In this chapter, we will use a simple model to explain how changes in the concentration of $CO_2$ affect the average temperature of the Earth's atmosphere.

The theme discussed extensively in the chapter on the History of Research is a subject I would like to explain using the simplest model possible. As mentioned in the chapter on the history of research, Hulbert developed an "atmospheric column one-dimensional model" in 1931, which accounted for the vertical mechanisms of the troposphere and stratosphere. However, the method introduced here is much simpler, neglecting vertical dynamics and essentially representing a zero-dimensional model. Furthermore, it is uncertain whether the method presented in this chapter has been published in past academic papers. Particularly, modern models describe extremely complex aspects of the entire Earth, which might be too intricate for beginners. Under the belief that such details can be omitted without losing the essence, a simplified model has been constructed.

The method introduced here is a simple model that extracts only the elements deemed "unignorable" from a common-sense perspective. As a conceptual framework for understanding the mechanism of global warming, it should be more than sufficient for the general public. (While this chapter explains using equations, the fundamental ideas are intended to be comprehensible even to middle school students. It's okay to skim over the mathematical transformations, as some parts may require knowledge of differentiation at a high school level.) The basic approach is to ignore as many detailed phenomena affecting Earth's weather as possible and focus solely on the major factors structurally influencing temperature.

• Energy Balance: How much energy enters the Earth, how much exits into space, and the difference accumulates as heat on Earth, roughly estimated as the greenhouse effect.
• Impact of Solar Radiation: How much would the Earth cool without the Sun? (Considering volcanic activity, the Earth would never freeze to absolute zero.)
• Effect of Greenhouse Gases: How much would the Earth cool without the greenhouse effect? How much is the Earth warmed by the greenhouse effect at its surface?
• Absorption Capacity of $CO_2$: Of the total energy received from the Sun, only a fraction is absorbed by $CO_2$, but how much exactly?

These concerns were formalized, and parameters were estimated based on empirical research to construct the model.

Energy balance and the influence of solar radiation

As illustrated in the chapter on Global warming and the energy balance of sunligh, a portion of the energy received from the sun remains in the atmosphere, while the majority is radiated back into space. Over short time intervals, all the incoming solar energy is eventually radiated back into space, maintaining the Earth's temperature at a constant level. If this energy balance were disturbed—for example, if 100 units of energy were received from the sun but only 99 units were radiated back into space—1 unit of energy would be trapped within the Earth, contributing to atmospheric warming by 1 unit. Conversely, if more energy were radiated into space than received from the sun, the atmosphere would cool accordingly.

The average temperature of Earth's atmosphere, denoted by $T$, is approximately $14{}^\circ C$ in Celsius, or approximately $287{}^\circ K$ in absolute temperature. Since $T$ varies with time $t$, it can be expressed as $T(t)$. Furthermore, to understand the relationship between the concentration of carbon dioxide in the atmosphere, $\rho_{CO_2}$, and the average temperature of Earth, we denote $T$ as a function of $\rho_{CO_2}$, expressed as $T(t, \rho_{CO_2})$.

If the incoming solar energy is balanced by the energy radiated from Earth to space, the temperature change would be zero. Therefore, in the short term, since this equilibrium holds, the atmosphere remains stable, leading to the following relationship.

The rate of change of temperature $(dT/dt)$ equals the effect of solar energy absorbed by the Earth's surface and subsequently raising the temperature of the atmosphere minus the effect of energy radiated from Earth to space causing temperature decrease, resulting in equilibrium: dT/dt = 0. Expressing this in mathematical terms, we have the following equation:

$$
\frac{dT}{dt} = \alpha(t)S - \beta (T(t) - T_0)=0 \tag{1}
$$

Here, let $S$ denote the energy per second [W] from the sun, $\alpha$ denote the ratio at which $S$ warms the atmosphere by emitting long waves after reaching the Earth's surface [$^\circ$K/W] (where $S$ is assumed to be approximately constant). Additionally, if there were no sun, the Earth would quickly freeze. In this scenario, there would be continuous emission of long waves from the Earth to space, and in the ultra-long term, it would converge to a constant temperature $T_0$. It is assumed that there is a cooling effect proportional to the difference between the current temperature $T(t)$ and $T_0$, and this coefficient is denoted by the constant $\beta$. This accounts for the possibility that even without the sun, the atmosphere and oceans could be warmed by volcanic activity or geothermal sources.

