An Analytical Formula for Potential Water Vapor in an Atmosphere of Constant Lapse Rate

Abstract

Accurate calculation of precipitable water vapor (PWV) in the atmosphere has always been a matter of importance for meteorologists. Potential water vapor (POWV) or maximum precipitable water vapor can be an appropriate base for estimation of probable maximum precipitation (PMP) in an area, leading to probable maximum flood (PMF) and flash flood management systems. PWV and POWV have miscellaneously been estimated by means of either discrete solutions such as tables, diagrams or empirical methods; however, there is no analytical formula for POWV even in a particular atmospherical condition. In this article, fundamental governing equations required for analytical calculation of POWV are first introduced. Then, it will be shown that this POWV calculation relies on a Riemann integral solution over a range of altitude whose integrand is merely a function of altitude. The solution of the integral gives rise to a series function which is bypassed by approximation of saturation vapor pressure in the range of -55 to 55 degrees Celsius, and an analytical formula for POWV in an atmosphere of constant lapse rate is proposed. In order to evaluate the accuracy of the suggested equation, exact calculations of saturated adiabatic lapse rate (SALR) at different surface temperatures were performed. The formula was compared with both the diagrams from the US Weather Bureau and SALR. The results demonstrated unquestionable capability of analytical solutions and also equivalent functions.

Full text

doi: 10.3319/TAO.2011.06.28.01(A)
* Corresponding author
E-mail: ali-varmaghani@uiowa.edu
Terr. Atmos. Ocean. Sci., Vol. 23, No. 1, 17-24, February 2012
An Analytical Formula for Potential Water Vapor in an Atmosphere
of Constant Lapse Rate
Ali Varmaghani *
IIHR-Hydroscience & Engineering, University of Iowa, Iowa, USA
Received 1 July 2010, accepted 28 June 2011
ABSTRACT
Accurate calculation of precipitable water vapor (PWV) in the atmosphere has always been a matter of importance for
meteorologists. Potential water vapor (POWV) or maximum precipitable water vapor can be an appropriate base for estima-
tion of probable maximum precipitation (PMP) in an area, leading to probable maximum flood (PMF) and flash flood man-
agement systems. PWV and POWV have miscellaneously been estimated by means of either discrete solutions such as tables,
diagrams or empirical methods; however, there is no analytical formula for POWV even in a particular atmospherical condi-
tion. In this article, fundamental governing equations required for analytical calculation of POWV are first introduced. Then,
it will be shown that this POWV calculation relies on a Riemann integral solution over a range of altitude whose integrand is
merely a function of altitude. The solution of the integral gives rise to a series function which is bypassed by approximation
of saturation vapor pressure in the range of -55 to 55 degrees Celsius, and an analytical formula for POWV in an atmosphere
of constant lapse rate is proposed. In order to evaluate the accuracy of the suggested equation, exact calculations of saturated
adiabatic lapse rate (SALR) at different surface temperatures were performed. The formula was compared with both the dia-
grams from the US Weather Bureau and SALR. The results demonstrated unquestionable capability of analytical solutions
and also equivalent functions.
Key words: Precipitable water vapor, Lapse rate, Troposphere, Meteorology
Citation: Varmaghani, A., 2012: An analytical formula for potential water vapor in an atmosphere of constant lapse rate. Terr. Atmos. Ocean. Sci., 23, 17-
24, doi: 10.3319/TAO.2011.06.28.01(A)
1. INTRODUCTION
The distribution of water vapor in the atmosphere has
always been noticed in different aspects. It plays a key role
in the balance of planetary radiation; it affects and reacts to
atmospheric motions; and it is a key component in many
features of atmospheric processes acting over a wide range
of spatial and temporal scales (Jade et al. 2005). It is particu-
larly material, in terms of the potential impact on climate
change and to assess long-term changes and decadal scale
trends of the atmospheric water vapor regime (Jacob 2001).
“Precipitable water vapor (PWV) is a measure of the total
water contained in a vertical column above the site. It is
commonly expressed as the resulting height of liquid water
if the entire vapor in the column were condensed” (Marvil
et al. 2006). Some studies have demonstrated that PWV es-
timates from ground-based global positioning system (GPS)
observations and meteorological data yield the same level of
accuracy as radiosondes and microwave radiometers (Jade
et al. 2005; Jade and Vijayan 2008). Many authors imple-
mented research to increase the precision of the technique
for GPS-based PWV estimation, typically exploiting a small
number of stations. Rocken et al. (1995) were the first to
show the concord between water vapor radiometer (WVR)
and GPS derived relative estimates of integrated water va-
por (IWV), with a level of agreement of about 1 kg m-2.
Svensson and Rakhecha (1998) assumed in their hy-
drometeorological method for determination of probable
maximum precipitation (PMP) that PMP will result from
a storm where there is an optimum integration of avail-
able moisture in the atmosphere and efficiency of the storm
mechanism. Factors that influence storm efficiency comprise
horizontal mass convergence, vertical velocity by frontal or

