Dynamical Analysis of the Global Warming

and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 ISRN Applied Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 International Journal of Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Journal of Function Spaces International Journal of Mathematics and Mathematical Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2013


Introduction
The standard approach for modelling natural and artificial phenomena in the perspective of dynamical systems is to adopt the tools of mathematics and, in particular, the classical integral and differential calculus.
Fractional calculus FC is a common expression that is used to denote the branch of calculus that extends the concepts of integrals and derivatives to noninteger and complex orders 1-9 .During the last decade FC was found to play a fundamental role in the modelling of a considerable number of phenomena 10-15 and emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations.As a consequence, nowadays, the application of FC concepts encompasses a wide spectrum 2 Mathematical Problems in Engineering of studies [16][17][18][19] , ranging from dynamics of financial markets 20, 21 , biological systems, 22, 23 and DNA sequencing 24 up to mechanical 13, 25-28 and electrical systems 29-31 .The generalization of the concept of derivative and integral to noninteger orders, α, has been addressed by many mathematicians.The Riemann-Liouville, Gr ünwald-Letnikov, and Caputo definitions of fractional derivative, given by 1.1 -1.3 , are the most used 32 : where Γ • represents the Euler's gamma function, the operator x is the integer part of x, and h is a time step.The Laplace transform applied to 1.1 yields where L and s denote the Laplace operator and variable, respectively.The Mittag-Leffler M-L function, E α t , plays an important role in the context of FC, being defined by This function establishes a connection between purely exponential and power law behaviours that characterize integer and fractional order phenomena, respectively.In particular, if α 1, then E 1 t e t .For large values of t, E α t has the asymptotic behaviour: The Laplace transform 1.7 permits a natural extension of transform pairs from exponential function and integer powers of s towards M-L function and fractional powers of s: The generalization promoted by FC leads directly to fractional dynamical models, but the fact is that neither their limits of application nor the methods and tools for capturing them seem to be well defined at the present stage of scientific knowledge.
This paper studies the complex dynamics characteristics of the global warming.It is believed that human activity is the main cause of such a phenomenon, and dramatic consequences to the planet are expected if the warming trend observed in the last century persists.The main goal is to analyse and discuss the characteristics of the global warming in the perspective of dynamical systems, which is a new standpoint in this context.It is shown that the application of Fourier transforms and power law trend lines leads to an assertive representation of the global warming dynamics and a simpler analysis of its characteristics.
The paper is organized as follows.Section 2 contextualizes the main subject.A heuristic approach to analyse the data from the meteorological stations in the time domain is proposed, and several characteristics of the global warming are exposed.Section 3 formulates the framework of the analysis in the perspective of FC and analyses the fractional dynamics of the system.Finally, Section 4 outlines the main conclusions.

Characteristics of the Global Warming
Earth is warming, and it seems that human activity and solar effects are the main probable causes 33-35 .Some impacts such as the record of high temperatures, the melting glaciers, and severe flooding are becoming increasingly common across the countries and around the world 36, 37 .Aside from the effect on temperature, warming leads to the modification of wind patterns, the development of humidity, and the alteration of the rates of precipitation.These phenomena are being the subject of intensive research due to major impact on social, economic, and health aspects of human life 38-40 .
Figures 1 and 2 show average temperatures computed for two decades separated by almost one hundred years.The white marks on the maps represent meteorological stations.Figure 1 is the contour plot of the worldwide temperatures corresponding to the period 1910-1919, and Figure 2 corresponds to the period 2000-2009.The temperature difference between the two decades is presented in Figure 3, showing that the northern hemisphere has been more affected by warming.
In our study, the Global Historical Climatology Network-Monthly GHCN-M , version 3 dataset of monthly mean temperature 41 , available at the National Oceanic and Atmospheric Administration, National Climatic Data Center NOAA-NCDC http://www .ncdc.noaa.gov/ghcnm/v3.php, is used.The current archive contains temperature records from 7280 meteorological stations located on land areas.However, few stations have long records, and these are essentially restricted to the northern hemisphere the United States and Western Europe .As the computation of the Fourier transform requires quite long time series, a sample of 210 worldwide meteorological stations, distributed as uniformly as possible, and having 100 years length records, was selected.Most stations of Africa, Alaska, Canada and the northern and southern regions of the globe do not meet the previous condition, which means that the results for these regions and also for sea areas , plotted on the maps, may be less accurate.
Each data record consists of the average temperatures per month.Some occasional gaps of one month in the data represented on the original data by the value −9999 are substituted by a linear interpolation between the two adjacent values.Moreover, although of minor influence, the distinct number of days of each month and the leap years are also taken into account.For the whole sample of meteorological stations, as the data is available for slightly different periods of time, depending on the station, the period from January 1910 up to December 2010 is considered for all cases.In this study, a heuristic decomposition of the time series is first proposed.The temperature signal from the ith meteorological station, T i t , i 1, . . ., 210, is approximated by the sum given in the following: where t is time, ω 2π/T represents the angular frequency, and T is one year.The coefficients a 0 and a 1 are the parameters of a trend line adjusted to the original data, T i t , using the least squares algorithm.This trend line is then subtracted from the signal T i t , and, for the result, the two first harmonics of the Fourier series are calculated.The corresponding coefficients are a 2 , a 3 and a 4 , a 5 , respectively.
Figures 5-11 show the mapping of the coefficients.Coefficient a 0 is closely related to the average temperature and highlights the warmer regions of the globe Figure 5 , whereas coefficient a 1 Figure 6 emphasizes the gradient of temperature increase.Consequently, Figure 6 is highly correlated with Figure 3.
The parameters of the first harmonic, namely, a 2 and a 3 , are depicted in Figures 7 and  8, respectively.The amplitude of the sinusoid Figure 7 unveils a strong mark centred in Siberia and a weaker, but also clear, mark in North America, respectively.Figure 8 represents the map of coefficient a 3 , corresponding to the phase of the sine function.As expected, northern and southern hemispheres are in phase opposition.The analysis and physical meaning of the coefficients a 4 and a 5 that correspond to the second harmonic of the heuristic approximation are more difficult and seem to point to a less significant meaning Figures 9  and 10 .Nevertheless, those parameters might also reveal relationships hidden in the data that can trigger a future comprehensive explanation of these phenomena.
It is important to notice that the heuristic approximation given by 2.1 captures most of the energy of the original signals.Moreover, the energy contained in the second harmonic is almost negligible.This is illustrated in Figures 11 and 12. Figure 11 represents the percentage of energy captured by the heuristic approximation with reference to the total energy of the original signals, revealing a percentage in the interval 86% 99% .Figure 12 represents the case of not including the second harmonic.We verify that it exhibits only slight differences when compared to the previous one.

