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Using Energy Conservation Concept and Basic Mathematical Modeling Techniques to

  Study the Effect of Greenhouses Gases to Earth's Climate System.

Abdulaziz B. M. Hame�∗� and Younis Ahamed. Abu Aash��

1. Department of Mathematics, Faculty of Science, Yobe State University, Nigeria

2. Department Physics - Faculty of Education, West Kordufan University, Sudan

  Emails: aziz.hamed@gmail.com,

ABSTRACT
The work, implemented the basic  mathematical modeling  techniques to Physical phenomenon 
(Conservation  of  Energy) base  on Stefan  Boltzmann  Law in to Earth's Climate  System using 
analytical method to  study  the  impact  of  climate  change,  effect  of  greenhouses  gases in  Earth's 
surface temperature, and spreading societal awareness of its dangers.

Earth's climate system is a complexity system, that is schematically made up of five components: the 
atmosphere;  the  hydrosphere  (oceans,  lakes,  and  other  bodies  of  water); the  cryosphere  (snow  and 
ice);  the  lithosphere  (land  surface);  and  the  biosphere  (all  living  things). These components  do  not 
exist  in  isolation;  they  are  interconnected  and  interact  at  several  levels,  either  directly  or  indirectly 
[1,16]. The system as a entire is powered by solar radiation and develops under the influence of its 
own internal dynamics through ocean currents and atmospheric circulation. On the other hand, there 
are  external  factors  which  drive  the  system;  these  are  called forgings,  include both  natural 
phenomena  such  as  cyclical  changes  in  the  Earth's  orbit  around  the  Sun,  volcanic  eruptions, 
variations  in  the  solar  output,  and  human-induced  (anthropogenic)  factors  like  changes  in 
atmospheric  composition,  human  activities,  and  so  on.  Climate  changing  is  one  of  the  greatest 
threats  towards  the  survival  of  mankind  on  the  earth, it  is  well  understood  that  human  beings 
activities  hold  the  main  responsibility  behind  these  climate  changing  issues  beside  the  natural 
disasters [1].

Theoretical models of the Earth's climate system most often involve systems of proportional models, 
ordinary differential and  partial  differential equations and  qualitative  behavior  of  their  solutions. 
Mathematical modeling is one of several approaches that have been used to study the climate system 
and  become  the  most  important  techniques  for  complicate  problems  in  modern  science.  However, 
enable  a  physically  based  estimate  the  range  of  future  climate  change,,  providing invaluable 
scientific information towards political and societal decision maker for well plan in order to protect 
our Planet.

According to the study, it has been found that, the Green house gases factor (0 ≤ β ≤ 1) and other 
relevant  human  activities play  a prominent role  in  Earth's  climate  change. Furthermore

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Mathematical calculation provided actual value for greenhouse factor β = 0.76, which lead to 

moderate temperature in Earth's surface all seasons. 

 

Key Word: Energy conservation, Mathematical Modeling, Greenhouses gases, Earth's 

Climate system, Global warming.  

 

1. INTRODUCTION 

Today climate change became an important issue in our planet, the study economic and 

social effects on societies as a whole and also its effect on global peace and security. Earth's 

climate system is a complexity system, that is schematically made up of five components: the 

atmosphere; the hydrosphere (oceans, lakes, and other bodies of water); the cryosphere (snow 

and ice); the lithosphere (land surface); and the biosphere (all living things). These 

components do not exist in isolation; they are interconnected and interact at several levels, 

either directly or indirectly.Climate changing is one of the greatest threats towards the 

survival of mankind on the earth. This motivates us to investigate climatic changing effects 

via fluctuations of the temperature, wind patterns and precipitation taking place in a given 

geoenvironment [1,12]. 

Practically the Sun is the main resource of energy that powers the Earth's climate system. 

This energy comes in the form of electromagnetic radiation, which originates from different 

depths in the Sun's interior. According to Physicist (Stefan Boltzmann), energy conservation 

theory, energy can change from one form to another but the total amount of energy remains 

constant. 

A complete climate model contains physical descriptions of all five components mentioned 

above and takes into consideration their coupling. Some components may be described in a 

simplified form or even be prescribed [16]. 

Mathematical model is a tool to describe physical phenomena, providing the world with a 

clear mathematics view on the current state of knowledge in climate system, climate change, 

weather predictions and its potential environmental and socio-economic impacts. 

