) can calculate the radiation flux at selected elevations in the atmosphere for specified constituents and conditions. It contains default values for several environments including the tropics and the 1976 Standard Atmosphere. Values for water vapor change rate and atmospheric temperature vary with altitude for different latitudes and seasonal conditions as shown in MODTRAN documentation [7].

Figures 2 and 3 are typical graphs showing how radiation flux absorbed by ghg other than water vapor (WV) is redirected to WV with increasing altitude. The redirection is quantified by the progressively increasing depth of the ‘notches’ at the characteristic wavenumber ranges for each ghg (except WV). Note that, for CO2, the redirection at 20 km is greatest and at 50 km and higher some of the flux returns to the CO2 molecules. Somewhere in the vicinity of 20 km some of the energy is redirected back to the CO2.

The ‘notches’ are evidence of energy redirection. Redirection is possible because of thermalization and reverse-thermalization in the gaseous atmosphere.


Energy absorbed by ozone begins to be significantly redirected to WV above 10 km, reaches a maximum below 50 km and stays at the maximum level above 50 km.

 Figure 2: 1976 Standard Atmosphere at MODTRAN6 default values.


Figure 3: Tropics atmosphere at MODTRAN6 default values.


Figures 2 and 3 are misleading; especially below about 10 km. Approximately 161 W/m2 of solar energy reaches the solid and liquid surfaces of the planet [8]. A few meters above the surface, the energy leaving the surface includes about 71 W/m2 from heat of vaporization of water (annual rainfall averages about a meter and what comes down had to have gone up). Another 17 W/m2 has been added by convective heat transfer, leaving 161 – 71 – 17 = 73 W/m2 in thermal radiation instead of the 269 W/m2 assumed at all altitudes by MODTRAN6 and shown in Figures 2 and 3.

With increasing altitude, the non-radiant flux is replaced with radiant flux and the solar energy that was absorbed by the atmosphere and clouds is incorporated. Most of this takes place by about 10 km so the graphs at 20 km and higher should be reasonably valid.

Most of the photons emitted by the water vapor molecules are at wavelengths different from the comparatively narrow band that CO2 molecules can absorb. Effectively, much of the terrestrial thermal radiation energy absorbed by CO2 (and other non-condensing ghg) is thermalized, redirected to, and radiated to space from water vapor.

At very high altitudes, temperature, molecule spacing and time between collisions increases to where reverse-thermalization to CO2 (and O3) molecules becomes significant as does radiation from them to space.

Figures 2 and 3 answer the question posed above of where (and when) the energy went which is ‘missing’ at the notches in TOA graphs. In Figure 1, the 18 W/m2 ‘notch’ is permanently redirected to water vapor and the 28 W/m2 ends up in the 600-740 cm-1 wavenumber range. At low altitude the tiny amount of energy absorbed by CO2 and much greater amount absorbed by water vapor are thermalized warming the low altitude atmosphere.

The water vapor content of the atmosphere diminishes rapidly as the temperature decreases with increasing altitude. At about 20 km it has declined to a level where emission from water vapor ceases to dominate and emission from CO2 molecules becomes significant. The result is part of the energy which had been redirected to water vapor at low altitude is, at high altitude, redirected back to the wavenumber range 600-740 cm-1. The ’redirection’ is not geometric because all wavenumbers refer to photons at essentially the same physical location.


References:
2. Relaxation time vs 440 & 816 K https://pure.tue.nl/ws/files/3478579/109243.pdf