Dynamics of the Global Warming
In this section, the global warming phenomenon is analysed in the perspective of a complex system that reacts to stimuli, being the response signals studied by means of the Fourier transform.The methodology is to obtain a representative signal as a manifestation of the system dynamics, process it with the Fourier transform, and, given the characteristics of the resulting spectrum, to approximate its amplitude by means of a power function.
In analytical terms, for a continuous signal x t , evolving in the time domain t, we have where F represents the Fourier operator, ω is the angular frequency, and j √ −1.The power law approximation is given by The parameters of the power law are the pair a, b to be determined by the least squares fit procedure.
Figure 13 depicts the amplitude of the Fourier transform obtained for the meteorological station Tokyo Figure 4 , where a peak value at the angular frequency ω 1.99 × 10 −7 rad/s that corresponds to a periodicity of one year is well visible.
At low frequencies Figure 14 , it is clear that the spectrum can be approximated by a power law with parameters a, b 292.1047, −0.8397 , leading to a fractional value of parameter b.
In the sequel, the values of a, b were computed for the whole sample of meteorological stations, using the least squares fit procedure.It was found that there exists a strong correlation between the two parameters.In fact, Figure 15 illustrates clearly the relation between log a and b.It can be seen that a straight line fits quite well into the data.Figures 16 and 17 depict the contour plots of log a and b, respectively.Therefore, we will concentrate our attention on one parameter only, namely, on b that represents the variation of the signal energy versus ω.
The map of parameter b Figure 17 reveals that climate changes are taking place in the northern hemisphere.Two large regions of Russia and Canada and, in a less extent, central Europe and Western Alaska are being the most affected areas.
As expected, Figure 16 is somewhat "redundant," since Figure 17, with parameter b, is sufficient to characterize the warming dynamics.We verify that abs b varies between 0 and 1 that can be viewed as the cases of white and pink noises, respectively.Therefore, we conclude that equatorial and south hemisphere regions exhibit more "correlated" variation, while the north hemisphere and the two poles have a more "erratic" variation of the temperature.
These results are of utmost importance because we can capture and analyse all information through a single map.In a different perspective, we should also note that the adoption of FC concepts captures large memory effects present in long time series, which is the case of Earth's warming.Therefore, these results encourage further research in this line of thought.

Conclusions
This paper analysed the global warming in the perspective of complex systems dynamics.The use of Fourier transforms and power law trend lines revealed fractional order dynamics characteristics of the phenomenon.While classical mathematical tools could be adopted, Mathematical Problems in Engineering

Figure 4 Figure 4 :
Figure 4 depicts the time evolution of the monthly average temperature of one typical station Tokyo, Japan, Lat 35.67 N, Lon 139.75 W , where three processes are visible, namely,i a continuous, almost linear, temperature increase, ii an annual periodic variation, and iii a "random" temperature variation that may be the symptom of a fractional dynamical behaviour.

Figure 5 :
Figure 5: Map of coefficient a 0 of expression 2.1 .

Figure 6 :
Figure 6: Map of coefficient a 1 of expression 2.1 .

Figure 7 :
Figure 7: Map of coefficient a 2 of expression 2.1 .

Figure 8 :
Figure 8: Map of coefficient a 3 of expression 2.1 .

Figure 13 :Figure 14 :
Figure 13: Amplitude of the Fourier transform of the monthly average temperatures of Tokyo.

Figure 15 :
Figure 15: Mapping of the power law parameters log a , b .

Figure 16 :Figure 17 :
Figure 16: Contour plot of parameter log a .