 

2. Problem Statement: 

Although much progress has been made over the decades, our Planet still faces multiple 

societal, economic and environmental challenges. Climate changing is one of the greatest 

threats towards the survival of animals and mankind on the Earth. Climate change has 

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already started to cause a wide range of physical effects with serious implications for 

investors and businesses. 

 

3. Objective of Study: 

The study focus on well known physics phenomenon (energy conservation theory base on 

Stefan Boltzmann Law) implemented to temperature of Earth's surface using essential 

mathematical modeling techniques in order to achieve the following:   

(i) Employ step by step mathematical model techniques to study the effect of greenhouse 

gases to Earth's climate and climate change.  

(ii) Mathematically, evaluate the suitable greenhouse gases factors which moderate Earth's 

surface temperature. 

(iii) Spread the weariness between the communities and countries about the activities that 

will release more greenhouse gases. Also its best practices in clean energy-related research 

and its subsequent contribution to the unanimous goal set in Paris Climate Change 

Conference 2020, of achieving net zero greenhouse gas emissions by 2050. 

 

4. Definitions and Concepts:  

4.1. Climate  

This is a word from ancient Greek “klima”, meaning inclination. Climate is commonly 

defined as the weather averaged or the statistics of weather over a long period. The standard 

averaging period is 30 years, but other period may be used depending on the purpose. It is 

measured by assessing the patterns of variation in temperature, humidity, atmospheric 

pressure, wind, precipitation, atmospheric particle count and other meteorological variables 

in a given region over long period of time. (climate definition in many references). 

 

4.2.Global Solar Radiation balance of the climate system 

Radiation is the transfer of energy by electromagnetic waves, which do not require a 

medium, such as air, for their transmission. Solar radiation is relatively short wavelength 

radiation emitted by the Sun. Thermal-infrared radiation is relatively long-wavelength 

radiation emitted by the Earth, atmosphere, and clouds. The Earth’s surface receives solar 

radiation during the day only, but its surface and atmosphere emit thermal-infrared radiation 

during day and night see [12]. The Sun is the only relevant energy source for the climate 

system on a temporal scale of less than about 10 �years. The different energy fluxes are 

Coming from the Sun, on average 341 W/m2 reach the top of the atmosphere, while barely 

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half of this is available for heating of the Earth’s surface. Major parts of the short-wave 

radiation are reflected by clouds or reflected directly on the Earth’s surface itself and are 

absorbed by the atmosphere. Incoming radiation contrasts with surface long-wave outgoing 

radiation of around 396 W/m2. Virtually Earth's climate system receives all its energy from 

the Sun. This energy comes in the form of electromagnetic radiation, which originates from 

different depths in the Sun's interior. As fact some of the energy is absorbed in the solar 

photosphere, further absorption in the Earth's atmosphere gives the solar spectrum at the 

Earth's surface its more ragged appearance, for more information see [1, 16].  

The points above give us main idea about the energy resource and energy balance, the 

mathematical explanations is Input Energy equal to Output Energy (Energy Conservation 

Law), and there are many factors affect the balance.    

  

4.3. Global warming 

Global warming is the increase in the average temperature of the Earth’s near-surface air and 

the oceans it can also be defined as a gradual increase in the overall temperature of the 

earth’s atmosphere generally attributed to the greenhouse effect caused by increased levels of 

carbon dioxide CO2, chlorofluorocarbons CFCs, and other pollutants [1,13]. 

In our point of view, the release of greenhouse gases is main factor that causing  the global 

warming which lead to increasing in the overall temperature of the earth’s  as well as climate 

change and  its Consequently such as (desertification, drought, flood, volcanoes, Tornadoes, 

Forest fires).         

 

GREENHOUSE GASES  

Greenhouse gases are thought to be the main contribution to climate change (The greenhouse 

effect). They are very efficient in trapping heat into the atmosphere; therefore, it results in the 

greenhouse effect. The solar energy is absorbed by the earth’s surface and then reflected back 

to the atmosphere as heat. Then as the heat goes out to space, greenhouse gases absorb a part 

of the heat. After that, they radiate the heat back to the earth’s surface, to another greenhouse 

gas molecule, or to space (The Green Effect). Daniela Burghila et al. stated in “climate 

change effect- where to next”, the biggest concern scientists have is about the emission of 

CO2 since it is about 75% of the total global emission of greenhouse gases [7, 9, 10]. 

