• Equation (5)
• Effective cloud area fraction
• Cloudy energy flow system
• Cloudy equations
• Decomposition of all-sky radiations
• Decomposition of convection
• Cause of warming?
  

But we know that in the Earth's clear-sky part, there is a sizeable imbalance, about 20 Wm-2 in the annual global mean. 

In its original form of Eq. (2), for example in Houghton (1977, Eq. 2.15), there is 2ϕ/π on the right-hand side:

being equal (in equilibrium) both with the incoming and the outgoing radiation. When we took it at the optical depth of two, χ₀* = 2, we got our Eq. (2), πBg = 2ϕ, and we took it as being equal to 2OLR. 

Same as in Goody and Yung (1989, Eq.9.5):

where F_S denotes the solar flux, equal to the effective emission.

But why we use 2OLR? Why our equation doesn't look like this 

πBg = Surface absorbed (SW + LW) = 2ASR?

Well, beacues there's an open atmospheric window, where a part of the surface LW emission escapes to space without eing absorbed by the atmosphere, and only LWCRE is gained back by the presence of clouds in the all-sky.

Therefore

Surface absorbed SW + LW ≠ 2ASR, 

but 

Surface absorbed SW + LW = 2ASR – WIN + LWCRE = 2OLR.

So we have a relationship:

2ASR = 2OLR + WIN – LWCRE           (Eq. 5)

In the clear-sky, as we have seen above from the detailed LBL-computation of Costa and Shine (2012), WIN (clear) = 65 Wm-2 with their model-OLR = 259 Wm-2, which, with our theoretical OLR(clear) = 266.8 Wm-2 would lead to WIN(clear) = 65 × (266.8/259) = 66.96 Wm-2. Notice that 66.7 Wm-2 is an integer position, 10/4 (on the sphere), that is 10 on the disk (where OLR (clear) = 40 and LWCRE = 4).

Therefore, in the clear-sky, Eq. (5) says:

2ASR(clear) = 2 × 40 + 10 – 4 = 86 on the disk, leading to

2 × 43/4 = 2 × 40/4 + 10/4 – 4/4 on the sphere. 

that is, ASR(clear) = 43/4 on the sphere = 286.81 Wm-2 and TOA Net clear-sky Imbalance = EEI (clear) = 3/4 = 20.01 Wm-2 (with the usual 1 = 26.68 Wm-2). Equation (5) provided us with the correct values of absorbed solar and TOA Net clear-sky imbalance, using only OLR and WIN as input parameters.

In the all-sky, if WIN (all) = LWCRE, in Eq.(5) we have ASR(all-sky) = OLR (all-sky), as ought.

(Our suggested theoretical interpretation is given here)

This equality, WIN(all) = LWCRE, as a consequence of Eq. (5), if true, has several technical (arithmetic) implications.

First, an effective cloud area fraction follows.

Under "cloudy" we always mean here the effective (IR-opaque) cloud area fraction.

WIN(all) = βeff × WIN(cloudy) + (1 – βeff) × WIN(clear) 

WIN(cloudy) = 0 by def.,

WIN(all) = 1, WIN(clear) = 10/4 =>

βeff = 3/5 = 0.6.

Note that Costa and Shine (2012) found WIN(all) = 22 ± 4 Wm-2, same as assuming an observed cloud area fraction βobs = 0.67.

WIN(all) = ( 1 – 0.67) × 65 = 21.45 Wm-2 in Costa and Shine (2012).

But evidently, the observed cloud area fraction is not opaque, so our proposed WIN(all) = 26.68 Wm-2 from Eq. (5) is not irrealistic.

From here, using the integer values, an idealized cloudy energy flow system can be derived, keeping in mind that the shortwave fluxes should be normalized to the effective (IR-opaque) cloud area fraction.

Cloudy energy flow geometry:

Decomposition of OLR(all-sky), G(all-sky) and convection into their clear-sky and cloudy contributions:

OLR (clear-sky)  = 266.80 Wm-2 = 10 units (= 6 cloudy units)

Clear-sky area fraction = 1 - βeff  = 2/5 = 0.4

Clear-sky contribution to OLR (all-sky) = OLR(clear-sky) × (1 - βeff) = 106.72 Wm-2 = 4 units

OLR(cloudy-sky) = 5 (cloudy) units = 222.33 Wm-2

Cloudy contribution to OLR (all-sky) = OLR(cloudy-sky) × βeff  =133.40 Wm-2 = 5 units

OLR(clear-sky) – OLR(cloudy-sky) = 1 (cloudy) unit = LWCRE (total).

