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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)
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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
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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
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International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 05
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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.
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