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International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 05 11 A Mathematical Model to Compute the Crucial Roles of Water Vapor in Global Warming Md. Hamidul Islam and Md. Abdus Salam Lecturer in Mathematics, Department of Electronic & Telecommunication Engineering, Prime University, Dhaka-1216, Bangladesh E-mail: [email protected], [email protected] Abstract-- Scientific evidences, personal as well as societal experiences have confirmed that our world has been getting warmer over the years, slowly at the start of the industrial revolution and appears to be much faster over the past three decades. CO2 is mainly responsible for that, along with other man-made gases such as methane, nitrous oxide, fluorocarbons etc. Water vapor has been hitherto neglected in greenhouse gases budget. According to physical law, warmer atmosphere can hold more water vapor before precipitation, which could make the Earth even warmer. A mathematical model is presented in this paper to show the actual effect of water vapor (WV) in global warming, in addition to the well known effects of carbon dioxide. Index Term-- Greenhouse gas, Effects of water vapor, Greenhouse effect, Global warming. 1. INTRODUCTION Mauna Loa, Hawaii 400 CO2 Concentration (ppmv) 390 380 370 360 350 MATHEMATICAL MODELS The mathematical model for global warming without the effect of water vapor will be presented first. The records of CO2 concentration rise [2] and temperature rise [3] of the world’s atmosphere covering the period 1959-2008 and 18802008 are respectively shown in Figs. 1 and 2. 0.6 Global Temperature Change (1880-2008) 0.4 0.2 0.0 -0.2 340 330 -0.4 320 310 1959 2. Departure from Long term-Mean (°C) Global warming is increasing in the average temperature of the earth’s near-surface air and oceans since the mid-twentieth century and its projected continuation. Global surface temperature increased 0.74±0.18°C (1.33±.32°F) [1] during the 100 years ending in 2005. CO2 is mainly responsible for that, along with other man-made gases such as methane, nitrous oxide, fluorocarbons etc. Analyzing the records of CO2 concentration [2] rise and temperature rise of the world’s atmosphere covering the period of 1970-2000 [3] it is seen that there is approximately linear relationship between them. Henceforth, the temperature is rising in non-linear manner with respect to the rise in concentration of CO2. This inspired us to looking for other factors which are inciting the temperature. The major non-gas contributor to the earth’s greenhouse effect is water vapor, also absorb and emit infrared radiation and thus have an effect on radiative properties of the greenhouse gases [4]. The presence of water vapor in the atmosphere acts as greenhouse gas and trapped long wave radiation emitted by the earth surface and absorb solar radiation and kept the planet warmer which improve the capability of atmosphere to admit more WV before condensation [5]. This non-linear effect of WV has to be taken into account in the mathematical modeling. 1964 1969 1974 1979 1984 1989 1994 1999 2004 Year Source: David Keeling and Tim Whorf (Scripps Institution of Oceanography). Figure.1 Record of CO2 concentration in atmosphere2 Since the rise in extraneous carbon dioxide gas in the atmosphere spewed from smoke stacks of fossil-fueled power plant has been identified as the main culprit of global warming [6], so we can express the temperature as a function of -0.6 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Year Figure. 2 Global Temperature Record3 carbon dioxide , that is differential equation, 111305-8686 IJBAS-IJENS © October 2011 IJENS . This leads us to a (1) IJENS International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 05 The term denotes the change of temperature per unit change in CO2 concentration in the atmosphere. This can be calculated more accurately by calculating the simple correlation coefficient between the annual mean temperature and annual mean CO2 concentration. Also the Figs. 1 and 2 shows the general trends during 19702000 are the increases of both CO2 and temperature in approximately linear manners. Over this 30 years periods the temperature rises by amount 1°F (0.55°C) while CO2 rises by about 43 ppm. This suggests us to consider the value of = . (2) The rationale and the mathematical model for the contribution of water vapor will now be discussed. It is an established scientific knowledge that when the air warms up, its capability of admit water vapor before condensation is higher. Thus a warmer globe is at the same time also a wetter globe, and this is being proved for example by the greener desert in Africa due to more humid atmosphere [7] .This extraneous WV absorbs more solar radiation from the sun and the from the earth, causing the globe to be warmer still, which renders it to admit even more WV. This seems to be a positive feedback loop without a restraining mechanism for equilibrium [8]. If this is true, it is very alarming and this inspires us to express the temperature as a function of and , . This gives us a partial differential equation, The coefficients presumably could be determined by the experiments, or by deciphering global meteorological data over the past years. Note that here is not the same as appeared in equations (1) and (2). It should be noted also that the average background of water vapor cannot be reduced by precipitation. As reasoned earlier, the amount of water vapor that could be present in the atmosphere also increases with its temperature; that is in equation (3) can be replaced by , where is another empirical coefficient which is related to vapor pressure of the atmosphere, it perhaps a function of itself but it will be assumed for the time being, for simplicity as a constant, which on substitution equation (3) becomes, (4) Now, by using the same data in figure 1 and 2 and by comparing with equation (2) it must be concluded that [ , say] If the value of and are known then the real contribution CO2 (hence WV) in global warming could be estimated through the value of ’s in conjunction with the interpretations of equations (1) and (3). As reasoned earlier, by observing the absorption bands of CO2 and WV, is evidently larger than . For a conservative estimate it is assumed here that is equal to . The value of can be estimated from a psychrometric chart as in [9]. The rational for the estimation is that the 12 background relative humidity is constant regardless of its increased temperature. This should be a plausible hypothesis since it should be natural adjustment for a warmer (thus dryer) atmosphere to absorb more WV to restore equilibrium level. For demonstration purpose the background relative humidity is assumed to be constant at 40%, and the starting dry-bulb temperature was assumed to be at 20°C, this fixes a point in the psychometric chart. The slope of the constant relative humidity curve in the psychometric chart at this point then used to estimate the value of . A rough estimation is done here by taking the secant’s slope obtained by incrementing the temperature by 5°C along the curve. The increase in humidity ratio (2.2 g H2O/kg dry air) over this temperature range is used to estimate the value of to be 704ppm/°C. Putting estimated e (=d) and f into equation (5), the value d is computed as, 3. D ISCUSSION It is seen that the value of in equation (6) is about 10 folds smaller than that computed in equation (2) for the case of ignoring water vapor contribution. By comparing equation (1) and (3), this suggests that extraneous CO2 contributes only about 10% in increasing the earth’s temperature while the much larger 90% is contributed by WV. Moreover, the value of is likely to be greater than that of . These two factors should contribute to increase the sensitivity in a non-linear manner which could result (3) into a much warmer globe than predicted by a linear model. It is thus suggested that experimental measurements should be carried out in a timely manner to determine the magnitudes and the functional relationships among , so that the seriousness of the role of increased background WV can be accurately assessed. Again from the relation = and equation (4), we have (7) (8) These indicate that as CO2 stabilizes ( the temperature and water vapor increase will also stop. The positive feedback effect of continued temperature and water vapor rises does not appear here. Unless, as earlier speculated, if is an increasing function of then it is seen that and will be increasing at an ever higher rate. (5) 4. CONCLUSION The mathematical model proposed in this paper and the numerical estimations of the models’ coefficients from the meteorological data and psychrometric chart predicted that global warming is caused about 90% by extraneous water vapor that can be additionally admitted by the warmer atmosphere and only 10% by carbon dioxide which is still the 111305-8686 IJBAS-IJENS © October 2011 IJENS IJENS International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 05 13 root cause of global warming. A non-linear effect is speculated which could possibly yield a much higher earth’s temperature than the predictions of most atmospheric models that do not take into account of this non-linear effect. As the mathematical model in this paper suggests that the water vapor has greatest contribution in increasing earth temperature so the temperature predictions taking into account the water vapor’s effect will be more accurate. The further study on water vapor could improve the analysis of a tropical cyclone vortex and the forecast of rapid intensification of a tropical cyclone. ACKNOWLEDGEMENT The authors would like to acknowledge Dr. M. Ali Akbar, Department of Applied Mathematics, University of Rajshahi, Bangladesh for his help in preparing the article. REFERENCE [1] [2] [3] [4] [5] [6] [7] [8] [9] IPCC (2007-05-04), Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Retrieved 2010-02-03. Source data published by the National Oceanic and Atmospheric Administration (NOAA), http:// co2now.org/Current-CO2/CO2Now/Current-Data-for-Atmospheric-CO2.htm. Retrieved 2010-05-03. Global temperature records, http://www.cru.uea.ac.uk/cru/info/warming/. Retrieved 2010-06-03 Kiehl, J. T., Kevin E. Trenberth. "Earth’s Annual Global Mean Energy Budget", 1997. Evans, Kimberly Masters. "The greenhouse effect and climate change". The environment: a revolution in attitudes, Detroit: Thomson Gale. ISBN 0-7876-9082-1., 2005 America's Climate Choices: Panel on Advancing the Science of Climate Change; National Research Council. Advancing the Science of Climate Change, Washington, D.C.: The National Academies Press. ISBN 0309145880, 2010. Hutchinson, C.F., Hermann, S.M., Maukonen, T. and Weber, J. Introduction: The “Greeting” of the Sehel. Journal of Arid Environments 63: 535-537, 2005. Held, I. M.; Soden, B. J. "Water Vapor Feedback and Global Warming1", Annual Review of Energy and the Environment 25: 441– 475, 2000. ASHARE psychometric Chart No.1, American Society of Heating, Refrigerating and Air1. Conditioning Engineers, Inc., Atlanta, USA, 1981. 111305-8686 IJBAS-IJENS © October 2011 IJENS IJENS