The major greenhouse gases in the earth’s atmosphere are: 

 Water vapour (H2O) 

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 Carbon dioxide (CO2) 

 Methane (CH4) 

 Nitrous oxide (N2O) 

 Ozone (O3) 

 Chlorolfluorocarbons (CFCs) 

 Hydrofluorocarbons (HFCs) [10]. 

As we mentioned that the release of greenhouse gases are very efficient in trapping heat into 

the atmosphere, Therefore, reducing the emission of these gases become the major goal in the 

study specially Carbon dioxide (CO2). 

 

2.5. Climate model 

A climate model is essentially a representation of the many interactions and dynamics within 

the climate which includes the atmosphere, ocean, land surface, and ice to make predictions 

of possible climate change for the future. Climate models are systems of proportional 

equations, differential equations based on the basic law of physics, fluid motion and 

chemistry. Mathematician are using mathematical models to modulating the scientific 

Phenomena in Earth's climate system to give good insight and predictions about climate 

change and its consequently implication.      

 

2.6. Theorem: First Law of Thermodynamics (Energy Conservation): 

The first law of thermodynamics state that the energy can neither be created nor destroyed, 

but can be converted from one form to another. In another form, during an interaction, energy 

can change from one form to another but the total amount of energy remains constant [5,8]. 

Therefore the first thermodynamics law is obtained on the experimental basis. In the other 

words, we can say that the energy of an isolated system is always constant. 

 

2.7. Mathematical Model 

 Models describe our beliefs about how the world functions. In mathematical modeling, we 

translate those beliefs into the language of mathematics [2, 3]. 

In the real world, these beliefs are phenomena in natural sciences or socioeconomic sciences 

along with some observations. 

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Here we designed the following diagram represents the general description of mathematical 

model techniques for all Phenomena in science and socioeconomics sciences step by step till 

test the validation of the model.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

The above diagram is the complete mathematical modeling process for any phenomenon in 

our real world. The process starts with the real world in Natural or Social Sciences along with 

with a physical system and some observations or an experiment notice. When the laws of 

physics that are thought to govern the behavior of the system are translated in mathematical 

terms, the result is what is called a mathematical model. By Understanding the Mathematical 

Concepts deeply such as the model givens, formulation and solution of the model. The 

mathematical model is subsequently analyzed for its properties and used to generate 

predictions about the behavior of the system in a changing environment. These predictions 

are tested against observations, so if there is no discrepancy between predictions and 

observations, the model will accept; otherwise, the model will refine or improve, returning to 

model formulation or givens by considering all the impressive factors in our model in order 

to achieve good results which agreed with real observations by repeating the process until the 

The Real World 

(Phenomenon) 

Social Sciences Natural Sciences 
 

The conceptual World 

 (Observations) 

Object / System 

Understanding the Mathematical Concepts   

Model Givens  

Variables/Parameter  
 

Mathematical 

Model Formulation 
Mathematical 

Model Solution  

Model Assumptions 

and Predictions, 

Model Test Invalid 

  

Valid/ Consist with Conditions and 

Principle of Assumption 

Improve by Revise the Model 

Givens and Formulation 

Decision 

    Apply  

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best result attain. Thus, mathematical modeling is an iterative process. Mathematical climate 

models enable a physically based estimate and predict the future climate change [4,5,6]. 

Therefore we applying the above diagram or flowchart to energy conservation theory in 

Earth's climate system in sequence to give an approximate solutions or predictions for 

sophisticate problem in Earth climate system, to help decision maker for well plan in order to 

protect our planet.  

 

2.8 Analytical Techniques: 

Energy Balance Model (Energy Conservation): In Earth's climate system, model builds a 

series of zero-dimensional energy balance models. The Earth's climate system is described in 

terms of a single variable, namely the temperature of the Earth's surface averaged over the 

entire globe. Conceptually  the Sun emits the radiation in the ultraviolet (UV) regime 

(wavelength less than 0.4 μm). This energy reaches the Earth's surface, where it is converted 

by physical, chemical, and biological processes to radiation in the infrared (IR) regime 

(wavelength greater than 5 μm). This IR radiation is then reemitted into space. In equilibrium 

state the Earth's climate the average temperature of the Earth's surface does not change, so 

the amount of energy received must equal the amount of energy re-emitted see [10]. A 

complete climate model contains physical descriptions of all five components mentioned 

above and takes into consideration their coupling. Some components may be described in a 

simplified form or even be prescribed [16,15,14]. 

