A Simple Physical Analysis of the Greenhouse Effect and Climate Change

Keywords

Greenhouse Effect, Blackbody Radiation, Climate Change, Solar Radiation, physical modelling, physical analysis

Abstract

This paper examines physical theories such as thermodynamics, quantum mechanics to develop parameters including absorptivity, reflectivity, and transmissivity of the atmosphere. It is on such parameters that the physical modelling equation has been established to quantitatively analyze the ongoing global warming and climate change situation. This theoretical research reaffirms the severe effect of the greenhouse gases that are already well known to contribute to global warming and climate change. This paper also reveals the great possibility that with more complex and advanced modelling based on multi-layer scenarios, such as that based on gradated layers of the atmosphere of this planet, a much more accurate and detailed analysis for a prediction of the future route of global warming processes will be possible.

Introduction

It is evident that, given the change of global temperature during the past few decades, the earth is becoming warmer; the interesting issue is, for many scientists, to determine whether such rises in temperature are due to anthropogenic causes or natural ones. There have been remarkable fluctuations of global temperatures throughout the long history of the planet Earth [1]. Ice ages and warm periods have existed alternately overriding each other, allowing in-between the interglacial period. Considering the past fluctuations, however, the abrupt rise of temperatures recently is considered to be part of a grand process in which the earth endures the rise in global average temperatures [2]. Several previous studies on the issue, which is considered the dominant theory, posit that present global warming may mostly be attributed to rapidly expanding human industrial activities [3]. To clearly understand global warming, however, it is necessary to differentiate the effects of principal solar radiation energy that warms the earth and the radiation from the earth out back to space. Normally these two types of radiation are ideally balanced, yet once this balance is broken due to greenhouse gases or water vapours, the earth’s temperature rises, and, as a result, a new balance is soon to be established [4]. This paper focuses on the exact balance of the solar and global radiation and the effect of the greenhouse gases in restoring new equilibrium on the earth. To that end, in this paper, a simple physical model has been created to explain the fundamental physics behind the radiation energy, blackbody radiation, which are then combined to delve into the resultant solar radiation. This research will also explore the effect of the earth’s atmosphere in absorbing global radiation on the global temperature of the earth’s surface. Finally, the model including reflectivity and transmissivity has been used to calculate and analyze the changes in the temperature of the surface of the earth.

Theory

Radiation Theory in Physics

Radiation is the energy transferred by electromagnetic waves. The intensity of the radiation emitted by an object is determined by its temperature and other properties. Radiation flux density is defined as the radiation energy per area and time [4]. Let E and F be radiation energy and radiation flux density, respectively, and equation (1) is obtained, in which A is area and t is time. Figure 1 illustrates the flux density
F = d² E/(dAdt)                                                                                      (1)

Figure 1. Radiation energy and flux density [4]
The radiation flux density depends on the direction of radiation. Yet intensity means the flux illuminated vertically onto the specific unit of area or the flux illuminated on a unit solid angle. Hence, equation (2) is given:
I = dF/(dΩ cos cosΘ)                                                                             (2)
where dΩ is the solid angle and Θ is the angle between the radiation ray and the vertical line to the surface.
Figure 2 illustrates the Solid angle:

Figure 2. Illustration of the solid angle [4]
If the flux is uniform in all directions (isotropic), then F=πI. Now consider the monochromatic flux of wavelength λ and let F_λ, I_λ be flux and the intensity for the particular wavelength. It is given that F=∫F_λdλ and I = ∫I_λdλ [4]. If frequency v is used instead of λ, F=∫F_vdv and I = ∫I_vdv, in which v = c/λ with c being the speed of light.
Here again, when radiation passes through matter, there is absorption, transmittance, and reflection. Let radiation with wavelength λ have absorptivity, reflectivity, and transmissivity a_λ, r_λ, and τ_λ, respectively. The following equation (3) can be obtained [5]
a_λ + r_λ + τ_λ = 1                                                                                       (3)

Theory of Blackbody Radiation

A blackbody is an imaginary object that absorbs all the electromagnetic waves radiated into it [6]. Hence, r_λ + τ_λ = 0 and a_λ = 1. Yet, even blackbodies emit some electromagnetic waves and thus show the frequency distribution of the radiation. Photons should be considered in calculating the frequency distribution. Since the photon is a boson, it satisfies Bose-Einstein Distribution with chemical potential 0, and equation (4) is obtained [7]. Figure 3 shows how Bose-Einstein Distribution is defined:

