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# The essential property of the greenhouse effect and Planck's radiation law

Motto: ”Things should be presented as simply as possible, but no simpler (Albert Einstein).

Problem: IPCC’s scenario SSP3-7.0 causes a radiative forcing of 7.0 W/m2, which increases the Earth's surface temperature by an average of 3.6°C.  The radiative forcing of 7.0 W/m2 initially decreases the outgoing longwave radiation (OLR) to 233 W/m2 (= 240-7.0). According to Stefan-Bolzmann (S-B) law, an increase in surface temperature increases the emission radiated by the earth's surface by about 20 W/m2. The increased amount of radiation on the Earth's surface of 20 W/m2 raises the outgoing radiation only to 240 W/m2, and the rest of the radiation, i.e. 20-7 = 13 W/m2, does not travel into space according to the IPCC. This problem of lost radiation has been raised by Howard Hayden, Professor Emeritus of Physics at Connecticut. According to him, with this proposal, the IPCC violates the well-known S-B law of physics.

In my opinion, it is more about Planck's law and how the Greenhouse (GH) effect works. This is also a violation of Einstein’s principle, that is, Hayden tries to put it too simply by forgetting the existence of the GH effect. This means that Hayden does not know the essential principle of the GH effect. Planck's law holds and the surface radiates that 20 W/m2 if the surface temperature rises by 3.6°C.  In addition, a couple of other things happen. Another issue is if the 3.6 temperature is correct or not.

First proof: the Earth's energy balance and the magnitude of the greenhouse effect

Qualitative proof comes from the Earth's energy balance and the greenhouse (GH) effect

Figure 1. The Earth's energy balance shows the energy sources of infrared radiation emitted by the atmosphere.

The Earth's energy balance shows that 395 W/m2 of the radiation emitted by the Earth's surface is absorbed by 155 W/m2. The magnitude of the GH effect according to Fig. 1 is the difference between the total energy input to the surface 510 W/m2 and the the net solar energy 240 W/m2, which is 270 W/m2. The same 270 W/m2 is the sum of three energy fluxes absorbed by the atmosphere: LW radiation 155 W/m2 latent heating 91 W/m2 and sensible heating 24 W/m2. This amount of 270 W/m2 plus the solar radiation absorbed by the atmosphere 75 W/m2 is radiated from the atmosphere to the surface. The relationship between the total GH effect and the LW radiation absorption is 270 / 155 = 1.74. It means that the GH effect is about 1.7 times greater than the LW absorption.

Another proof: an iteration-based GH model

I have conducted a study on how the greenhouse effect works based on an iterative mathematical model, which I have made a blog post about on my website, reference 1. The object of the iteration is the radiative forcing of 7.0 W/m2 in the greenhouse effect at the upper limit of the atmosphere, caused by, for example, carbon dioxide. At baseline, the Earth's surface radiates 392.2 W/m2, the GH effect absorbs 152.2 W/m2 and 240 W/m2 goes into space. The results of the iteration show that the radiation OLR initially going into space decreases by 7.0 W/m2, and when the temperature of the ground surface has risen by 2.07 degrees, it causes an increase in radiation from the Earth's surface by 11.48 W/m2. In the new equilibrium, the Earth's surface radiates 403 W/m2, the atmosphere absorbs 163 W/m2, and 240 W/m2 goes into space.

Atmospheric absorption has increased by 11.48 W/m2, with a ratio of 1.64 to an RF value of 7.0 W/m2. Iterations with different RF values showed that this ratio ranged from 1.58 to 1.65. As a rule of thumb, atmospheric absorption increases by about 1.65 times the RF value. In atmospheric science, this phenomenon is called Planck feedback. My iteration model is based on the basic phenomenon of the GH effect, i.e. the radiation caused by the amount of radiation absorbed by the atmosphere back to the earth's surface. Since the increase in the amount of energy in the atmosphere caused by the GH effect cannot go into space, it must return to the earth's surface, as shown in the energy balance picture.

Third proof: Earth's tropical climate zone

There are three climatic zones on Earth, namely tropical, subtropical and polar. The tropical zone is between latitudes -23.5 degrees and + 23.5 degrees, and it occupies 40% of the Earth's surface and has an average temperature of 26.5 degrees.  The tropical zone emits an average of 259 W/m2 of infrared radiation into space, which is 19 W/m2 more than the Earth's average (CERES data).

Figure 2. The amounts radiated into space by the climatic zones of the Earth.

Using the Earth's average emissivity coefficient of 0.95, Planck's law states that the amount of infrared radiation emitted by the Earth's surface in the tropics is 448 W/m2, i.e. 50 W/m2 more than the Earth's average. This means that in the tropics the atmosphere absorbs a total of 448- 259 = 189 W/m2, or 31 W/m2 more than the global average. The ratio of the increased absorption absorbed by the atmosphere of 31 W/m2 to the increased radiation going into space is 19 W/m2 = 1,63.

The conclusion is that the GH effect works convincingly in the same way, both for the increase in absorption caused by the GH effect and for the increased radiation on the Earth's surface caused by increased solar radiation, which also increases the absorption of the GH effect in the atmosphere. The Planck feedback effect works in the same way in both cases and explains the core event of the GH effect: the amount of radiation absorbed by the atmosphere returns to the Earth's surface and increases its temperature.

Conclusions

The IPCC scenario SSP3-7.0 does not contradict the S-B law, but the calculated temperature in the scenario is based on positive water feedback, which does not occur in climate. Therefore, my iteration model gives that scenario a temperature increase of 2.1 degrees instead of 3.6 degrees.

References

Attachment. Iteration as the greenhouse effect increases by 7.0 W/m2.