Ali Varmaghani
18
topographically induced lifting, and the rate of condensa-
tion of water vapor into droplets. At present, it is impos-
sible to assess the above-mentioned factors separately, and
therefore the observed highest rainfall is employed as an
indirect measure of storm efficiency. After the Banqiao and
Shimantan dams in China were built in the 1950s, estima-
tion methods of the inflow design flood for dam safety have
greatly altered with improvements in hydrometeorological
techniques. These new techniques exploit meteorological
theories and concepts to determine a design storm of a prob-
able maximum precipitation (PMP) magnitude (Svensson
and Rakhecha 1998). The PMP is then transformed into a
probable maximum flood (PMF) hydrograph after deduc-
tion of losses and determination of antecedent properties,
such as soil moisture content (Pilgrim and Cordery 1993).
The application of PMP to estimate the PMF has become a
standard for dam design in some countries where no risk of
overtopping can be approved: e.g., the USA (e.g., Riedel
1976; USNWS 1978; Hansen 1987), Canada (e.g., Gagnon
et al. 1970), China (e.g., Wang 1987; Pan and Teng 1988),
India (e.g., CWC 1972; Rakhecha et al. 1990), and Australia
(e.g., Kennedy 1982). Hence, the importance of maximum
precipitable water vapor or potential water vapor (POWV)
in an area is revealed.
Wang et al. (2009) demonstrated that a GPS survey is
an effective way of monitoring the PWV alteration. It can
continuously render both the temporal and spatial distribu-
tion of atmospheric water vapor. They utilized the time se-
ries data of GPS zenith tropospheric delays (ZTD), derived
continuously from 28 permanent GPS sites from 2002 to
2004, to analyze the change of precipitable water vapor on
the Chinese mainland. Valeo et al. (2005) performed studies
on PWV estimation - derived from GPS zenith wet delay
measurements - in conjunction with a basic snow evapora-
tion model to verify observations of snow evaporation in an
open urban area. Kumar et al. (2005) developed a simple
theoretical model for computing global insolation on a hori-
zontal surface. The input parameters for the model were the
latitude of the desired location and the amount of total pre-
cipitable water content in the vertical column at that loca-
tion. Jade et al. (2005) estimated the precipitable water va-
por from GPS data over the Indian subcontinent for a 3-year
period (2001 - 2003).
The total amount of water vapor in a layer of air is
often expressed as the depth of precipitable water lPWV even
though there is no natural process capable of precipitat-
ing the entire moisture content of the layer (Linsley et al.
1975). Marvil et al. (2006) calculated PWV by means of
two methods; in the first method, they represented the depth
of PWV as a fraction of sea-level water vapor density. An
atmospheric modeling program (ATMOS) was utilized to
compute the mentioned depth, lPWV, at 3.8 km for a range of
sea-level water vapor density
0
t
. They worked out a linear
fit between PWV and
0
t
through the following equation:
.l0174
PWV0
t=
(1)
where lPWV is the depth of precipitable water vapor in mm,
and
0
t
is sea-level water vapor density in gr m-3. In the sec-
ond method, the Magnus-Teten equation was used to deter-
mine the depth of PWV. The Magnus-Teten equation is as-
sessed at the dew point to estimate the local vapor pressure
of water. This is given by Eq. (2):
.
..logPT
T
2373
75 07858
vp
d
d
10 =++
(2)
where Pvp is the vapor pressure of H2O in hPa and Td is the
dew point temperature in degrees Celsius (Murray 1967).
This is applied to estimate integrated lPWV from an atmo-
spheric model as shown in Eq. (3):
.lPh21 10
PWVvp
h22
=-
^h
(3)
where h is the height in kilometers and Pvp (h) is the vapor
pressure of H2O in mmHg at height h (Allen 1973). Solot
(1939) proposed the following discrete formula to calculate
the amount of precipitable water in any air column of con-
siderable height:
.lqp00004
PWVha
D=/
(4)
in which lPWV is precipitable water in inches; pa is the air
pressure in millibars; and qh is the average of the specific
humidity at the top and bottom of each layer in grams per ki-
logram. Following the Solot equation, the US Weather Bu-
reau (USWB 1949) published charts of mean precipitable
water in the atmosphere over the United States. Figure 1
presents the depth of precipitable water in a column of sat-
urated air with its base at the 1000-millibar level and its
top anywhere up to 200 millibars, assuming saturation and
pseudo-adiabatic lapse rate. USWB (1949) also has pro-
vided tables for computing precipitable water in the atmo-
sphere over the United States.
2. METHODOLOGY
Knowledge of the vertical and spatial distribution of
moisture allows the calculation of the precipitable water in
an area. In order to compute the total amount of precipi-
table water lPWV in a layer between elevations 0 and z, it is
required to evaluate the following integral:
ldz
PWVv
z
0t=#
(5)
where
v
t
is the water content (vapor density) in the column