Suppose $T_0$ can be approximated near absolute zero, then $T_0 = 0 {}^\circ K$. But what is the reality? Let's consider a distant planet in the solar system, Uranus, for example. The average temperature of Uranus' atmosphere is approximately $-205{}^\circ C = 68^\circ K$.

Neptune, located about 19 times farther from the Sun than Earth, is estimated to receive about 1/400th of the energy received by Earth from the Sun. In this case, assuming no greenhouse effect in Neptune's atmosphere, the estimated surface temperature would be $53.2^\circ K$ (refer to the Appendix for calculation methods). Therefore, the difference of $14.79^\circ C$ between this estimated temperature and the actual surface temperature of Neptune corresponds to $T_0$. It is estimated that even without the Sun, Neptune would still be warmed by about 14 degrees Celsius due to geothermal heat and other factors.

Incidentally, the estimated temperature at the core of Uranus is around $5255^\circ K$, as estimated, which is not as hot as the surface of the Sun but still very high. The estimated temperature at the core of the Earth is around $5500^\circ K$, as estimated, which is roughly about the same level.

Even though cold planets have very hot interiors, does this mean we can assume Earth's $T_0$ to be about the same as that of Uranus? To verify this, we need to consider the composition of the atmosphere. In reality, Uranus' atmosphere consists of about 83% hydrogen, 15% helium, and 2% methane, and the greenhouse effect caused by methane may already account for the mentioned 14 degrees. Therefore, $T_0=14.79^\circ C$ may slightly overestimate $T_0$ as an estimate.

On the other hand, applying the same reasoning to Jupiter's moon Ganymede yields $T_0=3.77^\circ C$. This moon has an atmosphere composed solely of oxygen, resulting in almost no greenhouse effect. However, since Ganymede has only about twice the mass of the moon and is very small, its atmosphere is extremely thin, and the warming effect due to geothermal and volcanic activity is negligible compared to Earth. Therefore, $T_0=3.77^\circ C$ is considered to be an underestimate.

Let's assume that $T_0$ lies somewhere between $3.77^\circ C$ and $14.79^\circ C$ based on these considerations.

Here, considering $\frac{dT}{dt}=0$ in equation (1), we can rearrange it as follows.

$$
T(t) = \frac{\alpha(t)}{\beta}S + T_0 　\tag{2}
$$

With this, we have decomposed the average temperature of the Earth into the warming effect due to solar radiation and the warming effect due to volcanic activity or geothermal heat.

The Influence of Radiative Forcing

The term we need to understand to determine the extent of the increase in average temperature when the concentration of CO2 in the atmosphere rises is "radiative forcing." Radiative forcing refers to the change in the net radiation at the top of the Earth's troposphere (with positive sign indicating factors leading to temperature increase, such as radiation from space or greenhouse effect, and negative sign indicating factors leading to temperature decrease, such as radiation to space).

Interpreting radiative forcing using equation (2) above, an increase in the concentration of CO2 in the atmosphere results in an increase in the radiative forcing value, causing an increase in the value of α and thus leading to a rise in temperature. Now, what would happen if this forcing were to become zero? That would imply the complete absence of Earth's greenhouse effect. According to this source, the Earth's average temperature without the greenhouse effect would be approximately -18°C. Interestingly, this can be easily derived from Stefan-Boltzmann's law (details are provided in the Appendix). Accepting this premise, let's describe the relationship between atmospheric forcing $F(t)$ and temperature as follows.

$$
T(t) = a_0 F(t) +T_{ghe} + T_0 \tag{3}
$$

Here, $T_{ghe}$ represents the average surface temperature of the Earth if there were no greenhouse effect in the atmosphere. Here, $a_0$ is some constant chosen to match the left and right sides. To find $a_0$, let's consider the equilibrium state of the atmosphere. In equilibrium, $T(t)$ equals $T_{eq}$. For instance, assuming the average temperature of the Earth is $14^\circ C$, the radiative forcing of the atmosphere would be $F_{eq}$. Then, we have following.

