User:Conhegarty/Simple Mathematical Model of Observed Global Warming

Summary

Climate change is the single most important issue to be faced by mankind and everyone who is living on planet Earth today must be directly involved in its control. Only by having a basis knowledge of the mathematics that describe the processes that are giving rise to global warming can the seriousness of the problem be appreciated and the size of the concerted effort that is required to keep global warming under control be understood. Simple mathematical models are presented based on the conjecture that the observed increases in the annual average global temperature and the increases in carbon dioxide in the atmosphere are related to the scale of economic development since the start of the industrial revolution. Only the simplest mathematical and statistical procedures are used in the models, so that anyone with high school mathematics will be able to understand the calculations.

Simple Mathematical Model of Observed Global Warming

The industrial revolution started approximately 200 years ago (sometime between 1750AD to 1800AD). The technical mechanisms that became available at that time facilitated the evolution of society and human activities, by the controlled use of fossil fuels as a source of energy to do mechanical work. This exploitation of energy (e.g. in the mechanisation of manufacturing processes, as sources of automotive power, etc.) from that time formed the basis of a new economic paradigm for the development of commerce and industry and the growth of wealth. Since then, an ever-increasing set of new inventions, devices and processes have being introduced into our lives that increase our comfort and standard of living. Even today, this economic model of continuous growth in the production of goods and services with the associated consumption of fuel and resources continues to be the fundamental mechanism for increased prosperity and wealth.

However, all the developments since the start of the industrial revolution are also part of a very large-scale experiment that continues to be in progress on planet Earth. This experimental detail of this economic model involves the exponential growth of population, with concomitant use of planetary resources (e.g. mineral resources, etc.), alterations to land use and agricultural practice and the ever-increasing rates of consumption of fossil fuels. This economic model on which our present standard of living is based is not sustainable and it must be replaced with one that recognises the scientific constraints of the biosphere in which we exist.

The current global economic system has considerable influence on the processes that are operating within the biosphere and these have been the subjects of numerous reports by national and international agencies (including Intergovernmental Panel on Climate Change[1], UNEP [2], European Union [3], the World Bank [4], etc.). These influences are often considered to be “human activity” or “human influence on the environment”. The most conspicuous consequences of this economic model are the changes that are occurring in the climate system and this is an urgent issue of immediate importance for mankind. Droughts, forest fires, changes in weather patterns, loss of polar and glacier ice cover are also of immediate concern [5] .

All scientific reports predict adverse climate changes with significant global warming in the immediate future. Some of the most reliable predictive data on the adverse changes that are occurring to the biosphere come from very sophisticated mathematical models of the climate system. However, the mathematical descriptions of these models and the interpretations of their outputs require the expertise of specialists and are beyond the capabilities of most people. People are unwilling to consider the significance and importance of reports that use sciences with which they are not familiar or that they themselves do not understand.

Because of the seriousness of the issue of global warming, everyone who is living on Earth today must be directly involved in saving the sustaining biosphere that exists on this planet. The easiest way to gain an understanding of what is happening on the planet is to interact with a simple but accurate mathematical model of the changes that have already been observed. The value of the parameters in this mathematical model may then be altered in an experimental manner, to discover how the outputs from the model change with these alterations and the potential consequences for the real world.

Everyone must be encouraged to take an active interest in the very significant changes that are occurring on planet Earth and to be part of the solution to these problems. I therefore present here a very simple approach to the modelling of global warming (i.e. a part of the climate change issue) that is based on the observed annual average global temperatures. The level of mathematics used in the mathematical models is limited to that typically cover in high schools (i.e. second level colleges).

Mathematical Models

A mathematical model may be prepared to describe any physical system in terms of one or more simple mathematical equation. Each physical property of the real physical system that is being modeled has a corresponding parameter in the mathematical model. The numerical value of a parameter in the mathematical model corresponds to the specific state of the physical property to which it relates in the real physical system. Calculations may be performed using the equations in the mathematical model. When specific values are selected for the individual parameters in the model, the results of the calculations will correspond to the state of the physical system.

References to existing articles on mathematical models here!!