G(all-sky) = ULW– OLR(all-sky) = 160.08 Wm-2 = 6 units

G(clear-sky) = ULW – OLR(clear-sky) = 133.40 Wm-2 = 5 units

G(cloudy-sky) = ULW – OLR(cloudy-sky) = 177.87 Wm-2 = 4 (cloudy) units

Clear-sky contribution: G(clear-sky) × (1 – βeff) = 133.40 × 0.4 = 2 units = 53.36 Wm-2.

Cloudy-sky contribution: G(cloudy-sky) × βeff = 177.87 × 0.6 = 4 × 0.6 = 4 units = 106.72 Wm-2.

The cloudy atmosphere contributes twice to the all-sky greenhouse effect than the cloudless part.

Clear-sky net radiation at the surface = clear-sky non-radiative fluxes: 5 units = 133.40 Wm-2.

Cloudy-sky net radiation at the surface = cloudy-sky non-radiative fluxes: 2 units = 88.93 Wm-2.

Clear-sky contribution = 133.40 × 0.4 = 53.36 Wm-2

Cloudy-sky contribution = 88.93 × 0.6 = 53.36 Wm-2.

The clear-sky and the cloudy parts of the atmosphere contribute by the same amount to the all-sky convective fluxes.

If we assume that the clear-cloudy decomposition goes with the area fractions for the components of the convective flux separately, we would have:

All-sky evaporation = 80.04 Wm-2 = 0.4 × LH(clear) + 0.6 × LH(cloudy), 

with the above clear and cloudy convections we would find

LH(clear) =  SH(clear) =  5/2 units = 66.70 Wm-2.

LH(cloudy) = 2 units = 88.93 Wm-2; SH(cloudy) = 0.

In the cloudless atmosphere, evaporation and turbulent convection contribute the same amount to surface non-radiative cooling, while in the cloudy region both convection and evaporation contribute to cloud formation, but under the closed cloud deck, evaporation is the energy source with uplifting thermals practically zero. This is valid only in the global mean, if valid at all, 

Finally, the cloud-covered part of the surace radiates ULW × βeff = 400.20 × 0.6 = 240.12 Wm-2 energy, being equal to OLR(all-sky),

and the cloud-free surface radiates 400.20 ×0.4 = 160.08 Wm-2, being equal to the greenhouse effect.

On the distributions below, cloudy shortwave fluxes are weighted to a 3/5 effective cloud area fraaction.

With this assumption, the all-sky energy flow system is presented as the area-weighted sum of the clear and cloudy systems, each expressed in their own units:

This is the geometric background of planetary emissivity = 0.6, mentioned in the opening page (referred to Lennart Bengtsson, 2012).

IPCC AR6 data (WG1 Chapter 7 Fig. 7.2, "representing climate conditions at the beginning of the 21st century") verify the theory:

ULW = 398 Wm^-2, OLR(all-sky) = 239 Wm^-2 hence

planetary emissivity = 239/398 =  0.6005

greenhouse factor = (398 - 239) / 398 = 0.3995.

These values are observationally indiscernible from the geometric emissivity of 3/5 = 0.6 and greenhouse factor 2/5 = 0.4. 

No deviation, no enhanced greenhouse effect.

Notice also that reflected solar = 100 Wm^-2 and incoming solar = 340 Wm^-2 in that figure result in a planetary albedo of 100/340. 
This number (5/17) is identical to our theoretical geometric albedo of 15/51.

Cause of warming?

If the greenhouse effect seems to work under physical constraints, what migh tbe the cause of the observed warming?

"The changing nature of Earth's reflected sunlight" by Graeme Stephens, Maria Hakuba, Seiji Kato and others, Proceedings of the Royal Society A, (July 2022).

Abstract: 

" ... Here, we show that the global changes observed appear largely from reductions in the amount of sunlight scattered by Earth's atmosphere. These reductions, in turn, are found to be almost equally split between reduced reflection from the cloudy and clear regions of the atmosphere, with the latter being suggestive of reduced scattering by aerosol particles over the observational period... "