$$
T_{eq} = a_0 F_{eq} + T_{ghe} + T_0
$$

Solving for $a_0$ yields the following solution:

$$
a_0 = \frac{T_{eq} - T_{ghe}- T_0}{F_{eq}} \tag{4}
$$

Let's review the table described in the section on solar energy balance:

When reading this table, under clear sky conditions, the atmospheric radiative forcing is estimated to be $125 [W/m^2]$, while under cloudy conditions, it is $86 [W/m^2]$. Assuming that clouds cover approximately half of the Earth's surface at any given time, the weighted average of these two values is considered the effective equilibrium level, resulting in $F_{eq}=105.5[W/m^2]$. Additionally, the temperature remains constant at $T_{eq}=14 {}^\circ C = 287 {}^\circ K$.

Furthermore, if the radiative forcing $F(t)$ is zero, the atmospheric temperature is $T_{ghe} = -18 {}^\circ C = 255 {}^\circ K$. Additionally, considering the upper and lower limits provided for $T_0$, the constant $a_0$ in equation (3) is estimated to have a range of approximate values.

$$
a_{0min} = 0.26758 [{}^\circ Km^2/W] (T_0 = 3.77 {}^\circ C)
$$

$$
a_{0max} = 0.16313 [{}^\circ Km^2/W] ( T_0 = 14.79 {}^\circ C)
$$

$$
a_{0min} < a_0 < a_{0max} \tag{5}
$$

Relationship between Radiative Forcing of $CO_2$ and Concentration

Next, let's consider the radiative forcing $F$ concerning $CO_2$. Does the radiative forcing $F$ increase linearly with the increase in $CO_2$ concentration? Not necessarily. Of all the energy emitted from the Earth's surface into the atmosphere, only a portion within the absorption band of $CO_2$, which is the $15 \mu m$ band as shown in the figure below (enclosed by the dashed yellow line), can be absorbed by $CO_2$.

At the current equilibrium level, the radiative forcing is already around $26 [W/m^2]$ (based on the average values for clear and cloudy conditions), which means that it has already absorbed about half of the $15 \mu m$ band. No matter how much we increase the concentration of $CO_2$, it can only absorb up to about double that amount.

Therefore, considering ρ_CO2 as the concentration of CO2 in the atmosphere, let's explore the functional form of the forcing F(ρ_CO2).If we denote F_max as the maximum value of the forcing, from the above figure, F_max ≈ 52 [W/m^2] seems to be a reasonable estimate. When the CO2 concentration slightly increases, the forcing also increases slightly, but the magnitude of this increase depends on the difference between the current forcing and the maximum forcing (the remaining capacity for absorption). Hence, the following equation is obtained, where λ is a positive constant:

$$
\frac{dF(\rho_{CO_2})}{d\rho_{CO_2}} = \lambda (F_{max} - F(\rho_{CO_2}))
$$

Solving this differential equation yields the following equation:

$$
F(\rho_{CO_2})= F_{max} (1- \exp(-\lambda \rho_{CO_2})) \tag{6}
$$

Here, assuming $F_{max}=52[W/m^2]$ is double the current level. The $CO_2$ concentration at the time the reference paper was written in 1997 was $\rho_{CO_2}=363.04ppm$ approximately, so $\lambda = 1.909 \times 10^{-3} [ppm^{-1}]$ is estimated.

Here, $F_{other}=77.5[W/m^2]$, which is the average radiative forcing of greenhouse gases other than $CO_2$ (including water vapor), for clear and cloudy conditions. Using this result, we simulated the relationship between $CO_2$ concentration and the average atmospheric temperature, as shown in the following figure.

In the above figure, we provided estimates of observed Earth $CO_2$ concentrations and average temperature measurements. Even in modern times, the absolute values of estimated temperatures have statistical uncertainties, so please consider them as approximate reference values. Ultimately, it can be seen that the theoretical values obtained using the above model closely approximate the observed values.

However, the above discussion did not consider factors other than $CO_2$, such as the effect of water vapor greenhouse gases, which are also manifested with increasing temperatures. Therefore, the response to the increase in $CO_2$ concentration may be underestimated, especially in regions with high concentrations.