Mathematical Models of Economic Growth

Since 1500AD, records show that the population of the Earth has been growing exponentially. At that time, the structure of society was based on the old social order and food was grown using agricultural techniques based on human and animal power. It seems reasonable to assume that prior to the massive increase in industrial activities and associated use of fuels (i.e. coal and oil) since the start of the industrial revolution, mankind would have had a more limited impact on global warming during the last two hundred years. Within a short period after the start of the industrial revolution, the scale of the world’s new economy greatly exceeded that of the previously existing economy. The new economies of the world (based on the population growth, productivity, agricultural practice and changes in land use, industrial and commercial activities, fuel consumption per capita and the creation of wealth, etc.) started growing exponentially. I propose that a simple exponential model describes the size of the new economic activities at any time since 1800 AD as

Et = exp(kt) .... Equation (1)

where Et is the size of the economy in year t, k is the exponential growth factor for the economy and t is the time in years since the start of the industrial revolution. In this paper, the year 1800AD has been selected as the starting date for all models discussed.

The detailed mechanisms of all the individual processes that are operating within the global economies will not be discussed in detail here. Only the numerical value of the indicator parameters that describes the total observed integrated consequence of the sum of individual processes operating will be discussed. In figure 1, three graphs are shown to illustrate the growth in the size of the economy over time with different values for the rate constant, k.

In this paper, two simple models are prepared to describe physical properties of the environment that may depend critically on the size of economic development. These properties are (a) the annual average global temperature for each year and (b) the level of carbon dioxide gas in the atmosphere for each year. The former is an indicator of changes in the balance between incoming radiation to and outgoing radiation from Earth as a result of changes to the composition of the atmosphere that were caused by economic activity. The latter is an indicator of the level of the carbon dioxide waste product that has been released to the atmosphere as a result of the combustion of fossil fuels.

Data for Temperature Model

One of the most significant actions that allow an investigation of the change of temperature that has been taking place on planet Earth over the years has been the compilation of the table of annual average global temperatures since 1880AD that was prepared by NASA [6]. The data in this table are the most reliable tabulation of the temperature data that may be the subject of further analysis.

The data presented in the table are not the actual temperatures for each year, but rather the temperature anomaly multiplied by 100 (i.e. one hundred times the value by which the temperature in each year differed from a reference value). The reference value selected was the average value of the annual average global temperatures for the reference thirty-year period, 1951AD to 1980AD. During this reference period the annual average global temperature was 14oC. Any analysis of the trends in annual average global temperature can be based either on these temperature anomalies or on the actual temperature for each year that can be readily recalculated from these data. To simplify the description of the preparation of the simple model described here, the temperature anomalies reported in the NASA table have been converted to the observed annual average global temperatures for each year. Thus, in this analysis, the reported annual average global temperature is the variable of interest.

Based on temperature data calculated from the NASA table, the presentation in figure 2 was prepared. It is clear from this presentation that there is significant noise in the temperature data, with year-to-year variations and some periodic elevations and depressions over time within the data-set. As no uncertainty data were included in the NASA file from which the data was abstracted, no error bars are shown in the graphical presentation.

It is clear that a simple regression curve may be fitted to the data, most probably an exponential curve. From this initial assessment, the annual average global temperature on planet Earth appears to be giving a signal that is changing over time in some exponential fashion. These observed temperature data support the conjecture that the processes that are giving rise to the global warming started with the industrial revolution and are directly connected to the growth in the economic development since that time and that these are increasing exponentially in sympathy with these economic developments.

Mathematical Model for Observed Temperature Increase

Any mathematical model that described the observed temperature must consist of two terms. The first term relates to the temperature that would exist, in the absence of any increase in economic activities. The second term relates to the increment in temperature due to economic activities (i.e. human influence).

Tobserved = To (Absence of Economic Activity) + T1 (Due to Economic Activity) .... Equation (2)

In reality this second term may be a series of terms, one for each component of economic activity that has an influence on temperature. These individual processes may include the effects of carbon dioxide, methane, CFCs, ozone, oxides of nitrogen, water vapour, sulphur hexafluoride, NF3, albido, smoke, particulates, sea salts, dust & ash, etc. However, in the simple empirical model presented here, as a first approximation, the totality of these influences on temperature are compressed into the single second term in the equation. For nearly two hundred years a very simple instrument, a thermometer, has been used to monitor the progress of this experiment and the observed temperature is the signal that is recorded by that instrument. It is not possible to measure the individual terms (i.e. the first and second term) directly by observation, as only the temperature due to the combination of the terms can be measured.

