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Journal of Earth Science and Engineering 7 (2017) 175-180 doi: 10.17265/2159-581X/2017.03.005
Theoretical Model of the Tropospheric Pressure
Variation on the Height
Raul C. Perez
LIHANDO. CEDS, Dpto de Materias Básicas, Facultad Regional Mendoza, Universidad Tecnológica Nacional, Rodriguez 276,
Mendoza 5570, Argentina
Abstract: In the study and the research of the troposphere, the knowledge about its pressure variation on the height is necessary and important. Of course, exist the sounding to make this work, but not always it is possible to access to sounding data when is necessary, so then, it is important to has others alternatives methods in order to replace it. The troposphere is basically a fluid, susceptible to be studied under the fluid mechanics and thermodynamics of the ideals gases without loss generalities. So, it is possible to study of the atmospheric air like as a continuous perfect gas. These facts are important questions to study the troposphere under the laws and beginning Physics, using its respective equations, in order to get a theoretical model to simulate the behavior of its thermodynamics variables and parameters. Working on this line, it was developed a model in order to simulate the tropospheric pressure variation on the height from its measure on surface data. Key words: Atmospherics pressure, tropospheric pressure.
1. Introduction
It is possible to study a V volume to atmospheric air
like as a moist ideal gas, without loss generality. More
specifically, the atmospheric air can be considered as
a combination between two ideal gases: the dry
atmospheric air and the water vapor.
Under this conception, pressure p into the V volume,
it will be equal at the sum of the partial pressure of the
pa of the dry air and the vapor pressure e.
(1)
The dry atmospheric air has an equivalent equation
of the ideal gases, whose expression is:
(2)
where, is the number mol of the dry atmospheric
air and R is the gases universal constant to dry
atmospheric air, whose value is: 8.3143 Joule.K-1.mol.
So, the equation of vapor pressure is:
(3)
461 Joule/kg. K is the vapor water constant;
and is the vapor water density value.
Corresponding author: Raul Cesar Perez, Ph.D., physicist, research fields: earth, sea and atmospherics science.
2. Tropospheric Pressure
If the atmospheric air can be considerate like as an
almost stable perfect gas, thinking in its state of an
infinitesimal instant; it can be possible to study the
tropospheric pressure in a static situation.
So, a theoretical model about the behavior of the
tropospheric pressure at different height can be
developed.
2.1 Variation of the Tropospheric Pressure on the
Height
Under the supposition planted about to consider the
atmospheric air as a perfect gas, it is possible to obtain
a good approximation of the relation between the
atmospheric air pressure and its height on the sea
level.
For the fundamental law of the fluids mechanics,
the pressure variation on the height is shown in Ref.
[1]:
(4)
where, for the universal equation of the ideal gases,
D DAVID PUBLISHING
Theoretical Model of the Tropospheric Pressure Variation on the Height
176
the density is proportional to pressure to any height
for the relation:
(5)
Into Eq. (5) 0 and P0 are the density and
pressure on surface corresponding. Combining Eqs. (4)
and (5), it is obtained:
(6)
Rearranging the terms, Eq. (6) is becoming in:
gρ0
P0dy (7)
Integrating Eq. (7), from the pressure on surface
value p0 at the height y0 to the pressure value p at a
determinate height y, and it is obtained:
(8)
Solving the integral, we get the solution:
(9)
Using the standard value to seal level:
0 = 1.21 kg/m3, p0 = 1.01 105 Pa. And taken the
gravity acceleration value as g = 9.8 m/s2, we obtain: . / (10)
The pressure variation on the height presents an
exponential decay as shown in Eq. (10) and Fig. 1.
Fig. 1 shows the very good correlation between the
theoretical function Eq. (10) and the experimental data
obtained to sounding.
It is possible to appreciate, when Table 1 is
observed that atmospheric pressure value obtained
from the model is very accurate until 3,000 meters.
Above this height, the results present a difference with
a ten percent of error approximately.
One of the factors that influences the difference
between theoretical and experimental value, is that it
has been considered the gravity acceleration value as
constant equal at 9.8 meters for squad second. Then,
in order to have results with greater precision, it will
be better to consider gravity acceleration value as
variable with the height.
2.2 Theoretical Variation of the Gravity Acceleration
g on the Height
Isaac Newton had mathematically expressed into
two forms for the phenomena of the gravitational
attraction that perform the Earth on the bodies.
The first form is through the application of the
Newton’s second law. It is known that the force
attracts at the center Earths called weight respond at
the equation:
(11)
where, m is the mass of the volume V of the
atmospheric air studied.
