Parametric Sensitivity Analysis of a Mathematical Model of the Effect of CO2 on the Climate Change

Mathematical modeling is a very powerful tool for the study and understanding of the climate system. Modern climate models used in different applications are derived from a set of many-dimensional nonlinear differential equations in partial derivatives. The Climate models contain a wide number of model parameters that can describe external forcing that can strongly affect the behavior of the climate. It is imperative to estimate the influence of variations in parameters on climate change. The methods of 1-norm, 2-norm, and infinity-norm were used to quantify different forms of the sensitivity of model parameters. The approach applied in this research involves coding the given system of continuous non-linear first order ordinary differential equation in a Matlab solver, modifying and coding a similar program which is used for a variation of a single parameter one-at-a-time while other model parameters are fixed. Finally, the program is used to calculate the 1-norm, 2norm, 3-norm and infinity norm of the solution trajectories in the same manner. The study shows that the most sensitivity parameters in the model are the concentration of a suitable absorbent and the rate of inflow of absorbent in the absorption chamber.


Introduction
The global climate change is one of the biggest issues facing humanity today. Consequently, the prediction of future climate as well as changes in climate due to changes in natural processes and human-caused factors (e.g. greenhouse gas emissions) are issues that have deservedly received significant attention.
Mathematical climate models are mostly deterministic with a large-phase space dimension, containing a vast number of various parameters according to Palmer [1], Panchev and Spassova [2] Lea et al [3], Soldatenko et al [4] and Soldatenko and Chichkine [5] Equations that describe the evolution of the ECS and its processes are quite complicated. Therefore, in the majority of situations, we, unfortunately, cannot solve them analytically with an arbitrary set of initial conditions, even for very simple cases. We can only find an approximate solution using numerical methods. It was opined by McGuffie and Henderson [6], Barry and Hal [7] Goosse H [8] (2015) that in dealing with real-life or physical problems, mathematical modelling is always of great advantage because of its power to predict system behaviour and a clear insight of the important inputs and outputs. Mathematical models are of various forms such as deterministic, stochastic, fuzzy, and uncertain forms. However, there is a need to model problems arising from the Global warming due to carbon dioxide potentially affects human and the environment, the mitigation of carbon dioxide from the atmosphere is absolutely necessary. There is no dispute whether the presence of CO 2 in the atmosphere influences temperature rise. What is therefore an open question is the degree of impact of carbon dioxide has on the temperature. Researchers have carried out several investigations on ways to reduce the concentration of carbon dioxide into the atmosphere significantly by introducing liquid droplet Rosenwasser and Yusupov [9]. But Chang, et al [10], made a theoretical analysis to study the dynamics of carbon dioxide uptake by a liquid droplet and have observed the abatement in carbon dioxide concentration absorbed by liquid droplets. On their part, Han et al. [11] have established an experimental set-up to calculate the liquid phase mass transfer coefficient of carbon dioxide absorption by single water droplet. The result indicated that the mass transfer coefficient decreases as droplet formation time increases. Also, researches have been carried out to study the removal of carbon dioxide by using suitable absorbents such as sodium hydroxide, amines and aqueous ammonia solution (Bai and Yeh, [12]; Resnik et al, [13] Zeman and Lackner, [14], Stolaroff et al. [15]. It has also been discovered that the reduction in carbon dioxide concentration by aqueous ammonia solution is more predominant. As a result of this, Yeh and Bai [16] have made an experiment to compare the performance of two absorbents, aqueous ammonia solution and mono-ethanolamine (MEA), for scrubbing carbon dioxide in terms of high CO 2 removal efficiency and absorbing capacity.
Mathematical modelling represents a very powerful and effective instrument to study complex processes occurring in technical, economic, social and natural systems. State of the art mathematical models used in various branches of natural science are defined as a set of (partial) differential equations that contain a large number of parameters some of which can be inaccurate. Parameter errors and their time and space variability generate parametric uncertainty in mathematical models. It is therefore important to estimate the influence of parameter variations on the model output results. Sensitivity analysis, which is an essential element of model building and quality assurance, addresses this very important issue