Based on the concepts in equation (2), I propose that the equation that describes the temperature be based on the simple empirical equation,

Tcalc,t = To + Rf exp(t) .... Equation (3)

where Tcalc,t=t is the calculated annual average global temperature for each particular year, T0 is the temperature in the absence of economic activity, Et is the size of the economy in the year of interest, Rf is the increment in temperature per unit economic size and t is the time since the start of the experiment (i.e. 1800AD). Substituting the value for the size of the economy from equation (1) into equation (3) we get

Tcalc,t = To + Rf exp(kt) .... Equation (4)

The first term, To, defines the annual average global temperature in the absence of significant economic development. This variable will fluctuate on a year-to-year basis due to the dynamic behaviour of the weather and it is the source of the noise on the baseline of the plot in figure 3. However, the average value of this variable should be constant over long time periods and this appears to be the case from the plot shown in figure 4 (discussed below).

The second term, Rf exp(kt), relates to the increment in temperature due to the size of the economic development. Again, there is noise in this second term, but this is not considered further here.

Calibrating the Temperature Model

Before the model can be used, the values of the three parameters To, Rf and k that are used in the mathematical equation on which the model is based (i.e. equation (4)) must be defined. In figure 3, an exponential curve has been drawn through the observed temperature data for the period from 1880AD to 2008AD. The particular curve that best fits the observed data is the one where the overall total displacement of all observed data points from their corresponding calculated values on the curve is at a minimum. The statistical requirement to achieve this best fit is to have a minimum residual sum of squares. The residual sum of squares is calculated by taking the square of the difference between each individual observed annual average global temperature and the corresponding calculated value for that year from the mathematical model and adding these squares together.

Monte Carlo Method

It is necessary to find the values for the three parameters T0, Rf and k that are used in equation (4) that give the best fit of the regression curve to the observed temperature data. A simple computer program was prepared to search through a range of values for these parameters to find the residual sums of squares between the calculated temperatures from the model and observed temperatures for the period 1880AD to 2008AD. The random values that are selected for each of the parameters are restricted to values between the minimum and maximum values specified for the run in the program interface. Each time a residual sum of squares that is less than a previously calculated one, the values of the parameters that gave rise to that new minimum are printed as shown in table 1. A typical best Monte Carlo fit found after 1,000,000 trials (i.e. using different random values for the three parameters: T0, Rf and k) was To = 13.67 degC, Rf = 0.014 and k = 0.020.

It is important to realise that because the Monte Carlo method is based on selecting random values for the parameters of interest and then calculating the residual sum of squares for these values, the method drifts towards the a minimum residual sum of squares. However, each experimental run will find a slightly different using the method will find a slightly different minimum residual sum of squares and therefore a different set of values for the parameters of interest. This is illustrated by table 2, where on repeated running of eth program, the values reported in table 1, are found to be the fourth best fit. Nonetheless, all results are in close agreement with each other.

For the data used to prepare this plot, the total sum of squares was 7.74 and the residual sum of squares was 1.50. Thus, the regression curve (i.e. equation (4) using the values for the parameters found using the Monte Carlo method) is a good fit.

The estimates for the values of the three parameters and equation (4) are set up in a Microsoft Excel spreadsheet to allow visualisation of the experimental results and to support the preparation of graphical presentation of the output that is used here.

Trial No.	To	Rf	k	T_half	Resid_Sum_Sq	Comment
1	13.79164	0.026637	0.012058	57.48462	4.339994	Intermediate Data for Run No 1
2	13.73229	0.0192	0.01731	40.04427	1.934566	Intermediate Data for Run No 1
38	13.72023	0.014083	0.018482	37.5032	1.868611	Intermediate Data for Run No 1
85	13.67879	0.022477	0.017096	40.54547	1.631365	Intermediate Data for Run No 1
168	13.61702	0.016779	0.019506	35.53453	1.596135	Intermediate Data for Run No 1
381	13.69441	0.014203	0.019346	35.82912	1.550316	Intermediate Data for Run No 1
423	13.65385	0.016465	0.019212	36.07949	1.516468	Intermediate Data for Run No 1
967	13.65963	0.013764	0.020044	34.58151	1.504576	Intermediate Data for Run No 1
3875	13.65645	0.013865	0.019939	34.76412	1.499853	Intermediate Data for Run No 1
38958	13.65856	0.014178	0.019768	35.06497	1.497424	Intermediate Data for Run No 1
49539	13.66208	0.014745	0.019514	35.51984	1.496269	Intermediate Data for Run No 1
220135	13.67071	0.012695	0.020278	34.18173	1.49553	Intermediate Data for Run No 1
328202	13.66532	0.013793	0.019879	34.86752	1.495312	Intermediate Data for Run No 1
328202	13.66532	0.013793	0.019879	34.86752	1.495312	Final Result for Run No 1

Table 1: The output from the Monte Carlo program (collected during a single run of 1,000,000 random selections) that is sent to a file during the search for the minimum residual sum of squares. The Total Sum Squares = 7.740.

