Theory of Redirected Energy

rev 1/24/22,9/15/22,1/19/23,6/30/23,8/15/23,3/7/24,4/23/24,4/26/24,5/24/24,10/12/24,1/23/26,2/3

1. Top-of-atmosphere

Top-of-atmosphere (TOA) graphs of radiation flux from earth show ‘notches’ 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 (line-of-sight). 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. (Graph is from NASA [1])

On average, radiation emitted 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). Actual emissivity for earth’s surface is closer to 0.98. The red trace on Figure 1 shows the Planck spectrum for a black body (perfect emitter/absorber, emissivity = 1) at 294 K (69.5 F).

Part of the power (flux, energy-rate) radiated from the surface goes directly to space through the ‘atmospheric window’ in the approximate wavenumber range 770-1230 (13-8.13 microns) where no gas molecules (except ozone, O₃) significantly absorb radiation. Over the rest of the total range of significant terrestrial radiation (wavelength 5-2000 microns, wavenumber 2000-5 cm^-1) the radiation is absorbed by Long-Wave-InfraRed (LWIR) active gases (which, including water vapor (WV), are misleadingly called greenhouse gases (ghg)).

Height of the black line trace is the flux (energy-rate) as a function of wave number at the TOA at a single line-of-sight. This line accounts for the power that radiates from the surface plus that from latent and convective heat from the surface and solar radiation absorbed by clouds and WV in the atmosphere. The vertical distance on the graph from a curve representing radiation from the earth’s surface to the black line indicates the amount at each wave number that the radiation at TOA (energy rate) is less than the equivalent energy rate radiating from the surface. Energy is conserved so the rate is what changes, i.e. the flow of energy is slowed down as it passes through the atmosphere. The slowing is proportional to the logarithm of the number of molecules of the responsible ghg.

2. Atmospheric structure

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 about 4 Angstroms; 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 (within a few nanoseconds) shared with surrounding molecules both ghg and non-ghg. The sharing is thermal conduction in the gas. The process of absorbing radiation energy and sharing it with surrounding molecules is called thermalization.

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, called decay time, has been calculated for CO2 and averages from about 10 microseconds at sea level up to about 1.1 second at TOA [2, 3]. Decay time is very long compared to the relaxation time of a nanosecond or so which is the time required for thermalization to take place. I suspect that, early in thermalization, nearby ghg molecules are stimulated to emit photons that might sum to energy-rate different from that absorbed.

Power in Watts is defined as energy-rate by W = J/s where J is energy in Joules and s is time in seconds. One Joule for one second is one W by definition.

3. Steady-State Power Balance

As a starting point in the analysis, the flow of power is determined partly using estimates from other sources, some rational refinements, and an approximation of the power loss. A graphic that was copied from [15] is useful as a qualitative description of the various power flows. It is shown in Fig 1.1 with some revisions. Some power flows are more explicitly defined than at the NASA site and the values presented there are replaced by symbols (alpha characters) with values determined herein. All power magnitudes are in watts per square meter.

4. Calculation of the Magnitudes of Power (Energy Flow Rate)

Measured values of power from the sun vary only slightly depending on the solar cycle and slight other long-term variations in the ‘solar constant’. The value used here is 1365 W/m^2 of insolation. The power from the sun is intercepted by the planet according to the cross-section area but is distributed according to the surface area so this is divided by 4 to account for the ratio of surface area to cross-section area of a sphere.

Q = 1365/4 = 341.25 W/m^2

Part of this incident power is reflected. The fraction of the incident power that is reflected is often assumed to be determined by the earth’s albedo which, from what appears to be a credible source [13], is about 0.297. However, albedo is primarily diffuse reflection and does not include Fresnel-type low-incident-angle specular reflection [14] from the 71% of the planet that is covered by water. The fraction of the total power reflected from the planet is slightly higher than that determined by albedo alone and is about 0.308 (my calculation).

R = Q * 0.308 = 105.105 W/m^2

The amount of power reflected from the clouds and atmosphere was determined as required to maintain the same ratio of cloud + atmosphere reflection to surface reflection as is reported by Kiehl and Trenberth [15].