Materials and Methods
Consider an atmosphere affected by global warming gases containing CO 2 . To successfully model the phenomenon Mathematically, the following assumptions made by Shyam et al, (2014).
1. The rate of emission of carbon dioxide is constant.
2. There exists a threshold concentration of carbon dioxide below which harmful effects are insignificant. 3. The rate of introduction of liquid species in the atmosphere is in direct proportion to the difference of cumulative and threshold concentrations of carbon dioxide. Here, cumulative concentration stands for global average temperature of carbon dioxide in the atmosphere. 4. The rate of inflow of absorbent in absorption chamber is in direct proportion to the cumulative concentration of carbon dioxide. 5. The decrease in cumulative concentration of carbon dioxide is directly proportional to the cumulative concentration of CO 2 itself and the concentration of externally introduced liquid species. The decrease in concentration of CO 2 is also proportional to the product of cumulative concentration of carbon dioxide and the concentration of suitable absorbent. 6. During the interaction of carbon dioxide and externally introduced liquid species, particulate matter is formed which is removed from the atmosphere by gravity, lowering the concentration of carbon dioxide. 7. The natural depletion rates of carbon dioxide and externally introduced liquid species are assumed to be directly proportional to their respective concentrations. From the above assumptions, the following dynamical systems of nonlinear ordinary differential equations are formulated With the following conditions satisfied . The depletion of carbon dioxide due to interaction with absorbent is assumed to be in direct proportion to the cumulative concentration of carbon dioxide and that of absorbent and by the same amount absorbent is also used. The constant 1 η is the interaction rate coefficient of carbon dioxide with absorbent. All the above constants are assumed to be positive.
Numerical Simulation In this section, we perform a numerical simulation of

Method Analysis
The approximate model formulation was constructed on the basis of these simplifying assumptions. The method of sensitivity analysis which we have used in this study has been adapted from recent research reports of Ekaka-a et al [17]. A brief sketch of this method is as follows: STEP I: Code the given system of continuous non-linear first order ordinary differential equation in a Matlab programming language.
STEP II: Modify and code a similar program which is used for a variation of a single parameter one-at-a-time while other model parameters are fixed.
STEP III: Design an appropriate ODE45 Runge-Kutta scheme which will simulate the program in step I and step II STEP IV: Use the program in step III to calculate the 1norm, 2-norm, 3-norm and infinity norm of the solution trajectories in the same manner, use the same program to calculate the four popular norms of the differences of the solution trajectories.
STEP V: Based on the original parameter in the step I, calculate the cumulative percentage effect on the solution trajectories due to variation of each chosen parameter at a time when other parameters are fixed.
STEP VI: Interpret the result quantitatively. That is, the parameter which when varied a little and produces the biggest cumulative effect on the solution trajectories is called a most sensitive

Results and Discussion
Efforts shall be made to present and discuss our results which we have achieved in this study.        From Figures 1-4, it is convenient to classify the identified parameters with respect to their degree of sensitivity and relative significance.

Discussion of Results
From our results which have been presented in the previous section, we observe that the parameters c K and 1 λ which represents be the concentration of a suitable absorbent and the rate of inflow of absorbent in absorption chamber respectively can be classified as the most sensitive parameters using the 1-norm and 2-norm estimated sensitivity values while the other parameters can be classified as relatively least sensitive parameters.

Conclusion
The sensitivity of the parameters in the model shows that the higher the degree of the concentration of a suitable absorbent and the rate of inflow of absorbent in absorption chamber, the higher the quantity of carbon dioxide in the atmosphere which implies that the more of these you have, the more the temperature rise in the atmosphere which is in total agreement with result in recent literatures that when the carbon dioxide concentration goes up, temperature goes up and when the carbon dioxide concentration goes down, temperature goes down. This study has shown the more sensitive parameter in the model that can bring about climate change.