Trial No.	To	Rf	k	T_half	Resid_Sum_Sq	Comment
423544	13.67087	0.012798	0.02021	34.29645	1.495245	Final Result for Run No 4
903160	13.66728	0.013565	0.019937	34.76675	1.495248	Final Result for Run No 7
558411	13.66834	0.013104	0.020113	34.46343	1.495257	Final Result for Run No 9
656484	13.66572	0.01397	0.019796	35.01402	1.495332	Final Result for Run No 2
95961	13.66959	0.013071	0.020099	34.48613	1.495358	Final Result for Run No 3
443667	13.67407	0.012294	0.020397	33.98247	1.495374	Final Result for Run No 8
328202	13.66532	0.013793	0.019879	34.86752	1.495312	Final Result for Run No 1
455939	13.67435	0.012381	0.020337	34.08234	1.495595	Final Result for Run No 10
403203	13.66266	0.01401	0.019805	34.99871	1.4956	Final Result for Run No 5
928608	13.66236	0.014798	0.019534	35.48412	1.495674	Final Result for Run No 12
637162	13.65761	0.015246	0.019429	35.67544	1.49589	Final Result for Run No 11

Table 2: The minimum sum of squares that were found by running the program a number of times varied between runs. However, the results are general agreement with each other and are close to the true minimum sum of squares. The Total Sum Squares = 7.740.

Output from the Temperature Model

Consider again the graphical presentation in figure 3. In this figure, the best fit plot was drawn using the regression curve (equation 4) and the values for the parameters derived from the first run of the Monte Carlo method (To= 13.67 degC, Rf = 0.014 and k = 0.020). Despite the dispersion in the observed data, the calculated temperatures in the curve are a good fit to the observed temperature data. The Monte Carlo run indicates that the background annual average global temperature (i.e. present since 1800AD) in the absence human activity is 13.67 degC.

We can use the second term in equation (4) to estimate what the annual average global temperature would have been in each year in the absence of an exponential economic model of growth. This estimate of the background temperatures is calculated by subtracting the calculated component for the increment in temperature due to economic activity (i.e. the second term, Rf exp(kt)) from the observed annual average global temperature anomaly reported by NASA. The result is shown in figure 5. It is clear from this graphical representation that in the absence of human activity there would have be no significant alteration to the pattern of the natural variation in the annual average global temperatures throughout the period from 1880AD to 2008AD. In particular, no significant increase in temperature over that period is evident in the absence of a term linked to economic growth. Also, from a visual inspection of the plot, there appears to be no significant alteration to the pattern of variability in the data across this time period. The calculated background temperature data in this plot should be compared with the observed temperature data shown in figure 2. This observation is supportive of the assumptions in the conjecture that the increase in temperature is linked to the exponential economic model.

Small Temperature Changes Early in the Model

In any experiment where a signal is being used to monitor some changing property, the limit of detection of the change is defined as three times the standard deviation of the root mean square noise on the baseline of the signal. Thus, if the change in a signal is small, it may be impossible to identify a change in the property being monitored (i.e. in this article, the annual average global temperature) when the noise level (i.e. the random fluctuations in the individual annual readings) in the data is large. Thus, a change may be in progress for some considerable time before it can be detected with certainty.

Without doing any calculations, it is clear from the plot in figure 2 that the observed range (i.e. maximum to minimum difference) in the temperature is approximately 0.3oC for the period between 1880 and 1920. As the range corresponds to approximately six standard deviations, the limit of detection for a change in annual average global temperature is approximately 0.15 degtC.

The values derived from the Monte Carlo exercise for To, Rf and k gave rise to the long period during which the increase in temperature due to economic activity were less than the naturally occurring noise on baseline. It was only by 1940AD that the actual annual average global temperature has increased above the limit of detection. This is clear from the data in figure 6, where the regression curve of the calculated annual average global temperature from the model is plotted with the scatter plot of the derived background temperature data in the absence of economic activity taken from figure 5. Thus, the unequivocal emergence of the signal of economic influence on temperature was probably as recent as 1950AD. It is because of these particular circumstances that the importance of controlling the quality of the atmosphere by controlling emissions of greenhouse gases was not recognised earlier.

Predicting the Future from the Temperature Model

An interesting property of exponential curves is that there is a unique “doubling interval” that depends on the value of the rate constant, k. This doubling interval is similar in character to the half-life, T1/2, in radioactive decay. For each exponential curve, the following relationship holds: T1/2 = ln(2)/k, where T1/2 is the doubling interval and k is the rate constant for the curve.

