How "Greenhouse Gases" are Mathematically Contrived
The claim that greenhouse gases add heat to the atmosphere hinges upon the Stefan-Boltzmann constant. The result is so absurd that corrections had to be made later (explained below).
The Stefan-Boltzmann constant indicates how much radiant heat is given off by a surface at a particular temperature. Simple observations indicate that about 20 times too much radiation is projected by the Stefan-Boltzmann constant. Without such excessive radiation, the claimed greenhouse effect would not exist. Since numerous other lines of logic also show that the greenhouse effect cannot exist, they show the absurdity of the quantities produced by the Stefan-Boltzmann constant.
The Stefan-Boltzmann constant indicates that so much radiation is given off by the surface of the earth that it would give up all energy from the sun at an absurdly low temperature of -18°C (0°F), if it all radiated into space. But since the average temperature of the earth's surface is 33°C higher (15°C or 59°F), only greenhouse gases could add the extra heat needed to get to present surface temperatures, according to global warming alarmists.
With the Stefan-Boltzmann constant used by climatologists, the surface of the earth must be giving off 390 watts per square meter of radiation at its average temperature of 15°C (59°F). To get their numbers to balance, climatologists have only 24 W/m² leaving the surface by conduction and convection. That's 6% as much conduction and convection as radiation, even though the earth's surface has a lot of wind moving across it. Cooling fans would never be used if only 6% increase in cooling could be achieved. Fans remove far more heat than radiation alone would.
Based on the Stefan-Boltzmann Constant
This is the Kiehl-Trenberth model. (Jeffrey T. Kiehl, Kevin Trenberth (1997): Earth’s Annual Global Mean Energy Budget; in Bulletin of the American Meteorological Society, Vol. 78, No. 2/1997, S. 197-208)
The Stefan-Boltzmann constant is this:
W/m² = 5.67051 x 10-8 x K4
With the Stephan-Boltzmann constant, the surface of the earth must be giving off 390 watts per square meter of radiation at its average temperature of 15°C
Radiation from sun onto earth's surface:235 - 67 = 168 W/m².
Radiation from atmosphere to earth: 324 W/m².
Total on earth's surface: 324 + 168 = 492 W/m².
From surface by conduction (air rising): 24 W/m².
From surface by evaporation: 78 W/m².
From surface as radiation: 390 W/m².
Of the 390 W/m²: 40 W/m² directly into space.
Net radiation from surface to atmosphere: 350 - 324 = 26 W/m².
Net energy from surface to atmosphere: 24 + 78 + 26 = 128 W/m².
From sun to atmosphere: 67 W/m².
Emitted from atmosphere to space: 128 + 67 = 195 W/m².
Total into space: 195 + 40 = 235 W/m².
Alarmist climatologists use this procedure to show that the numbers can be balanced when using the Stefan-Boltzmann constant, and the greenhouse effect is supposed to be a necessary method of getting the surface temperature up from the -18°C which liberates 235 W/m², based on the Stefan-Boltzmann constant, to 15°C.
To account for the extremely high radiation indicated by the Stefan-Boltzmann constant, there had to be a lot of radiation interacting with the earth's surfacespecifically 324 W/m² going from the atmosphere back to the surface. This amount left almost nothing for conduction, convection and evaporation. The 390 W/m² being emitted from the surface included 40 W/m² going into space and 350 W/m² going into the atmosphere. The 324 W/m² coming back out of the atmosphere and onto the surface had to be less than the 350 W/m² going in. The 324 W/m² left almost no space for conduction, convection and evaporation, because most of it had to be used to create the 390 W/m².
An important thing to notice about alarmist science is how sloppy everything is. Throughout the subject, there are contradictions. That isn't how science is supposed to work. When things don't look right, you find out what the problem is. You don't say the science is settled. Climatologists pushed themselves into a corner with fake numbers and false claims, and they can't remove the resulting contradictions.
Balancing ridiculous numbers was more important to alarmist climatologists than a credible logic. Throughout the global warming issue, logic is sacrificed to absurd claims and fake mathematics including falsified data. It also means physicists made up the Stefan-Boltzmann constant off the top of their heads with no relationship to objective reality. It rationalizes fake math and numbers for greenhouse gases, which requires a lot of radiation, but contradicts logic and evidence.
An Approximate Correction
If the Stefan-Boltzmann constant were reduced to one twentieth, at 15°C (59°F) only about 5% of the heat leaving the earth's surface would be radiation, while the remainder would leave as conduction, convection and evaporation, which is more in line with what is really happening. No greenhouse gas effect would be involved.
Instead of 390 W/m² radiation given off by the earth's surface at the average global temperature of 15°C, the amount would be about 20 W/m². About half would go into the atmosphere and half around the atmospheric gases and into space, which is 10 W/m² in each.