A Formula for PWV in Atmosphere of Constant Lapse Rate 19
(Bras 1990). Specific humidity, qh, is the mass of water,
v
t
,
per unit mass of moist air,
m
t
, and can be determined by
Eq. (6):
.
..
qpe
e
P
e
0378
0622 0622
h
m
v.
t
t
==
-
(6)
where e and P are vapor pressure and total atmospheric
pressure in millibars, respectively (Bras 1990). The amalga-
mation of Eqs. (5) and (6) yields:
.lPedz0622
PWV
m
z
0
t
=#
(7)
Fig. 1. Depths of precipitable water in a column of air of any height above 1000 millibars as a function of dew point, assuming saturation and
pseudo-adiabatic lapse rate (USWB 1949).

Ali Varmaghani
20
Saturation vapor pressure for water in millibars can be es-
timated by:
..
..
expeT
T
6107835 86
17 2693882273 16
s=-
-
l
l
^h
;E
(8)
where
Tl
is the air temperature in degrees Kelvin (Potter
and Colman 2003). Therefore, saturation vapor pressure in
Pascal can be shown as follows:
..
.
expeT
T
6107823729
17 2693882
s=+
`
j
(9)
where T is the air temperature in degrees Celsius. The pa-
rameters e and es are related to each other through the fol-
lowing equation:
ere
100s
=
(10)
in which r is relative humidity or the ratio of the vapor den-
sity (or pressure) to the saturation vapor density (saturation
vapor pressure) at the same temperature (Bras 1990).
Air in the atmosphere follows, reasonably well, the
ideal gas law, which for a unit mass is:
PRT
1
m
t=l
(11)
where R is gas constant in square centimeters per square
second per degree Kelvin, and
Tl
is ambient air tempera-
ture in degrees Kelvin (Bras 1990). The relation between
degrees Kelvin (
Tl
) and degrees Celsius (T) is:
.TT27315=+
l
(12)
The troposphere, the lowest layer of the air, is charac-
terized by a nearly uniform decrease in temperature. Most
of the weather changes in the air are limited to this layer. Its
average thickness reaches about 11.3 km (Donn 1965). The
temperature variation in the troposphere is assumed to be
linear (or piecewise linear):
TT z
0a=-
(13)
where T and T0 are ambient temperature at elevation z and
surface temperature, sequentially. The rate of cooling α is
called the ambient lapse rate and usually varies between 5
and 8°C km-1 (Bras 1990). The depth of potential water va-
por, lPOWV, is obtainable when the entire layer becomes satu-
rated (r = 100). Inserting Eqs. (9) through (13) in Eq. (7),
assuming r = 100 results in the following equation:
0.622 610.78 237.29
.
.
exp
lRTz
Tz
Tz
dz
17 2693882
27315
POWV
z
0
0
0
0
#
a
a
a
=-
+-
-
+
^^^
hhh
;E
#
(14)
The integrand in Eq. (14) is merely a function of alti-
tude. Having solved the Riemann integral in the above-men-
tioned equation by means of three variable substitutions, the
potential water vapor in an atmosphere of constant lapse
rate is obtained by:
*! *!
ln lnlR
keA
A
nn
AA
B
B
nn
BB
e
POWV
a
nn
n
nn
n
b
0
0
10
0
1
a
=+
-+-
-
33
==
ccmm
;E
//
(15)
where lPOWV is the depth of potential water vapor in a column
of air with height z in terms of mm. a, b and k are the equa-
tion constants; A, A0, B and B0 are the equation parameters,
presented in Table 1, which are functions of T, T0 and α. T
and T0 should be in degrees Celsius, and α is in terms of
°C km-1. A simpler integral form of lPOWV is:
lR
ka b
xa
edx
xb
POWV
x
l
l
0
a
=-
--
^^^
hhh
#
(16)
.
lT
aT
23729
0
0
0
=+
(17)
237.29
lTz
aT z
0
0
a
a
=-+
-
^^hh
(18)
Saturation vapor pressure over water can be approxi-
mated within 1% in the range of -50 to 55°C by:
.. .eT33 863900073808072
s
8
.+
^h
6
.. .T000001918480001316-++
@
(19)
Table 1. The value of the parameters in Eq. (15).
Parameter Value
a17.2693882
b131.5430392
k379.90516
A(-237.29a) / (T0 - αz + 237.29)
A0(-237.29a) / (T0 + 237.29)
B[(a - b) (T0 - αz) - 237.29b] / (T0 - αz + 237.29)
B0[(a - b) T0 - 237.29b] / (T0 + 237.29)