It is a fundamental property of an exponential curve that the increment in temperature during each doubling interval is exactly that of the accumulative increase in temperature from that at the starting time of the curve to the beginning of that doubling interval. Based on the estimated rate constant, k = 0.020, the first estimate of the doubling interval of the increment in temperature due to human influence in the empirical equation is approximately 34.5 years. This assumes that the processes that are giving rise to the increment in temperature are continuing apace.

At present, in 2008AD, the increment in temperature, due to the economic model term is equal to the temperature rise since 1800AD (from 13.67 degC in 1880AD to 14.52 degC in 2008AD, i.e. an increase of +0.85 degC). This same increment in temperature will be added to the current annual average global temperature during the next 34.5 years (i.e. the temperature will be approximately 15.3degC by 2043AD). This temperature in 2043AD represents an increase of 1.4OC over the average value for temperature between 1951 and 1980 and over 1.7 degC above the starting temperature in 1800AD. This is very close to the 2.0 degC limit advocated by the European Union for an acceptable increase in temperature from pre-industrial times.

All things continuing apace, and presuming that the model continues to be applicable, the added increment in the subsequent doubling period will be a further +1.7oC. (i.e. the temperature will be approximately 17.1 degC by 2077AD). And again the temperature will probably increase to approximately 20.5 degC by 2110AD. These predicted temperatures are broadly in line with recently reported pessimistic outputs from the sophisticated mathematical models. These predicted annual average global temperatures from this model are summarized in table 3. It should be noted that the mathematical model studied here is the most optimistic possible and does not consider or include any term for a time lag in the temperature response to changing atmospheric composition. See Note 3.

Year AD	Annual Average Global Temperature
1800 AD	13.67 degC
2008 AD	13.67 + 0.85 = 14.52 degC
2043 AD	14.52 + 0.85 = 15.37 degC
2078 AD	15.37 + 1.7 = 17.07 degC
2113 AD	17.07 + 3.4 = 20.47 degC

Table 3: Predicted future annual average global temperatures from the economic model using equation (4) and the values of the parameters determined by the Monte Carlo method and a calculated doubling period of 35 years (to the nearest whole year) . The 2 degree C limit over pre-industrial temperatures recommended by the European Union is 15.67 OC.

Limitations of the Temperature Model

Because the preparation of the table of annual average temperatures required a significant manipulation of the basic raw data, a more detailed statistical investigation of these raw data is required to give the most reliable values to the three parameters To, Rf and k. Because the signal due to the size of the economy is now significantly greater then the noise, and the signal is above the limit of quantitation (defined as ten times the standard deviation of the RMS noise on the baseline), as more data for the coming years becomes available, more reliable determination of the values of these parameters will be available.

It is important to note that in this simple model the parameters Rf and k are each treated as having single composite values covering the numerous parallel and inter-related processes that are operating simultaneously within the biosphere. Expansion of this model to include the influence of the individual components of the processes operating in the economic model is beyond the scope of this simple study.

It is also a requirement that the mechanism of the economic model that are giving rise to increasing temperature continue apace and that no new mechanisms are introduced (e.g. melting of Tundra ice and subsequent release of methane).

Mathematical Model for Observed Carbon Dioxide Increase

In 1896AD, Svante Arrhenius was the first person to calculate that a doubling of the level of carbon dioxide in the atmosphere would lead to an increase in temperature of approximately 5 degC, due to greenhouse effects of that gas (Arrhenius 1896AD). His value is in close agreement with current estimates (i.e. in the range 2 degC to 4 degC) derived from computer models. He also reported that the increase in temperature as a result of increased levels of carbon dioxide in the atmosphere would also increase the concentration of water vapour (also a greenhouse gas), thereby amplifying the temperature increasing effect of carbon dioxide alone. To explore the observed data on atmospheric carbon dioxide, a new model was prepared. The carbon dioxide levels in the atmosphere account for only approximately one quarter of the greenhouse increase in temperature, while the water vapor concentrations account for approximately one half of the greenhouse warming. Other gases are responsible for the final quarter of the warming effect. Nonetheless, I will take the concentration of carbon dioxide to be a surrogate for all greenhouse gases. An inspection of the plot of the levels of carbon dioxide in the atmosphere against time (Keeling Plot 2006), indicate an exponential component in the model [7]. The same conjecture about dependence on the size of the economic model were made for the levels of carbon dioxide in the atmosphere. Again there are two terms, one relating to the level of carbon dioxide in the atmosphere in the absence of economic activity and the other related to the increasing size of the economy over time. The equation proposed is

