By Markus Ott

I assume that most readers are aware that the 33°C overall greenhouse effect (GHE) is the product of an arithmetically and physically incorrect calculation. It inappropriately averages the insolation evenly over the entire surface of the Earth, as if it were flat and not a rotating sphere, and then uses this average of approximately 240W/m² to calculate the mean surface temperature for an earth without an atmosphere via the Stefan-Boltzmann law, see page 97 in AR4 here. This physically incorrect mean-value calculation produces a mean surface temperature for an earth without an atmosphere of -19°C. Based on this incorrect result an overall GHE of 33°C (14 – -19) is postulated to explain the measured mean near surface temperature of about 14°C.

Here I present a model by Uli Weber that produces a surprisingly good estimate for the mean near surface temperature of the earth simply by fixing the problems in the arithmetic of the calculation described above.

To avoid this incorrect mean-value formation, a model is developed, that allows the irradiated power to be calculated at every point on the planet’s surface. Using the Stefan-Boltzmann law one then establishes a surface temperature distribution function T(ϴ) that assigns a temperature to each point on the planet’s surface. With this surface temperature distribution function T(ϴ), one can then calculate the mean surface temperature of the planet “practically error-free.”

This sounds quite hypothetical. But thanks to the Lunar Diviner Experiment, we have detailed data on the surface temperature of the Moon, that can be used to calculate the surface temperature distribution function T(ϴ) described above for the Moon.

With this data, our moon becomes the ideal model for a celestial body without an atmosphere.

When analysing the temperature data from the Lunar Diviner Mission, it turns out, that on the sunny side of the Moon, the surface temperatures are very well described by the Stefan-Boltzmann law when the angle of incidence of the solar irradiation is taken into account (Figure 1).

According to Williams et al. 2017, the surface temperature distribution function T(ϴ) is as follows.

where ϴ = solar elevation angle, α = 0.11 (albedo of the moon), and S₀ = 1368W/m² (insolation or solar constant). σ is the Stefan-Boltzmann constant.

Near the poles, the solar elevation angle ϴ (Teta) approaches 90°. This means that the cosϴ approaches zero and very little power is radiated. The surface there is very cold. At the equator, when the sun is at its highest point and ϴ = 0°, cosϴ = 1. There the full power S₀ reaches the surface and (1-α)S₀ is absorbed. The highest surface temperature is measured there.

Concentric circles with the same angle of incidence and thus the same temperature are thus formed around the point of the sun’s highest point (Figure 2).

The shadow side of the moon is not considered further. It simply cools down overnight and is heated up again the next day.

The moon rotates about 28 times slower than the earth. This means that the shady side has much more time to cool down than the night side of the Earth. After sunrise, however, the lunar surface also has more time to heat up. The diurnal variation in temperature is therefore more symmetrical than on Earth. The daily maximum is reached at the highest point of the sun (Figure 3).

If one compares the diurnal variation of the lunar surface temperature with corresponding terrestrial measurements (Figure 4), one finds that the temperature variations measured on the Earth’s surface differ from the lunar values primarily in that the maximum temperature is reached only after the solar maximum (noon) and that the difference between day and night is smaller. These differences are mainly due to the fact, that the Earth rotates about 28 times faster than the Moon and that the heat capacity of the Earth’s atmosphere and surface slows down the heating and cooling of the Earth’s surface.

Despite these differences, the diurnal variation of the Earth’s surface temperature is more similar to the diurnal variation of the Moon’s surface temperature than to the homogenous -19°C surface temperature assumed in the 33°C-GHE-model. Therefore, let’s take a look at what comes out when the surface temperature distribution function of the Moon is transferred to Earth.

When transferring the “moon model” to earthly conditions, we simply replace the albedo of the moon with the albedo of the earth. This means that although we calculate without an atmosphere, the reflection of sunlight on clouds and water surfaces is still taken into account.

with ϴ = solar elevation angle, α = 0.3 (albedo of the earth) S₀ =1368W/m² (insolation or solar constant). σ is the Stefan-Boltzmann constant.

This temperature distribution function T(ϴ) divides the sunny side of the earth into concentric circles with the same surface temperature (Figure 5).

Uli Weber describes a numerical solution that calculates the global average temperature via this distribution function. The calculation effort is manageable. Anyone can do the calculation themselves using Excel. The result is 14.03°C and is surprisingly close to the global average temperature of about 15°C “measured” by the IPCC.

A detailed description of the calculation can be found here: “Anmer­kungen zur hemisphä­rischen Mittelwert­bildung mit dem Stefan-Boltzmann-Gesetz,“ ­(Notes on hemispheric ­averaging using the Stefan-Boltzmann law), Uli Weber.

Because it is not particularly difficult in this case, I will also demonstrate a “proper” integral solution for Uli Weber’s hemispherical approach.

Here, the concentric ring with the same angle of solar irradiation becomes an infinitesimally narrow area element dA (see Figure 6).

The area-averaged mean temperature of the sunlit hemisphere T_m is calculated by forming the area integral of the surface temperature distribution function T(ϴ) over the sunlit hemisphere and then dividing by the area of the hemisphere.

With an area-averaged temperature of T_m ≈ 15°C, the integral solution provides an average temperature of approximately 15°C as specified by the IPCC and determined by weather station measurements.

Result: If you place sunshine on only one side of the Earth, as in real life, and use the results of the Lunar Diviner experiment to calculate the average surface temperature of the Earth, the “computed” overall atmospheric greenhouse effect disappears.