Figure 3. Bose-Einstein Distribution [11]
Before substituting, here are some simple explanations of variables in statistical thermodynamics
k_B is the Boltzmann constant, a value that links the temperature of a gas, like the atmosphere, with the average kinetic energy of its particles
β = 1/ k_B T defines thermodynamic beta (or coldness) as the reciprocal of thermodynamic temperature
h = h/2π defines a form of Planck’s constant, which describes the angular momentum of a particle, a photon in this case – all values of angular momentum are quantised, which means they must be a multiple of h.
ω = 2πf defines angular velocity as multiplied by the frequency (rotations per second).
n_k1a = 1/(e^βhω – 1)                                                                                   (4)
where n_k1a  is the Bose-Einstein Distribution Function of a photon, as described by the graph above.
Thus, the average number of photons with a wave vector between k and k + dk is given by:
(5)
The mean energy of photons with frequencies between ω and ω + dω is given by the product of the number of photons and the energy of a single photon, hω , as follows [9].

Therefore, the total average energy density over all is given by:

The radiation power density, which is defined as the energy per unit volume and unit time, of photons whose frequency is between ω and ω + dω is given by eq (8), when using the fact that the flux radiated from the surface of a black body is related to the energy density as

, where = flux = energy/(area × time) per frequency, involving c/4 for solid angle integration, finally resulting in the total radiation power density F as

where is known as the Stefan-Boltzmann constant. When the blackbody is not completely ideal, then the flux density has a constant, and the equation becomes [10]:
F = εσT⁴ ,                                                                                                   (10)
which is known as the Law of Stefan-Boltzmann [7], and ε is known as the emissivity of blackbody which has a value between 0 and 1, with the value closer to 1 for more complete blackbody. Figure 3 illustrates spectra made by blackbody radiation and figure 4 illustrates Stefan-Boltzmann law with black body radiation

Figure 3. Spectra from a blackbody [12]

Figure 4. Stefan-Boltzmann law with black body radiation [13]

Modelling

Solar Radiation and the Surface Temperature of the Earth

The temperature of the earth is determined by how much energy comes in from space and returns to space. When incoming radiation is greater than the earth’s radiation energy, the global temperature rises. To clarify this, let I_solar be the intensity of solar radiation per area of the earth’s surface, which is approximately 1,370 W/m² [4]. Then the area of the earth illuminated by solar radiation is given by the product of I_solar  and πR_Earth², which is the cross-sectional area of the earth with radius R_Earth. Figure 5 illustrates the structure of the earth’s absorptivity, transmissivity and reflectivity.

Figure 5. Structure of the earth’s absorptivity, transmissivity and reflectivity [14]
Let r_Earth be the reflectivity of the earth, and equation (11) is established:

Now if the intensity of solar radiation onto the earth’s surface is I_solar, and the earth reflects it with reflectivity , power density incident onto the surface of the earth is (1 – r_Earth) I_solar . Hence, radiation power incident onto the global surface will be approximately 0.3. [7]
On the other hand, the power of the radiation the earth emits back into space is the product of the flux density F_out and the surface of the earth. Now, assume that the earth is a blackbody and has no atmosphere. Then according to the Law of Stefan-Boltzmann, the following equation is obtained [8]:

where T_earth is the temperature of the surface of the earth.
Hence, the power density emitted by the earth is given by:

For the earth to have an energy balance, φ_in = φ_out must be satisfied, and thus this yields:

or if this equation is solved for the surface temperature of the earth, (15) can be obtained as follows:

With ε = 1, for which the earth is an ideal blackbody, the surface temperature of the earth is approximately 255K or -18C. The average surface temperature of the earth is known to be about 15C, and thus there is an approximately 33C difference. [4] For this to be rationally explained, the greenhouse effect of the atmosphere should be considered.

Results

The Greenhouse Effect

It is well known that nitrogen and oxygen gas, which compose a majority of the atmosphere, neither absorb nor emit radiation energy, yet trace gases like water vapour, carbon dioxide, and methane gas absorb a significant portion of radiation energy, causing the greenhouse effect.
Figure 6 shows how the greenhouse effect is created on the earth:

Figure 6. Greenhouse effect made by heat trapped by greenhouse gases in the atmosphere [15]

Figure 7. Single-Layer Model in which there is only a single layer of atmosphere on the Earth [7]
As seen in Figure 7, In the Single-Layer Model, solar radiation enters the earth and the earth’s radiation is emitted out to space and the atmosphere of the earth reabsorbs the earth’s radiation and emits it up and down, as shown, if there is only one layer of atmosphere specified, which passes short-wavelength solar radiation without any absorption. In contrast to this, the earth’s atmosphere is mostly confined to infrared rays and thus the layer absorbs the rays and simultaneously emits the radiation of atmospheric temperature back both into outer space and down to the surface [5].
For radiation equilibrium to be established in this situation, there must be energy balance within the atmospheric layer.