CO2calc,t = CO2initial + K(t) .... Equation (5)

where CO2calc,t=t is the calculated level of carbon dioxide for each particular year, t, predicted by the model, CO20 is the level of carbon dioxide in the absence of economic activity, Et is the size of the economy in the year of interest, K is the increment in level of carbon dioxide per unit economic size and t is the time since the start of the experiment (i.e. 1800AD). Substituting the value for the size of the economy from equation (1),

CO2calc,t=t = CO20 + K exp(kt) .... Equation (6)

Although carbon dioxide is not the only greenhouse gas, at the present time it is the most significant one. For this reason, the level of carbon dioxide will be considered this model as the surrogate for all greenhouse gases.

Data for Carbon Dioxide Model

The data on the levels carbon dioxide that are used here were measured at Mauna Loa in Hawaii (Keeling et al 2008) [8]. This is a remote site that gives representative data for background levels of carbon dioxide in the atmosphere. The plot of season data set for carbon dioxide levels at this site are known as the “Keeling Curve” (12) which illustrates the exceptional sensitivity of the analytical method used to measure the concentrations of carbon dioxide in the atmosphere.

Output from Carbon Dioxide Model

The output from the Monte Carlo method to find the values of the parameters used in equation (6) are shown in table 3. Because detailed data for carbon dioxide is only available since 1956, the Monte Carlo method was constrained to using an initial level of carbon dioxide of approximately 276 ppm that was reported for the pre-industrial era (say, 1800AD). An excellent correlation is achieved. The total sum of squares was 22,000 and the residual sum of squares was 38.

The best estimation of the values of the three parameters in equation (6) used in the model, that were found using the Monte Carlo method, were CO20= 275ppm, K = 1.55 and k = 0.021. The corresponding doubling period was T1/2 = 33.7 years. Thus, Monte Carlo run indicates that the background level of carbon dioxide (i.e. since 1800AD) in the absence human activity would be approximately 275ppm.

Interestingly, the Monte Carlo analysis of the carbon dioxide model gives rise to an economic model, exp(kt), where the rate constant is k = 0.02065. This value for the rate constant k in the carbon dioxide model is in close agreement with the rate constant derived for the temperature model (also, k = 0.020). The closeness of agreement between the economic component in both these models is better illustrated by the doubling periods of both models (i.e. T1/2 = 34.5 years for the temperature model and T1/2 = 33.7 years for the carbon dioxide model. For this reason, I will declare that these doubling periods are so close as to be identical within the bounds of the experimental error based on the data used for this experiment. Thus, we can consider that the exponential model (i.e. the economic component in both models) to be identical (as a first approximation) for both the temperature model and the carbon dioxide model.

Trial No.	Initial CO2	K	k	T_half	Resid_Sum_Sq	Comments
355696	275.0977	1.553543	0.020558	33.71589	37.71207	Final Result for Run No 1
967786	275.1507	1.537879	0.020611	33.63026	37.7691	Final Result for Run No 12
386120	275.015	1.594105	0.020435	33.92021	38.07574	Final Result for Run No 8
735080	275.2525	1.51472	0.020682	33.51468	38.15058	Final Result for Run No 9
257123	275.3315	1.505786	0.020705	33.47696	38.40702	Final Result for Run No 2
41319	275.3877	1.511078	0.020681	33.51655	38.54006	Final Result for Run No 3
249231	275.2658	1.569125	0.020494	33.82184	38.72508	Final Result for Run No 4
431959	275.6979	1.510064	0.020665	33.54146	39.20316	Final Result for Run No 11
464285	275.1421	1.528773	0.020628	33.60226	39.75985	Final Result for Run No 5
531243	275.2144	1.477631	0.020814	33.30214	39.79028	Final Result for Run No 6
678959	275.9874	1.439575	0.020908	33.15302	40.43405	Final Result for Run No 7
889390	275.7149	1.507395	0.020662	33.54615	40.98831	Final Result for Run No 10

Table 4: Again, the minimum sum of squares that were found by running the Monte Carlo program for carbon dioxide a number of times varied between runs. However, the results are general agreement with each other and are close to the true minimum sum of squares. The Total Sum Squares = 20021.5938408163.

Again, using the concept of a doubling period for the exponential economic component in the carbon dioxide model, it is possible to predict the future levels of carbon dioxide in the atmosphere, presuming that the energy use of the growing economy continues apace. These predictions, based on a doubling period of 34 years, are summarised in table 5.