The other form is under the postulation of the
Universal Gravitation law, whose equation is:
(12)
where, G = 6.673 10-11 N.m2.Kg-2 is the universal
gravitation constant and r the distance between both
mass.
Considering at m1 the Earth mass MT = 5.972 1024
Kg and m2 the at the m mass of the atmospheric air
volume.
Combining Eqs. (11) and (12), and rearranging
therm, the next equation is obtained:
.6.3210
3.9851106.3210
(13)
where, y is the vertical distance from the land surface
to height to which wants to calculate the gravity
acceleration value; and 6.32106 meter corresponds at
the average radio of the Earth.
Eq. (13) is the mathematical expression of the
gravity acceleration variation on the height respect to
a reference system with its origin is in the Earth
center.
Replacing this expression into Eq. (7), it proceeded
to integrate and it is possible to obtain an expression
to the atmospheric pressure on the height:
. ,. (14)
Fig. 1 Theo“Física”.
Table 1 Val
12/08201 Pr(m
Presion M
939 929 925 923 880 850 821 728 700 607 528 460 434 400 362 322 289 269 217 180 160 129 104 88
939292928885827370625448453936343129252421171513
Eq. (14) r
to the heigh
place respec
The abov
The
retical and ex
lue comparison
res. mb)
Altura
odelo (m)
39 29 25 23 81 52 24 35 09 21 47 85 52 97 67 42 14 97 54 42 10 78 51 33
704 792 828 846 1,233 1,514 1,791 2,747 3,056 4,163 5,225 6,238 6,820 7,916 8,562 9,170 9,876 10,35011,64812,07413,25314,63316,00917,069
represents th
ht y in meters
ct to the sea le
ve equation w
oretical Mode
xperimental gr
n between the m
a 01/01/2010
Real
0 8 4 3 3 9 9
933 925 924 896 859 850 836 740 700 609 528 441 410 400 300 289 250
he atmospheri
s; also y0 is t
evel.
was used in o
el of the Trop
raphics of the
model results a
Pres. (mb)
A
Modelo (
933 925 924 892 857 848 834 739 699 612 535 455 427 418 328 319 284
77711112345677991
ic pressure v
the height of
order to simu
pospheric Pre
atmospheric p
and real measu
Altura 29/09/2009
(m) Real
704 778 787 1,079 1,410 1,500 1,641 2,658 3,119 4,235 5,352 6,714 7,250 7,430 9,450 9,704 10,670
930925920910850792762700653576500460424400344334300297283250220
value
f the
ulate
the
plac
of t
sou
essure Variat
pressure vs. he
ure from the d
/ Pres. (mb) Modelo
930 924 919 909 848 790 760 698 661 580 507 469 436 414 364 355 324 322 309 280 253
atmospheric
ce of Argenti
the 2015; it
unding. These
tion on the He
eight Resnick,
different sound
Altura 23/20
(m) Re
704 752 798 892 1,475 2,071 2,394 3,101 3,558 4,660 5,790 6,430 7,046 7,480 8,567 8,780 9,530 9,600 9,928 10,770 11,610
949369392589585082470970062559500416400333326300276250
pressure at m
ina and Chile
was compare
e comparisons
eight
R., Halliday,
ding.
/03/ 10
Pres. (mb)
eal Modelo
8 6 1 5 5 0 4 9 0 5 8 0 6 0 3 6 0 6 0
948 936 931 926 896 853 828 719 711 641 616 524 446 432 370 364 340 317 293
many heights
e in the day
ed with real
s are shown i
177
D., Krane, K.
Altura
o (m)
704 806 850 902 1,172 1,590 1,839 3,016 3,115 3,975 4,310 5,670 7,010 7,290 8,580 8,726 9,300 9,875 10,550
s to different
September 4
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pospheric presmeasure for sou
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ssure value onunding and theof 2016.
9
n e
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Theoretical Model of the Tropospheric Pressure Variation on the Height
180
(3) The power of the model lies in the fact that it
can calculate the approximate tropospheric air
pressure value to any height with only its measure on
surface.
(4) Also it is possible to simulate the tropospheric
pressure of the sounding at any time on any place.
References
[1] Pérez, R. C. 2011. Dinámica Atmosférica y los Procesos Tormentosos Severos. LAMBERT Academic Publishing (LAP). GmbH & Co. pp. 17-9.
[2] Rogers, R. R. 1977. Física de las Nubes. Ed. Reverté, 67-9.
[3] http://weather.uwyo.edu/upperair/sounding.html.