Also, the energy balance should be established on the earth’s surface, giving equation (18) as follows:

Therefore,

Finally, likewise, energy balance should also be established on the entire planet.
Since I_{up,atmosphere} = I_in,solar ,                                                           (20)
                                                          (21)
If the earth is assumed to be an ideal blackbody, by equation (21), T_atmosphere is about 255K. Then since T_ground = 2^1/4 T_atmosphere , T_ground becomes about 303K. This is 30C, which is 40C higher than the case in which the earth’s atmosphere was assumed not to exist. This means the earth’ atmosphere plays an essential role in maintaining a habitable temperature on the earth. Now, it is assumed that there is reflection and transmission, rather than complete absorption in the atmosphere. Let reflectivity, transmissivity, and absorptivity r, τ, a, and then a + r + τ = 1. Now with this definition, in the Single-Layer Model mentioned above, I_{up,atmosphere} can be replaced with I_{up,atmosphere} + τI_up,ground , and I_{down,atmosphere} with I_{down,atmosphere} + rI_up,ground.
Thus, the equation in this equilibrium condition changes to:

Therefore, finally equation (22) and (23) will be obtained.

Table 1 shows the calculation results when reflectivity r and transmissivity τ varies from 0.1 to 0.9 under the condition that ε = 1, with r in the rows and τ in the columns.

Table 1: Surface temperature of the Earth depending on reflectivity and transmissivity.

Discussion

In Table 1, it is evident that, when τ – r = 0.2, the result becomes closest to 288K which is the very present average surface temperature of the earth. It is also known that the higher concentration of carbon dioxide in the atmosphere reduces the transmissivity of the atmosphere of the earth. Table 1 also affirms the fact that as transmissivity decreases, the earth’s surface temperature rises. This shows that greenhouse gases such as carbon dioxide and water vapour are the most plausible cause of global warming of the earth. The calculation based on the no-atmosphere model has revealed that surface temperature of the earth was -18C, which is 33C lower than the known value of the average surface temperature, 15C [5]. This difference explains well the essential role of the atmosphere for the stability of the earth’s temperature. The calculation based on the single-layer-model shows that surface temperature was 15C higher than the known value [5]. This error may be mostly due to the assumption that the absorptivity was chosen to be 100%. This model was modified so that reflectivity and transmissivity parameters are included in the equation and resulted in the fact that the accepted value of the surface temperature of the earth was established when the difference between transmissivity and reflectivity becomes 0.2. It is clear that as reflectivity increases or transmissivity decreases, the atmospheric temperature increases and that the reduction of transmissivity of the earth’s radiation due to the increase of the greenhouse gases becomes the cause of global warming on the planet.

Conclusion

The modelling of climate change has so far been applied to global warming of the earth based on the theory of solar and earth’s radiation under the assumption that the earth is a quasi-ideal blackbody. Two models are introduced: one is for the assumption that there is no atmosphere and the other one for the single-layered atmosphere. With the comparison of the two models, it has been reaffirmed that the role of earth’s atmosphere is critically essential for the stability of earth’s surface temperature and negative effect of greenhouse gases on the earth’s self-adjusting climate system and hydrological cycles. With the more advanced models such as 2 or 3 layered will be expected to allow more specific and better results for more precise analysis and prediction. Simple as it may, the Single-layered Model explains quantitatively well the behaviour of earth’s climatological cycle and is expected to open new challenges to more advanced research with the advent of AI technology.

Acknowledgements

I would like to thank my beloved parents who never cease to encourage me when I face unexpected difficulty. I also hope to extend my gratitude to my physics teacher, who helped me finish the paper even with his utmost busy schedule.

Variable	Interpretation	Value	Unit
E	energy of photon		J
h	Planck’s constant	6.62608×10-34	J.s
c	speed of electromagnetic radiation	2.99792458×108	m.s-1
v	frequency of electromagnetic radiation		Hz
I	intensity of radiation
T	surface temperature of object		K
A	surface area of object		m2
σ	Stefan-boltzmann constant	5.6696×10-8	W.m-2.K-4
P	power emitted from hot object		W
	emissivity of blackbody
a	absorptivity
r	refelectivity
kB	Boltzmann constant	1.38066×10-23	J.K-1
	trasmissivity		m.K
k	wave vector		m-1
F	radiation flux
	Power density		W/ m2
RE	radius of the Earth		m
	Bose-Einstein Distribution Function of a photon
	1/ kB T	1.36×103	W.m-2
		0.30

Table 2. Summary of the physical quantities used in this paper

References

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About The Author

Yong Woo Choi attends Brea Olinda High School in Brea, CA. He has been especially intrigued by linking pure theories to practical applications, a strong motivation to get down to serious research on modeling a real practical scientific issues such as global warming and climate change by using simple elemental math and physics to the level beyond the knowledge of secondary education.