A = R/ (1+23/79) = 81.4 W/m^2

Part of the remaining power is absorbed by clouds and atmosphere. I selected a value of 48% for this to be consistent with the absorbed power rate assessment by Kiel & Trenberth. An additional factor of 0.6 accounts for the observation that clouds cover about 60% of the planet. This is absorbed at the 1.34- and 1.87-micron (centered at wavenumbers 7463 & 5618) bands by water vapor and thermalized.

B = (Q-A) * 0.48 * 0.6 = 74.84 W/m^2

The incoming power that gets through the atmosphere is simply that which is not reflected or absorbed by the clouds and atmosphere.

C = Q - A - B = 185.01 W/m^2

The part of this that is reflected by the surface is simply that part of the total reflection that was not reflected by the clouds and atmosphere.

D = R - A = 23.71 W/m^2.

The power that is absorbed by the surface is the part of C (the entering power that got through the atmosphere) that is not reflected.

E = C - D = 161.31 W/m^2.

The power leaving the surface as convection (thermal), evaporation (latent) and radiation from the surface that directly leaves the planet are all as presented on the K & T chart except G which is based on average global precipitation.

F = 17 W/m^2

G = 78 W/m^2

J = 40 W/m^2

The radiation from the surface is gray-body radiation as calculated by the Stephan-Boltzmann equation. Most (71%) of the surface of the planet is covered by sea water with an emissivity of about 0.995 at water temperature. Part is from snow with emissivity of about 0.99 (clean) and the rest is from land that is mostly about 0.99 but with a few small local areas that are substantially lower. The over-all average emissivity from the surface is taken to be 0.98. The average global temperature is taken to be 288 K as typically reported. The S-B constant, σ, is 5.6697E-08 W/m^2 /T^4. (Temperature in degrees Kelvin)

U = 0.98 * 5.6697E-8 * 288^4 = 382.26 W/m^2

Clouds are fine particles of liquid water or ice and thus also radiate according to the S-B equation. Clouds cover about 60% of the planet and have an average emissivity [16] of about 0.5. Their average temperature was determined to be 258 K in an entirely different analysis that produced an average global temperature of 288K. The thin air and low temperature at high altitude means that there is very little water vapor so radiation up from the clouds nearly all gets directly to space. The small part that doesn’t is ignored resulting in:

P = 0.6 * 0.5 * 5.6697E-8 * 258^4 = 75.36 W/m^2

The same power flux as from the top of the clouds exists also from the bottom of the clouds. The down flux encounters substantial absorption prior to reaching the ground. K & T report 40 W/m2 or 40/396 = 10.1 % of the radiation making it from the surface through the ghg to space. A calculation using data from Barrett [17] for 60% cloud cover produces 10.8 %. Since about half of the source radiation is stopped by clouds in going from the surface to space and not stopped by clouds when going from clouds to the surface, the fraction getting to the surface is about 21% of that going from the surface directly to space.

K = 0.21 * P = 15.83 W/m^2.

The total power being radiated from the planet must be the amount from the sun minus the amount reflected.

M = Q - R = 236.10 W/m^2.

The power radiated from the atmosphere must be the total for the planet minus that just from the clouds to space minus that which goes directly from the surface to space.

N = M - P - J = 120.78 W/m^2.

Energy must be conserved so the rate entering the atmosphere must be slowed. The energy-rate (power) is slowed by about 33 ms due to ‘random walk’ of photons up through the atmosphere. This equates to a power loss of

H = (M – J)/ (1-0.033) –(M-J) = (236 – 40)/ (1.0.033) = 6.692 W/m^2

The net power that is returned to the surface is

I = U + G + F + B -M – H -K = 293.3 W/m^2.

Figure 1.1: Power flow in the atmosphere.

