The greenhouse effect occurs in the atmosphere. It can't be described by a few simple equations. The science of climate is not well developed. Climate models don't work.
The paper described in this summary is on estimating the change in the greenhouse effect occurring in the atmosphere with additions of carbon dioxide. It has 15 figures, 5 tables and 92 equations, many of them requiring a strong command of integral calculus.
Physicists William Happer and William van
Wijngaarden determined the current levels of atmospheric carbon
dioxide and water vapor are almost completely saturated. In radiation
physics, “saturated” means adding more molecules will not cause more warming.
Selected quotes follow:
The Abstract:
“The atmospheric temperatures and concentrations of Earth’s five most important, greenhouse gases, H2O, CO2, O3, N2O and CH4 control the cloud-free, thermal radiative flux from the Earth to outer space. ... (cooling off the planet)
" Doubling the current concentrations of CO2, N2O or CH4 increases the forcings (global warming) by a few percent. These forcing results are close to previously published values even though the calculations did not utilize either a CO2 or H2O continuum."
...The W & H paper covers the five most important greenhouse gases, examined from the standpoint of a cloud free atmosphere. Clouds reflect some of the incoming radiation from the sun, but also reduce the outgoing radiation from earth to space. From the greenhouse perspective, the nighttime warming effect of clouds is eliminated, and the greenhouse effect is considered alone.
... Using the language common to those who study this field, both CO2 and water vapor are saturated; increasing these gases will not have a significant impact on the climate of the earth. Such an effect can be described by a logarithmic function ..."
Dependence of Earth’s Thermal Radiation
on Five Most Abundant Greenhouse Gases
https://arxiv.org/pdf/2006.03098.pdf
W. A. van Wijngaarden
Department of Physics and Astronomy,
York University, Canada,
and
W. Happer
Department of Physics,
Princeton University, USA
June 8, 2020
"The temperature record from 1850 to the present shows the average surface temperature of the Earth has increased by about one degree Celsius. The Intergovernmental Panel on Climate Change (IPCC) attributes most of this temperature rise due to increasing greenhouse gas concentrations associated with anthropogenic activity. The average concentration of CO2 in the atmosphere has increased from 280 ppm to over 400 ppm largely due to the combustion of fossil fuels. Concentrations of N2O and CH4 have also risen substantially since the start of the industrial revolution."
... This paper examines the effect of greenhouse gas concentrations on thermal radiation for the case of a clear sky. It considers the five most important naturally occurring greenhouse gases: H2O, CO2, O3, N2O and CH4."
... Radiative forcing depends strongly on latitude ... Near the wintertime poles, with very little water vapor in the atmosphere, CO2 dominates the radia-tive forcing. The radiation to space from H2O, CO2 and O3 in the relatively warm upper atmosphere can exceed the radiation from the cold surface of the ice sheet ...
... at current concentrations, the forcings from all greenhouse gases are saturated. The saturations of the abundant greenhouse gases H2O and CO2 are so extreme that the per-molecule forcing is attenuated by four orders of magnitude with respect to the optically thin values. Saturation also suppresses the forcing power per molecule for the less abundant greenhouse gases, O3, N2O and CH4, from their optically thin values, but far less than for H2O and CO2.
... the overlap of absorption bands of greenhouse gases causes their forcings to be only roughly additive. One greenhouse gas interferes with, and diminishes, the forcings of all others. But the self-interference of a greenhouse gas with itself, or saturation, is a much larger effect than interference between different gases. Table 4 shows that for optically thin conditions, the forcing power per molecule is about the same for all greenhouse gases ... "