15°C surface: 20 W/m² radiation = 10 into space, 10 into atmosphere.
Sun's energy onto surface: 168 W/m².
Energy from the sun would heat up the surface and air around it through conduction, convection and evaporation.
Claimed from sun to atmosphere: 67 W/m², then back into space.
From atmosphere to space: 158 + 67 = 225 W/m².
Same up and down: 225 W/m² onto surface.
Total into space: 225 + 10 = 235 W/m².
Total onto surface: 168 + 225 = 393 W/m².
Conduction, convection and evaporation from the surface:
Total into atmosphere: 373 + 10 + 67 = 450 W/m².
Total leaving atmosphere: 225 up, 225 down = 450 W/m².
This means conduction, convection and evaporation heat the atmosphere at 373 W/m² with no greenhouse effect involved.
The usual claim is that half of the radiation from the atmosphere goes upward into space and half goes downard onto the surface. One reason why less would go downward is reflection from the surface. As the angle increases, more reflection occurs. How much should that be? Maybe 30%. Then radiation onto the surface would decrease from 225 to 158, and conduction, convection and evaporation would decrease from 373 to 261. Due to complexities, there is no science which will say exactly what the numbers should be.
A second reason why there might be less radiation downward than upward is that the absorbing gases create a barrier to motion for half of the radiation from the atmosphere. Convection (with turbulence) carries ground heat rapidly upward about 300-500 meters (and farther upward more slowly), while absorbing gases only allow half of the radiation to go about 5-50 meters. This could reduce downward radiation by another 20-40%.
The 10 W/m² radiating from the surface of the earth and being picked up by molecules in the atmosphere do not create a greenhouse effect, because there is no difference between heat entering the atmosphere through radiation and that entering through conduction, convection and evaporation. The heat entering from one method is subtracted from heat entering by another source. All heat is the same. Explained Here.
Attempts at Refinement
The NASA Energy Budget" indicates that 30% of the sun's energy is reflected back into space without heating the earth. That's 70 w/m² of the 235 w/m² sent to earth by the sun. We will use the NASA concepts for energy entering the earth, as these numbers are much more realistic than those of the Kiehl-Trenberth model.
One of the concerns is that the amount of radiation from the atmosphere onto the surface of the earth must be reduced substantially. Reducing this amount is in line with the SBC being way too large.
Only 120 w/m² gets to the surface of the earth from the sun, as indicated by the NASA chart. The amount radiating onto the surface from the atmosphere (at night) should be much less, so we give it a factor of 70 w/m². This amount is much less than the 164 w/m² radiating from the atmosphere into space. This difference is realistic due to reflection, convection upward and thinner air upward. Convection distances are short (approximately one km), but saturation is much shorter (100 m for all greenhouse gases). Very significant is that the upper atmosphere is lower pressure (one tenth as much at the top of the troposphere), which means radiation can move upward and escape ten times easier than it can flow downward.
Radiation from the surface is 1.6% as much as conduction and evaporation (3/187), which is realistic for radiation from such a cold surface (15°C, 59°F).
There is no analysis which says what temperature should result from energy moving around. The temperature equilibrates, and only measurement tells what it does.
The only basis for claiming that a greenhouse effect occurs is an absurdly high amount of radiation indicated by the Stefan-Boltzmann constant which cannot be explained in any other way. Without such extreme radiation, no explanation is needed.
The Kiehl-Trenberth model was produced in 1997. The NASA Energy Budge appears to have been produced in 2007. They cover the same subject but are vastly different. The main problem is that Kiehl and Trenberth were forced to show radiation to be 79% of the energy leaving the surface of the earth, with conduction only 5%. Nothing could be more preposterous. Any known material would vaporize before 16 times as much energy would leave as radiation compared to conduction in air.
This absurdity was forced onto the analysis by the Stefan-Boltzmann constant. When an apparent correction was made, the SBC was not used, but still, 41% of the energy leaving the surface of the earth was indicated to be radiation, with 14% as conduction, which is still 3 times as much radiation as conduction. It's unlikely that any white hot metals would do that, even if no air blows across them.
The entire concept of greenhouse gases creating global warming depends upon the SBC in several places. The Kiehl-Trenberth model was a simple application of the SBC. There was no choice but to do so, as such constants cannot lie; they are embeded in a large part of physics. Why then did the NASA analysis turn the SBC into a lie? Probably because the lie was too incredible to look at. Would not someone have noticed the discreptancy? Some critics did. But being invisible to the public, whistle blowing is a losing battle, as it is everwhere else. In fact, there is nothing but contradictions and discrepancies in the subject of global warming, as applying science to the subject is impossible, and all results are contrived by starting at the desired end point.
One of the things these energy flow-charts show is that the Stefan-Boltzmann constant is too absurdly high to apply to the subject.