5. Power assessment for the atmosphere:

1. (75 W/m^2) The power entering the atmosphere from the sun (absorbed by WV in the 1.38- and 1.97-micron bands,

2. (78 W/m^2) 3. latent heat from hydrologic cycle,

3. (17 W/m^2) convection,

4. (382 W/m^2) radiation from the surface at emissivity of 0.98 and T = 288 K,

5. (˗293.3 W/m^2), back radiation to surface,

6. (˗16 W/m^2) radiation from clouds to ground thru atmospheric window,

7. (˗40 W/m^2) radiation from surface to space thru atmospheric window.

This all adds up to 202.7 W/m^2 of unreduced power. The photon energy travels from ghg molecule to ghg molecule in a ‘random walk’ up thru the atmosphere. The average path length of the ‘random walk’ is approximated by calculation [10] to be a million 10-meter steps. This path is traversed at the speed of light in the atmosphere in about 33 ms. This slows the energy-rate to 0.967 * 202.7 = 196 W/m^2 which matches the power leaving the atmosphere. Water vapor increase increases the path length of the random walk and contributes to the increase to the misleadingly named GHE and the increase in average global temperature.

6. Atmospheric water vapor

A common observation which shows that WV increase contributes to global warming is that cloudless nights cool faster and farther when absolute WV content of the atmosphere is lower. Clear nights cool faster and farther in the desert than where it is humid.

Below the tropopause (below about 8 to 16 km depending mostly on latitude; more at the equator) WV decreases from an average of about 8,000 ppmv (0.8%, 4% near the equator) at ground level, because of the low temperature (about negative 50 °C), to 203 ppmv at most, at the tropopause (assume about 12 km). Saturation vapor pressure of ice at ˗50 C/total pressure at 12 km = 3.94 Pa/19400 Pa = 0.000203 = 203 ppmv (at saturation). Thus, at the troposphere and above, radiation to space is mostly from CO₂ and other IR active molecules that do not condense in the atmosphere. Increased CO₂ in the extremely thin air there actually counters planet warming.

In addition to the decline in WV ppmv due to temperature decline, is the decline resulting from pressure decline with altitude of 19.4/101.1 = 0.191. The total WV molecule population gradient from surface to tropopause is thus about 8000/203 * 101.1/19.4 = 206 to 1.

Radiation from WV molecules below about wavenumber 600/cm can only be absorbed by other WV molecules. This steep WV molecule population gradient means that much of the outward directed radiation from WV molecules will make it all the way to space. This is demonstrated by the ‘hash’ in flux intensity observed in this wavenumber range measured at the top-of-atmosphere as shown on Figure 1. At about 2 km and higher, outward directed photons from WV molecules can make it all the way to space.

7. Thermalization and redirection

Thermalization causes some of the energy absorbed by CO₂ molecules to be redirected by gaseous thermal conduction to WV molecules. The Schwarzschild equation [12] accounts for this with “as radiation passes through an isothermal layer, its monochromatic intensity exponentially approaches that of blackbody radiation corresponding to the temperature of the layer.” Much of the radiant energy along a line-of-sight is ‘redirected’ with respect to wave number so CO2 molecules become transparent to it.

The redirection mechanism is summarized as follows: Much of the radiation energy that is absorbed by CO₂below the tropopause is shared with all molecules (thermalization) and emitted to space by WV molecules. This is discussed further in Sect 10 of [10].

Above the tropopause the remaining energy is conducted from non-ghg molecules to all ghg molecules for eventual radiation towards space. For lack of a better term, call the conduction of energy from non-ghg to ghg molecules and radiated from them reverse-thermalization.

Ghg in the warmed air can significantly 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.

8. Water Vapor Factor

Water vapor is a transparent gas that, molecule for molecule, is more effective at absorb/emit of earth-temperature infrared radiation (IR) than carbon dioxide (at low altitude, absorb lines for CO₂overlap). From Jan 1988 thru Dec 2024 NASA/RSS accurately measured and reported monthly the global average WV as Total Precipitable Water (TPW). The anomaly data are reported at [11]. The nominal value is about 29 kg/m^2 so the trend from Jan 1988 thru Dec 2024 is about 1.5 % per decade.

