Basic Model of Greenhouse Effect: Climate Sensitivity=0.2 C

Model of an Earth-atmosphere system absorbing energy from the Sun, which is partially transferred to the ground surface thermodynamically and emitted to outer space partially from the ground through an "atmopsheric window" and partially from the atmosphere. The effect of "closing the window" is at most 2 C as the maximal "total greenhouse effect" of all "greenhouse gasses", thus  with a "climate sensitivity" as a 10% change from doubled CO2 emission of at most 0.2 C, which is so small that it can never be observed. The model indicates that CO2 global warming alarm of 3 C is false alarm. 

The standard basic model of the Earth's energy budget predicts a "total greenhouse effect" of 33 C as the difference between the presently observed global Earth surface temperature of + 15 C and a projected temperature of - 18 C of an Earth with fully transparent atmosphere without any "greenhouse gas", or an Earth without any atmosphere at all like the Moon, cf. PS2 below.

The projection of - 18 C results from a direct application Stefan-Boltzmann's law as the radiative equilibrium temperature of a black body (or grey body) at the same distance to the Sun as the Earth.

This model is extremely simplistic but yet is used as a starting point for projections of alarming global warming from small changes in the radiative properties of the atmosphere by human emission of CO2. A 10% change of  a "total greenhouse effect" of 33 C is presently sending an alarm of 3 C propagated to humanity by IPCC.

Following Einstein one should always seek a model which is as simple as possible, but not too simple. Viewing the above 1-stage model as too simple, we consider instead the following 3-stage model of an Earth-atmosphere system "without greenhouse gas effect":

1. Radiative absorption of 240 W/m2 out of incoming 340 W/m2 from the Sun with absorptivity 0.7 by Earth+atmosphere system.
2. Thermodynamic transfer of 240 W/m2 from Earth+atmosphere to Earth ground surface (without loss).
3. Radiative emission of 240 W/m2 from ground directly to outer space through "fully open atmospheric window" with emissivity 0.7 at 280 K = 7 C according to Stefan-Boltzmann $240 = 0.7\times\sigma\times 280^4$ with $\sigma = 5.67\times 10^{-8}Wm^{-2}K^{-4}$.  

We compare with present observation of ground temperature of 15 C with emission of 40 W/m2 directly from the ground to outer space, with thus "1/6-open atmospheric window".

Extrapolation to fully closed window, then predicts a ground temperature of 15+ 8/5 C, say 17 C.

The total effect of closing the present atmospheric window of 40 W/m2 by massive emission of a greenhouse gas, thus could at most cause a warming of 2 C.

We have thus with a less simplistic model arrived at a "total greenhouse effect" of 2 C instead of the 33 C by the too simple model.

Incidently, this connects to the so-called "2-degree goal", but now with the 2 C as the maximally possible "total greenhouse effect", with thus 10% changes of size 0.2 C as an upper estimate of climate sensitivity as global warming by doubled CO2. Reducing 33 C to 2 C reduces global warming alarm to nothing.

The maximal change of global temperature under varying radiative properties of the atmosphere including clouds and "greenhouse gasses", could thus be estimated to be at most 2 C. This conforms to the observed temperature variation during the last 10.000 years after the last ice age, see below where the alarming dotted 3 C prediction reflects the too-simple-model.

Note that the crucial element in the not-too-simple model is the thermodynamic step 2 separating emission from absorption resulting in a double reduction with absorptivity=emissivity= 0.7, to be compared with one reduction with absorptivity=emissivity in the too-simple-model.

Notice further that the thermodynamics including lapse rate et cet is included in the observation of 15 C with 5/6 closed window.  The model thus isolates the crucial role of the "greenhouse effect" in the CO2 global warming scenario through a variable "atmospheric window".

PS1 The too-simple-model is used to predict the "total greenhouse effect" to be 33 C and is also used for the basic prediction of CO2 global warming of 1 C by Stefan-Boltzmann from an estimated "radiative forcing" of 4 W/m2 from doubled CO2 from an estimated 10% broadening of the "ditch" of the CO2 OLR-spectrum.  For a comparison with the not-too-simple model, see the next post.

PS2 The max and min temperatures of the Moon at the equator is 390 K and 120 K, respectively, which gives a mean of about 255 K = - 18 C.