 

Mathematical Formulation: To well understand the Mathematical concept about our 

physical phenomena (Energy Conservation Theory) in Earth's climate system, we have to 

classify the model components, such as units, variables, and constants, these factors may 

influence direct or indirect in our model [11], 

 T, the temperature of the Earth's surface averaged over the entire globe, in Kelvin (K) 

or Celsius scale,( variable). 

  R, the radius of the Earth, parameter (constant). 

 A, the energy flux density (also referred to as the energy flux)|the amount of energy 

(W) flowing through a flat surface of area 1��. From satellite observations we know 

that the energy flux from the Sun is  � =  1367.6����. parameter (constant). 

  σ, sigma, Stefan Boltzmann constant; its value is 5.67 × 10���������parameter. 

 The Earth's viewed as a disk from  the Sun 

 The area of the disk as seen by the Sun is ���. 

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 The amount of energy from the Sun to the Earth (disk) is the Incoming Energy (W)  

��� = ����  

 As fact, all bodies radiate energy in the form of electromagnetic radiation. 

 The temperature of the body is essential in our study, because the amount of energy 

radiated deepened on it. 

 In Physics, shows that black - body radiation is given by the Stefan-Boltzmann law, in 

���� units [11]. ���(�) = ���.  

 The area of the Earth's surface is 4���  

 The amount of energy radiated out by the Earth surface is outgoing energy(W) 

���� = 4������  

 

The Model Formulation and Solution: 

The First Law of Thermodynamics speaks about the conservation of the energy: ''Energy 

cannot be destroyed or created; it can only be transformed''. At thermal equilibrium, the 

incoming energy (temperature) must equal to outgoing energy and with use Stefan 

Boltzmann Law, such that  

��� = ����  

Therefore the mathematical model represent the following equation 

 ���� = 4������  ⇒ � = 4���   

TO solve the equation for T, thus � = (
�

��
)

�

� 

Where � = 5.67 × 10��and � = 1376.6 

The value of � = (
����.�

�×�.��×����
)

�

� ≈ 278.7� 

 

Interpretation: The Earth's temperature increase if the incoming energy is greater than 

out coming energy and decrease if the incoming energy is lower than out coming energy. 

Whoever the Earth's temperature remains constant if the incoming energy balances the out 

coming energy and the planet said to in thermal equilibrium, this as normal understanding.  

The value of � =  278.7�   which equals to 5.5 degrees Celsius, but actual average 

Earth's temperature approximately 16 degrees Celsius, this is a huge difference between 

the predictions (the calculated value) and observation (the actual value). 

 

Decision: According to the interpretation above, the model*1* is invalid. Therefore the 

result should be rejecting and the model must be improve. 

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Model*2*   

In order to get a better model, returning to the model givens, formulation and give deep 

insight to the problem. In model *1* we omitted a number of important factors. These 

factors represent incoming energy from Sun reflected back out to space. Snow, ice, and 

clouds, for example, reflect a great deal of the incoming light from the Sun. We use the 

term albedo ('' The fraction of the incoming solar energy scattered by Earth back to space 

is referred to as the planetary albedo"). to measure the Earth's reflectivity.   

 

Model *2* formulation: 

By adding the reflectivity factor into the constant of above model*1* 

 �: albedo. The Earth's average ������ is about 0.3, which means that roughly 70% of 
the incoming energy is absorbed by the Earth's surface. 