Given that at ground level average global WV is about 0.8% or 8,000 ppmv (parts per million by volume) (Ref 34, 35 of [10]), the increase in WV molecules in 3.6 decades is about 0.015 * 8000 * 3.6 = 432 ppmv. From Mauna Loa data at [9] the CO₂ increase in that time period is 423 - 349 = 74 ppm. Per ideal gas laws, ppm = ppmv. With that, at ground level, WV molecules have been increasing 432/74 = 5.8 times faster than CO₂ molecules. Thus, at ground level, regardless of the initial source of warming, WV molecules have been increasing about 5.8 times faster than CO₂ molecules. The idea that CO₂ starts the increase is ludicrous considering the comparatively large fluctuations in surface temperature.

Radiation from WV molecules can be in any direction but, because of the steep decline with altitude of the population gradient of WV molecules, the distance traveled by a photon before it encounters another WV molecule is greater towards space than towards earth so the prevailing direction of IR flux is towards space. This is shown on a Top of Atmosphere (TOA) graph of radiation flux vs wavenumber by the jagged line below about wavenumber 600. Because of the characteristic absorb/emit signature of every gas, no other gas can significantly absorb or emit radiation in the wavenumber range occupied by WV. The line is jagged because radiation that reaches TOA/space is from WV molecules at different temperatures/altitudes.

9. Top-of-atmosphere with black body overlay

Fig 1.5 is a TOA graph with overlaid constant temperature curves for black body radiation at noted temperatures. For any specified atmosphere, these are also constant elevation curves. The temperature and associated altitude for standard atmosphere are shown in the upper right corner of Fig 1.5.

Figure 1.5: Typical TOA radiant emission. (U Chicago version of MODTRAN [8]

At about 2 km and higher, the outward directed radiation from WV can make it all the way to space. Below the tropopause, much of the energy absorbed by CO₂ and other IR active molecules is redirected with respect to wave number via thermalization to WV molecules. This mitigates any warming from increased CO₂ in the troposphere (or any other IR active gas that does not condense at earth temperatures).

The end result is that CO₂ does not cause significant climate change, sequestering it is an expensive mistake and the Green New Deal would have no significant effect on climate.

10. 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 WV change rate and atmospheric temperature vary with altitude for different latitudes and seasonal conditions as shown in MODTRAN6 documentation [7].

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

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. At low altitude the tiny amount of energy absorbed by CO₂ and much greater amount absorbed by WV are thermalized; contributing to warming the low altitude atmosphere.

The WV content of the atmosphere diminishes rapidly as the temperature decreases with increasing altitude. Above the tropopause it has declined to a level where emission from WV ceases to dominate and emission from CO₂ molecules becomes significant. The result is most of the residual energy not emitted to space by WV at low altitude is, at high altitude, redirected back to the wavenumber range 600-740 cm^-1 and emitted to space by CO₂. The ’redirection’ is not geometric because all wavenumbers refer to photons at essentially the same line-of-sight.

Energy redirection helps explain why average global temperature tracks average global WV and not CO₂ [6].

References:

1. NASA/GISS TOA graph https://www.giss.nasa.gov/research/briefs/2010_schmidt_05/

2. Average elapsed time to emit a photon https://sealevel.info/Happer_UNC_2014-09-08/Another_question.html

3. Average elapsed time to emit a photon http://rabett.blogspot.com/2013/04/this-is-where-eli-came-in.html

4. Time between gas molecule collisions http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/frecol.html

5. MODTRAN6 calculator http://modtran.spectral.com/modtran_home#plot

6. Climate change drivers http://globalclimatedrivers2.blogspot.com

7. MODTRAN6 defaults http://modtran.spectral.com/modtran_faq

8. MODTRAN calculator http://climatemodels.uchicago.edu/modtran/

9. Mauna Loa data for CO2: https://www.co2.earth/monthly-co2

10. Water Vapor vs CO2 for Planet Warming, Sect 1 of: https://watervaporandwarming.blogspot.com