Year AD	Model Level of Carbon Dioxide
1800 AD	275 ppm
2006 AD	275 + 105 = 380 ppm
2040 AD	380 + 105 = 485 ppm
2074 AD	485 + 210 = 695 ppm
2108 AD	695 + 420 = 1115 ppm

Table 5: Predicted future levels of carbon dioxide in the atmosphere from the economic model using equation (6) and the values of the parameters determined by the Monte Carlo method and a calculated doubling period of 34 years (to the nearest whole year) .

Implications of Identical Economic Models

Because we have determined that both the temperature model (using equation (4)) and the carbon dioxide model (using equation (6)) models depend on the same economic model (using equation (1)), where the rate constant, k, has the same value within the experimental error of the methods used in this article (i.e. k = 0.02), the ratio of the proportionality constants can be calculated to give the increment in temperature per increment in the level of carbon dioxide. In the preparation of the models, Rf was defined as the temperature increment per unit economic growth and K was defined as the carbon dioxide increment per unit economic growth. Therefore, the ration Rf/K is the increment in temperature per increment in carbon dioxide level. The sensitivity of temperature to rising levels of carbon dioxide may be calculated as Rf/K = 0.014/1.55 = 0.009OC rise in temperature per 1 ppm rise in CO2.

Because we judged that the economic models that are used in both the temperature model and the carbon dioxide model were identical, we have established a quantitative linear algebraic relationship between an Increment in the levels of carbon dioxide in the atmosphere and an increment in the annual average global temperature. It is therefore possible to prepare a new table relating the predicted annual average global temperature as a function of the concentration of carbon dioxide in the atmosphere.

Carbon Dioxide in Atmosphere	Annual Average Global Temperature
275 ppm	13.6 degC (Year 1800AD)
380 ppm	14.5 degC (Year 2008)
485 ppm	15.4 degC
590 ppm	16.3 degC
695 ppm	17.2 degC

Table 6: Predicted future annual average global temperatures as a function of the level of carbon dioxide in the atmosphere. This table was derived for the sensitivity of an increase of 0.009C per unit ppm increase in the level of carbon dioxide in the atmosphere. See Note 3.

Significance of Calculated Experimental Results

The significance of the data presented in table 6 for the future security and well-being of planet Earth and its population cannot be over-emphasised. It is clear from the data derived from the models that if "Business as Usual" scenario continues in the development of the economy following the exponential increase in the use of fossil fuels, the temperature on planet Earth will have risen by more than the 2 degC limit above pre-industrial temperatures set by the European Union by mid-century (i.e. shortly after 2040AD from table 5). This 2 degC limit is the maximum increase in temperature that can be tolerated before serious irreversible damage will be caused to the biosphere. There is direct evidence available to support these terrible predictions for future temperature increase. Detailed records are available for the period 1956AD to 2006AD, where data for both the annual average global temperature and the annual average levels of carbon dioxide in the atmosphere are available. These data are paired on a year by year basis. A plot of these data is shown in figure 7, as well as the sensitivity line for the dependence of temperature on levels of carbon dioxide derived from the models (i.e. the ration Rf/K from the models that shows that the calculated temperature rises 0.009 degC for each increment of 1 ppm in the level of carbon dioxide in the atmosphere). From visual inspection of this plot, it is clear that the trend inn observed data has the same slope as the calculated sensitivity (i.e. 0.009 degC per ppm CO2).

Conclusions and Comments

From all the foregoing considerations, it is clear that the annual average global temperature is determined by the level of carbon dioxide in the atmosphere (including the summation of the carbon dioxide equivalent of all other greenhouse gases, including methane, CFCs, nitrogen oxides, sulphur hexafluoride, nitrogen trifluoride, etc.), The concentration of carbon dioxide (including the carbon dioxide equivalent contribution of other greenhouse gases) in the atmosphere is the critical controlling factor.

Although the calculations performed here are first order approximations, it is nonetheless clear from the simple experiments performed using a personal computer that the proposed action by the governments to merely limit the rate of emission of carbon dioxide are grossly inadequate and seriously defective. The level of carbon dioxide that will exist in the atmosphere will determined the annual average global temperature. Global temperatures will continue to rise as long as further greenhouse are added to the atmosphere. Zero emission of carbon dioxide is required by approximately 2040AD or whenever appropriate maximum level of carbon dioxide in the atmosphere that us required to stay below the target maximum temperature (i.e. 15.67 degC), which ever is the sooner!