A Formula for PWV in Atmosphere of Constant Lapse Rate 21
where es is in millibars and T is ambient temperature in de-
grees Celsius (Bosen 1960). Comparing Eq. (9) with Eq.
(19) implies the maximum error of 4.6% occurring at -50°C.
The diagram published by the US Navy Weather Research
Facility, illustrating the physical structure of the atmosphere,
shows the minimum possible temperature in the troposphere
around -55°C (Donn 1965). Therefore, in order to find an
indefinite integral for Eq. (14), another approximation of
Eq. (9) was made in the range of -55 to 55°C within maxi-
mum and mean error of 4.5 and 0.5%, respectively:
.* .* .eTTT6423 10 2047 10 00003015
s
96 65 4
.++
--
....TT T002642 1434444610 78++++
32
(20)
where es is in Pascal and T is ambient temperature in degrees
Celsius. Having inserted Eq. (20) in Eq. (14), instead of
Eq. (9), and solving the integral, potential water vapor in
mm is readily obtained by the following formula:
lRAT TBTT CT TDTT
1
POWV zzzz
6
0
65
0
54
0
43
0
3
a
=--+ -+ -+ -
^^^^hhhh
;
.
.
lnET TFTT GT
T
27315
27315
zz
z
2
0
2
0
0
+-+-++
+
^^
c
hh
m
E
(21)
where A, B, C, D, E, F and G are the equation constants,
presented in Table 2. Tz is the ambient temperature at height
z in degrees Celsius. The other parameters in Eq. (21) are as
defined earlier. Eqs. (16) and (21) are applicable as long as
temperature variation in the desired limit is linear (i.e., the
assumption of constant lapse rate).
3. DISCUSSION AND CONCLUSION
In this study, it was revealed that analytical computa-
tion of POWV relies on a Riemann integral solution over
a range of altitude whose integrand is merely a function of
altitude. The solution yielded a series function which was
circumvented by approximation of saturation vapor pres-
sure in the range of -55 to 55 degrees Celsius, and a formula
for POWV in an atmosphere of constant lapse rate was suc-
cessively proposed.
The formula [i.e., Eq. (21)] was verified and compared
with the diagrams (Fig. 1) published by USWB (1949). A
comparison was optionally carried out at different heights
for the temperatures 10, 20 and 28°C. The gas constant, R,
was taken 287 cm2 sec-2 °K-1. Table 3 compares lPOWV ob-
tained from Eq. (21) with USWB diagrams. It is necessary
to state that USWB assumptions for lapse rate value (α) are
not explicitly mentioned since their calculations were on the
basis of pseudo-adiabatic lapse rate. Comparing Eq. (21)
with Eq. (16) at normal meteorological conditions implies
maximum and mean error of 0.066 and 0.008%, respective-
ly. For instance, an error of 0.063% is observable between
the two formulae at sea level temperature 10°C with ambi-
ent lapse rate 6°C km-1 in a 10 km saturated layer [lPOWV is
obtained 21.873 mm from Eq. (16)].
An atmosphere of constant lapse rate only occurs when
the atmosphere is moisture free. When the air is fully satu-
rated the atmosphere becomes unstable, and in an adiabatic
process, vertical temperature gradient of standard atmo-
sphere follows saturation adiabatic lapse rate (SALR):
cL
dT
dw
g
s
ps
a=
+
^h
(22)
where αs is SALR in °C km-1; g is acceleration due to grav-
ity (m s-2); cp denotes heat capacity of dry air at constant
pressure in J kg-1; L represents latent heat of condensation in
J kg-1, and ws is mixing ratio of the mass of water vapor to
the mass of dry air (Wallace and Hobbs 1977). The second
term in the denominator of Eq. (22) indicates the presence
of water vapor. In order to compare POWV obtainable from
constant lapse rate and POWV from SALR, Exact calcula-
tion for SALR was performed in the sense that accelera-
tion due to gravity, g, was obtained through the following
formula:
gG
Rz
M
e
e
2
=+
^h
(23)
where G is gravitational constant, equal to 6.673 * 10-11
N m-2 kg2; Me is the earth’s mass (5.98 * 1024 kg); Re is the
earth’s radius which is approximately (6.38 * 106 m); and z
is the altitude in m (Bourg 2002). Heat capacity of dry air at
constant pressure, cp, in J kg-1 was precisely determined by:
cR TT TT
p
234
ab cdf=++++
^h
(24)
where T is in Kelvin, and α, β, γ, δ, and ε are constant and
have the values of 3.653, -1.337 * 10-3, 3.294 * 10-6, -1.913 *
10-9 and 0.2763 * 10-12 for air respectively (Moran et al.
2011). Latent heat of condensation of water vapor (L) was
approximated by:
Table 2. The value of the constants in Eq. (21).
Constant Value
A6.65851E-10
B3.639415922E-08
C3.44569192613E-05
D-7.0714633E-03
E3.342085312
F-1798.139526
G491541.7167

