On the Walker’ Model for the Carbon Dioxide in the Earth’s Atmosphere

Climate change is a wicked problem because it is hard to say what the problem is, and to define it clearly. However, we know that global temperature rise correlates with increasing levels of atmospheric carbon dioxide [1] and [2]. In this paper, we analyze a model for the carbon dioxide developed by Walker in [3] with several source terms. Our numerical results show that the burning fossil fuels have an effect on the carbon dioxide in the earth’s atmosphere and the climate change problem, one of the major global challenger of our time.


Introduction
All of us know that Mathematics and Physics have traditionally shared their interests and developments; many are the historical examples in which physical phenomena have motivated original mathematical theories and vice versa. On the contrary, the connection between Mathematics and Life Sciences (Ecology, Zoology, Physiology…) have been rather scarce, perhaps because they have kept themselves in a prudently descriptive. This does not mind that biologists have ignored the mathematical progresses (Biostatistics is a common subject in all the curricula of the university degrees in Biology), but if we do not consider this tool F. Vadillo to arrange laboratory data or samples, it was very scarce connection between Mathematics and Biology until a very few years ago. A proof of this affirmation is that the main scientific paper which deals with this topic, Journal of Mathematical Biology appeared in 1974, and the development of Computation Sciences allows to study the planet Earth using the universal trilogy: modeling, analysis and control proposed by Prof. J.L. Lions in [4]. In fact, the application of Mathematics in Life Sciences is a speciality which is growing at a spectacular pace. Besides the classical [5] and [6], more recent texts, for example [7]- [21] explain other models in Physiology, Genetic, Cellular Biology, Climatology … or [22], [23] and [24] develop stochastic differential equation models applied in Epidemiology, Predator-Prey Problems, Chemical Reactions … and even [25] uses Variational Methods in Molecular Modeling.
The carbon dioxide (CO 2 ) is a colorless and odorless greenhouse effect gas which affects climate because it traps the hot that otherwise would dissipate to the space; thus, it is directly associated with the Earth's temperature. In other dead planets, the whole atmosphere is almost exclusively composed by CO 2 ; for instance, Venus' atmosphere consists in a 98% of that gas and the temperature on its surface reaches 477˚C.
At the beginning of the XX century, three of every 10,000 earth's atmospheric molecules contain CO 2 ; so one can say that, although it was not a very abundant gas, it was important because it did prevent our congelation. If the CO 2 percentage in the atmosphere became twice as much, some scientists consider that the average temperature of our planet would go up by between 3˚C and 6˚C, with consequences not difficult to imagine.
The carbon dioxide cycle can be summarized in the following scheme call the byogeochemical cycle in Figure 1.
It is therefore evident that the burning fossil fuels has a effect on the carbon dioxide in the earth's atmosphere. Nowadays, the coal combustion power stations are the ones which produce the highest rate of CO 2 by energy unity. In Geographic magazine, the emissions of this greenhouse effect gas produced from other fuels as petroleum, natural gas, biomass … are much smaller; in the USA, for instance, an 83% of the CO 2 emissions is calculated to come from the coal combustion power stations.
The situation becomes worrying when we know that, nowadays, the biggest electricity generation source is the coal combustion. The USA consume nowadays some 1094 million tons of coal which generate approximately the 50% of the electric energy they consume. Which happens in other parts of the world? Here the Asiatic giant does appear: China consumes about 1531 million tons of coal and its demand will grow more than twice as much towards the year 2025 in order to satisfy the industry and consumers. According to the sources commented above, worldwide coal consumption will increase by a 56% by the year 2025, with the resulting carbon dioxide emissions. This paper is organized as follows. In Section 2 we describe the Walker' Model for Carbon Dioxide in the Earth's Atmosphere, in Section 3 awe computed its numerical solutions and finally in Section 4 we draw the main conclusions.
Our numerical methods were implemented in Matlab, the codes that we have used are available on request. The experiments were carried out in an Intel(R) Xeon(R) CPU ES-2620 0 @ 2.00 GHz 16 GB of RAM.

The Walker' Model for Carbon Dioxide in the Earth's Atmosphere
The seawater has a much greater storage capacity for carbon dioxide than the atmosphere because this gas is contained in seawater in three different forms: as solute carbon dioxide gas (CO 2 ), as hydrogen carbonate (HCO 3 ) and as carbonate (CO 3 ). When part of the CO 2 in the water is converted into HCO 3 and CO 3 , seawater has a much greater storage capacity for carbon dioxide than the atmosphere.
The model developed by J. C. G. Walker in [24] simulates the interaction between the exchange of the carbon dioxide stored in three media: atmosphere, the shallow ocean and the deep ocean. The model is also commented in reference [26].
and the algebraic equations for the equilibrium of CO 2 between the atmosphere and the shallow ocean

Numerical Results
The five differential Equations (1)  According to the sources commented above, it is foreseen that in the following 25 years the electricity consumption become twice as big and, unless the governments begin right now to build thousands of expensive nuclear reactors, this implies that coal and natural gas will satisfy almost the whole demand. In the second example, we will suppose that the emissions will remain constant In the third example considered, we will suppose a more optimistic situation in which an international agreement is reached in order that by the end of this century the CO 2 emissions may be a 90% smaller than the unavoidable emissions of the year 2030. The situation is represented in Figure 4. Evidently, the situation gets better, since the gas quantity is stabilized. It should cause no surprise to observe the plain in the figure, because carbon dioxide is a very stable gas that remains long time in the Earth's atmosphere; as somebody said, the CO 2 our grandmothers caused when they turned on their kitchens is probably still going around.

Conclusions
In the light of these results, it might be opportune to make several brief comments: 1) Since the mathematical models always take what is regarded as the most relevant information and does ignore many other details, the conclusions ought to be always prudent; as Lord May of Oxford says in [29], we are not vaccinated against the excesses. In any case, these numerical results do not contradict the observations, which would give a certain warranty to the model. The specialists are those who will have to discuss the details and extract the conclusions. However, slow and steady change simply leads to habituation, not action.
2) This model describes a serious and very worrying situation which remind the polemic speech of the mathematician James Lovelock in [30] in the year 2004, defending the necessity to produce electricity in nuclear centrals; it should not be forgotten that Lovelock is the writer of the famous book [31], where he arguments that Earth is only an organism which he called Gaia, and that he is a known ecologist militant as well. Obviously, in the decisions other kinds of considerations take part: economical, demographical, political … to cope with a change in climate.
Studying the world that surrounds us, the Nature, has occupied the best thinkers from the most remote times, as Voltaire says in his philosophical dictionary: Tous les raisonneurs depuis Thalès, et probablement longtemps avant lui, ont joué colin-maillard avec toi; ils ont dit: ''Je te tiens''! et ils ne tenaient rien. Nous ressemblons tous à Ixion; il croyait embrasser Junon, et il jouissait que d'une nuée (All reasoners since Thales, and probably long before him, have played at blind man's buff with you (Nature); they have said: "I have you''! and they had nothing. We all resemble Ixion; he thought he was kissing Juno, and all that he possessed was a cloud).