By the way, when using this model to calculate the average atmospheric temperature for $CO_2$ concentrations doubling from pre-industrial levels of $280ppm$ to twice that, it results in a temperature of $16.35^{\circ} C$, which represents an increase of approximately $ 2.6^{\circ} C$ from the pre-industrial temperature of $13.76^{\circ} C $ in 1891. This result aligns closely with the scenarios of "high" and "very high" $CO_2$ emissions as described by the recent IPCC report. Therefore, the assumptions underlying the IPCC's claims seem to be well understood based on the discussion above.

The key point here is that complex models used in modern weather forecasting, like those used in modern weather forecasts, are unnecessary for estimating the average surface temperature. It has been confirmed that it is sufficient to know only the energy flux from the sun and the net amount of energy emitted from the Earth's surface to space (energy balance).

Sample code

The above can be described in Python code as follows.

import numpy as np
import math
import matplotlib.pyplot as plt

###############################################################################################
T_absolute = 273.15   # Conversion between absolute temperature and Celsius temperature.
T0 = 3.77      # Convergence temperature in the absence of the Sun　
Tghe = -18 + T_absolute # Surface temperature of the Earth without the greenhouse effect　K
Teq = 14.454 + T_absolute   # Average temperature at the time of 1997.  14.454C
Feq =105.5  # Average Radiative Forcing of Earth W/m^2　
###############################################################################################

###############################################################################################
F_rho = (32 + 24) / 2  # Radiative forcing of CO2 in 1997 W/m^2
Fmax = F_rho * 2     # Radiative forcing of the atmosphere in 1997. [W/m^2] Doubling is the upper limit estimate.
rho = 362.9        # The concentration of CO2 at the time of 1997.　[ppm]
F_other = ( 75 + 10 + 8 + 51 + 7 + 4) / 2 # 77.5 [W/m2]
###############################################################################################

_lambda =  (1 / rho) * math.log(Fmax / (Fmax - F_rho)) #0.001899
print("lambda: ",_lambda)

obsx=[280, 316.9, 362.9, 372.4, 402.9, 415.7]  # [ppm] observed in 1891,1960, 1997, 2002, 2016, 2021 respetively
obsy=[13.764, 13.974, 14.454, 14.544, 14.894, 14.764] # celsius

def T_co2(rho, T_0 = 0.0) -> float:
    a0 = (Teq - Tghe - T_0) / Feq
    #print("a0: ",a0)
    return a0 * (Fmax * (1- np.exp(- _lambda * rho )) + F_other) + Tghe + T_0 - T_absolute

rho_co2 = np.linspace(0, 610, 100)

fig = plt.figure(figsize=(10, 5))
plt.rcParams["font.size"] = 16
ax = plt.subplot(111)

T0 = 3.77
T = T_co2(rho_co2, T0)
ax.plot(rho_co2, T, color = "C0", 
        label="$theory[{}^{\circ}C] (T_0=$"+"{})".format(T0) )

T0=14.79
T2 = T_co2(rho_co2, T0)
ax.plot(rho_co2, T2, color = "C0", linestyle = "--", 
        label = "$theory[{}^{\circ}C] (T_0=$"+"{})".format(T0) )

ax.set_title("Effect of $CO_2$ concentration on air temperature")

ax.plot(obsx, obsy, color = "orange", 
        label = "$observation[{}^{\circ}C]$", marker = "o")

ax.set_ylabel("$Temperature [{}^{\circ} C]$")
ax.set_xlabel("Concentration of $CO_2$ [$ppm$]")
ax.grid(which="both",color="gray",linewidth=0.2)
ax.legend()

ax.fill_between(rho_co2, T, T2, facecolor = 'C0', alpha = 0.5)
t = 560
print("Earth average Temperature at CO2 level of {}[ppm]: {}[C]".format(t, T_co2(t,3.77)
                                                            

Access to the series

• The Truth of Global Warming: Part 1 - Do You Truly Understand "Global Warming"?
• The Truth of Global Warming: Part 2 - History of Research
• The Truth of Global Warming: Part 3 - History of Earth's Atmosphere
• The Truth of Global Warming: Part 4 - Global warming and the energy balance of sunlight
• The Truth of Global Warming: Part 5 - Verification of global warming through mathematical models
• The Truth of Global Warming: Part 6 - The impact of population growth on global warming