    
The Model *2* Build: 

The amount of energy reaching the Earth is incoming energy (W) 

     ��� = ����(1 − �)  
The amount of energy radiated out by the Earth is Outgoing energy (W) 

E��� = 4πR�σT�  

 

The  Model Solution: 

According  to conservation of the energy and thermal equilibrium state, the incoming 

energy equal to out coming energy and Stefan Boltzmann Law such that  

   ��� = ���� ⇒ ����(1 − �) = 4������ ⇒ �(1 − �) = 4��� 

   Therefore � = �
�(���)

��
�

�

�
= �

����.�×�.�

�×�.��×�����

�

�
≈ 254.9�  

 
Interpretation: Although we consider the ������ factor or the energy absorbed by the  

Earth's surface, but the value of temperature � = 254.9�, this equivalent to 

−18.25 degrees Celsius, its prediction of the temperature value at equilibrium is worse 

than the prediction of Model*1* ,the difference is still wide mathematically. 

 

Decision: By test the model *2* , we rejecting the result and revise the model*2* in order 

to refine or improve the predictions.   

 

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Model *3*: The result in model 2 did not give the expected accuracy value, so it's better           

     to look carefully to another factors  The only option is to look where we might have 
overlooked something in the model. In this cycle, we focus on the outgoing radiation,    

which come from chemical reaction, nature and human activities.   .  

 

Mathematical Formulation: 

  Greenhouse gases come from chemical reactions, nature such as Volcanoes or human       

activities such as burning fossil fuels, like carbon dioxide (CO�,H�O, CH�, NO� ) methane, 

and water, as well as dust and aerosols have a significant effect on the properties of the 

atmosphere.  

 

Model*3* build and solution:  

Experimentally β ,Greenhouses factor ranging 0 ≤ β ≤ 1  

Incoming energy (W): E�� = πR�A(1 − ρ)  

The amount of energy radiated out by the Earth is Outgoing energy (W): 

    E��� = 4πR�βσT� 

 

Model*3*Solution:  

According  to conservation of the energy and thermal equilibrium state, the incoming   

energy equal to out coming energy and  Stefan Boltzmann Law,  such that  

  E�� = E���,  

   Thus  T = �
�(���)

���
�

�

�
≈ 282.9K  

 

Interpretation: 

The value of β = 0.66 gives a climate model that correctly predicts the current global 

average temperature T ≈ 282.9K, which equivalent to 9.64 degrees Celsius, this gives 

reasonable approximate value to the Earth's surface temperature. Mathematical calculation 

provided actual result for greenhouse factor β = 0.76, this may lead to moderate 

temperature in Earth's surface all seasons. 

The effect on the outgoing radiation is difficult to model because it depend on the 

Greenhouse gases mostly depend on human activities which are increasing daily. 

Meanwhile, it's manageable, if there is a strong determination to reduce this factor to 

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reasonable value, since its human factor,. However many conferences recommended, 

reducing the greenhouse gases which mainly belong to human activities. 

 
Result Discussion: Step by step mathematical model techniques has been implemented 

Overall result in model1*1*, model*2* and model *3*, it's clearly indicate the effect of 

greenhouse gases factor β which range ≤ β ≤ 1 very huge. However, this factor is 

controllable or reducible to appropriate value because mainly depend on human activities.     

 

Conclusion: Concept of energy conservation, Stefan Boltzmann Law, and step by step 

mathematical modeling techniques have been implemented, concentrated on the 

inconsistency of the results between the prediction and observations. On the other hand 

the model has been improved by considering the other factors, which can affect the 

results.  

The study shows, the effect of greenhouse gases to the Earth's climate system, by 

increasing Earth's surface temperature, which causes many problem, such as 

desertification, drought, flood, volcanoes, Tornadoes, and Forest fires. Mathematical 

calculation provide an appropriate value to greenhouses factor to be  β = 0.76 in order to 

get the Earth's surface temperature 16 degrees Celsius.   

The impacts of climate change are expected to grow more severe over the coming years 

and decades unless the industrial countries take the initiative seriously to reduce the 

greenhouses gases and spreading the wariness between the societies and communities 

about the advantage and disadvantage of climate change and it's consequently. 

  . 
                 

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[2] Glenn Marion, An Introduction to Mathematical Modeling, Bioinformatics and Statistics 

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[4]. John A. Trangenstein. Numerical Solution of Hyperbolic Conservation Laws,Department 

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[5] Goosse H., P.Y. Barriat, W. Lefebvre, M.F. Loutre and V. Zunz (2010). Introduction to 

climate dynamics and climate modelling. http://www.climate.be/textbook, 2010. 

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[6]. Roger K. Smith and Wolfgang Ulrich. LECTURES ON NUMERICAL 

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