That future temperature must be protective of the biosphere and suitable for the comfort and survival of the whole human population. Not only must the global economy become carbon neutral at the earliest possible time, but also every other possible mode of action to reduce the rate of temperature increase must be considered and used. These actions on a global scale must include (a) the reduction and reversal of deforestation, (b) the elimination of the industrial use of all chemicals that have a significant green-house effect (e.g. CFCs, SF6, NF3, etc.), (c) the alteration to the structure of cities to make them sustainable and carbon neutral, (d) altering agriculture practice to be more environment friendly in terms of the carbon and nitrogen cycles, etc. However, a complete instantaneous change of lifestyle for all persons living on Earth and the abrupt removal of carbon based fuels that are necessary for economic activity cannot be made in the immediate future. Therefore, the mechanisms of the current economic system will continue to operate in the “business as usual” mode of current economic activities for the some time to come.

Warning: If temperature are allowed to continue to rise, other mechanisms may come into play to further increase temperatures that are not considered in the models considered here. There are considerable quantities of methane currently stored as methane hydrate under present climatic conditions in the frozen Tundra that will be released if global temperatures increases. It will be necessary to capture and use this methane that will otherwise be released to the atmosphere causing significant further global warming. Methane is a much more potent greenhouse gas than carbon dioxide. The details of the mechanisms that are operating to cause this increase in temperature are not defined here. It is always possible that the mechanisms that are operating at present to cause the elevation of temperatures due to economic growth and human activity may change.

It appears likely from the conjectures proposed in this article that the cause of the present state of global warming (i.e. the present annual average global temperature) is due to the cumulative effect of all economic development to date (including industrial, commercial, agricultural, transport, fuel use, etc.). Then that same measure of economic effort that brought us to the present level of wealth and prosperity will probably be required to reverse the problem (i.e. at least an equal cost and effort to limit or reverse the cause of temperature increase will be required). If the exponential growth represented in this economic model continues to be applicable in the future, then the economic effort to reverse the scale of the problem global warming will double with each doubling period. Unless immediate efforts are made to address the global warming problem, at some time in the very near future, the scale of the effort that will be required to control the rate of global warming will be beyond the level of economic resources available to solve the problem.

Notes

In order to facilitate access to references materials, only articles that are readily available on the internet are cited.
Because Microsoft Excel calculates in double precision mode, the many digits that are displayed for each answer gives the impression that very precise results are obtained. In reality the results are significantly less precise and in this exercise the temperature data are supplied by NASA to the nearest 0.01 degC and the results of calculations based on the data cannot be more precise.
Holdren has reported that the observed temperature observed at any time is only approximately 65% of the increase from pre-industrial times that are calculated from sophisticated mathematical models. The further increase in temperature is a result of the delayed response (i.e. thermal inertia) of the planetary system to changes in radiative forcings caused by changes in the composition of the atmosphere and the altering albedo of the Earth's surface.

References

Arrhenius, Svante: Philosophy Magazine and Journal of Science, 5th Series, April 1896, pp237-276. Reprint of paper on greenhouse effect at http://www.globalwarmingart.com/images/1/18/Arrhenius.pdf See also http://en.wikipedia.org/wiki/Svante_Arrhenius

European Union from http://ec.europa.eu/environment/climat/eccp.htm

Hansen, J et al: PNAS 103:14288-293 (26 Sept 2006) doi:10.1073/pnas.0606291103 See also http://pubs.giss.nasa.gov/docs/2006/2006_Hansen_etal_1.pdf

Holdren John P: The John H Chafee Memorial Lecture on Science and the Environment 2008 from http://www.ncseonline.org/conference/Chafee08final.pdf

IPCC, International Panel on Climate Change, Fourth Synthesis Report 2007 from http://www.ipcc.ch/ipccreports/ar4-syr.htm. Also see a wide range of other reports from the home page http://www.ipcc.ch/.

Keeling R F, Piper S C, Bollenbacher AF: Atmospheric CO2 values (ppmv) from http://cdiac.ornl.gov/ftp/trends/co2/maunaloa.co2

Keeling Curve from http://cdiac.ornl.gov/ftp/trends/co2/maunaloa.co2

NASA from http://data.giss.nasa.gov/gistemp

Organisation for Economic Cooperation and Development from http://www.oecd.org/department/0,3355,en_2649_34361_1_1_1_1_1,00.html

Temperature data taken from http://data.giss.nasa.gov/gistemp/tabledata/GLB.Ts+dSST.txt

United Nations from http://UNFCCC.int/2860.php

United States Environmental Protection Agency from http://www.epa.gov/climatechange/index.html