Why does Uli Weber’s hemispheric Stefan Boltzmann approach work so well?

In addition to his “Hemisphärischer Stefan-Boltzmann-Ansatz” for calculating the mean near-surface Earth temperature”, Uli Weber has published a series of articles explaining this model on the EIKE website.

Here I try to break down his detailed explanation to the absolute essentials.

The “measured” mean near-surface temperature of the earth (NST = Near Surface Temperature ≈ 15°C) is the result of averaging, in which a time- and area-averaged average is formed from the temperature data of a measurement network spanning the globe. This mean value is of course heavily dependent on the method used to calculate the mean value. This means that it makes little sense to haggle over half degrees here. Further information on the determination method can be found here.

Before we look at why Weber’s model is able to calculate the near-surface mean Earth temperature so well, let’s look at how well the temperature distribution function describes the lunar surface temperature over the course of a day-night cycle. For this purpose, we compare the diurnal variation of the surface temperature measured at the landing site of Apollo 15 with the values calculated for the same location.

In Figure 7, the measured daily variation of the lunar surface temperature (in blue) and that calculated using the distribution function (in red) are superimposed.

It is noticeable that the measured and calculated values agree very well during the lunar day. During the night, however, there are considerable deviations between the measured and calculated surface temperature. This deviation of approx. 77°C (-196°C measured instead of -273°C calculated) is due to the fact that the calculation is based on the assumption that the surface temperature is caused exclusively by solar radiation and that the part of the moon not exposed to the sun must therefore have a temperature of 0K (-273°C). However, as the surface of the moon warmed up during the day and stores some heat overnight, the theoretical value of -273°C is never reached.

This 77°C deviation appears far less dramatic if we calculate the heat flux from the lunar surface, which corresponds to the measured -196°C.

This results in a heat or radiation flux of approx. 2W/m². Compared to the total energy fluxes occurring during a day-night cycle, this is negligible.

It can therefore be seen that the temperature distribution function T(θ), due to the low heat capacity of the lunar surface, describes the daily temperature variation on the moon pretty well.

If the temperature distribution function is used to calculate the ground-level Earth temperature for a vertical solar incidence (θ = 0), a value of almost 90°C results.

with θ = 0° and α = 0.7 results in

 Which equals ~ 360K or 87°C

Almost nowhere on Earth is such a high temperature reached near the ground.

Similarly, the -273°C calculated for the night side is never reached.

The temperature distribution function T(θ) calculates the maximum achievable temperature for each location, assuming that the power radiated by the sun is immediately and completely converted into IR radiation. Uli Weber calls this value the Stefan-Boltzmann temperature equivalent to distinguish it from the actual measurable temperature.

On Earth, such a direct and complete conversion into IR radiation is not possible (due to thermalization, evaporation, …). Here, most of the absorbed solar radiation is converted into sensible heat. This heat drives global circulation and charges the very large heat reservoirs in the climate system. The heat content of the global circulation is approximately 4 orders of magnitude greater than the energy absorbed from the sun in one day.

Heat content of the oceans ≈ 50,000 days of solar radiation

or

Heat content of the oceans ≈ 100,000 times the heat required to prevent a nighttime cooling as observed on the moon (source).

The calculated extreme temperature values are therefore never reached and the night side cooling can be neglected. For a more detailed explanation of why 87°C is not reachable on Earth’s surface due to evaporation and circulation see Sud, et al.

This results in a surprisingly simple convection model. The heat input takes place via solar radiation. The irradiation occurs with S₀ (solar constant 1368W/m²) over a circular area with the size πR² (R = earth radius). Approx. 30% of this radiation is reflected by the earth back into space (albedo α ≈ 0.3). This results in a heat input of approx. 957W/m² over an area of πR².

This heat input is converted into sensible heat, distributed over the entire Earth by means of global circulation and then radiated back into space over the entire surface of the Earth (4πR)².

Similar to the “33°C greenhouse effect model”, this results in an average outgoing long wave radiation of approx. 240W/m². Assuming thermal equilibrium, this results in:

Irradiation = radiation

However, in contrast to the “33°C greenhouse effect model,” the average outgoing long wave radiation is not used to calculate the average surface temperature, a procedure that ignores the rules of arithmetic. In Uli Weber’s model, the “temperature genesis” takes place directly and locally on the site of irradiation, before the mean value is formed.

At the respective irradiation point, the solar radiation is almost completely converted into sensible heat. This conversion into sensible heat means that the irradiated power (P) or the associated heat quantity (Q) becomes a linear function of the temperature.

“Locally”, for a very small surface element A, the following then applies (with t =time, C=heat capacity of the surface element A, σ = Boltzmann constant):

or

or

After being converted into sensible heat, the radiated power becomes part of the global circulation, ensuring proper “arithmetically correct” averaging.

This is shown in the following graphic, which was taken from Uli Weber’s article “Der hemisphärische Stefan-Boltzmann-Ansatz ist kein reines Strahlungsmodell – Teil 1” (31 December 2023, Figure 8).

The takeaway from this short and relatively simple analysis is that the presence of atmosphere and oceans makes Earth warmer than a comparable planet without atmosphere and oceans. But this higher mean surface temperature is not caused by back radiation from greenhouse gases. The higher mean surface temperature is the result of the enormous heat capacity of the oceans and atmosphere and their ability to distribute the absorbed heat over the planet.

Andy May lightly edited this post for clarity.

Download the bibliography here.

Uli’s writings on this subject go back to 2017, see the bottom of this post for a list and links.