Ali Varmaghani
22
...LTTT00000614342 0 00158927236418
32
=- +-
.2500 79+
(25)
where T is in degrees Celsius (Rogers 1976). ws in Eq. (22)
is defined by:
.wpe
e
0622
s
s
s
=-
(26)
where p is total pressure. Total pressure at elevation z, p, is
dependant of lapse rate and was obtained up to 11 km from
International Standard Atmosphere (ISA) model:
pp T
z
1R
g
0
0
d
a
=- a
`
j
(27)
where Rd is dry air gas constant (Iribarne and Godson
1986).
The mentioned equations indicate that saturation adia-
batic lapse rate is a function of temperature and pressure
while temperature and pressure are functions of SALR too.
Therefore, a trial-and-error layer-by-layer algorithm was
performed in MATLAB to compute SALR. The equations
also imply that precise calculation of SALR is feasible only
if the troposphere is divided into several sub-layers pro-
vided that the premise of constant lapse rate for each layer
maintains the accuracy of computations. The height of each
layer was conservatively assumed 10 m - yielding less than
1% error for the value of SALR if the entire troposphere
shows the variation of at most 10°C km-1. Therefore, the
troposphere was divided into 1100 layers. It should be noted
that Eq. (27) was considered to be piecewise valid for satu-
rated air: for each layer, its own value of SALR was used
in Eq. (27).
Figure 2 aptly reflects the diagrams of SALR vs. al-
titude for three surface temperatures. As can be seen, tem-
perature gradient or SALR in the troposphere increases with
altitude and generally has the lower values at higher sur-
face temperatures (T0). As observable in Fig. 2, the range
of SALR lies between 3.6 to 9.8°C km-1 for normal surface
temperatures. POWV obtained from the assumption of con-
stant lapse rate was compared with POWV from SALR at
15°C surface temperature. Average value for constant lapse
rate was chosen in the sense that both POWVs become equal
at the tropopause (see Fig. 3). Those two POWV curves in
Fig. 3 were generated by Eq. (21). The result demonstrated
that the assumption of constant lapse rate produces maxi-
mum error of 5.36% at altitude of 4.44 km in comparison
with values of lapse rate due to SALR at 15°C surface tem-
perature. The average error was proved to be 3.24% at the
mentioned atmospheric condition.
The obtained result in this study may offer that POWV
estimation based on constant lapse rate does not lead to un-
satisfactory errors, and hence encourages analytical solu-
tions. The superiority of the proposed analytical formulae
over USWB tables and diagrams, as well as their high ac-
curacy, is that mathematical relations are not limited while
tables and diagrams just cover some specific values. USWB
diagrams also present POWV merely above the sea level
(where the base altitude is zero) while the suggested equa-
tions are suitable not only above the ocean but also above
mountainous zones and hence they are more appropriate
for grid-based models. The results achieved in this study
demonstrated flexibility and high capability of analytical
solutions beside discrete and empirical methods. For future
studies, it is suggested that new contributions are made to
estimate latent heat of vaporization for broader ranges since
Eq. (25) estimates L within the range of -40 to 40°C which
does not cover the variation of atmosphere temperature.
Table 3. A comparison between Eq. (21) and USWB diagrams for lPOWV.
Height
(km)
lPOWV (in)
Error
(%)
lPOWV (in)
Error
(%)
lPOWV (in)
Error
(%)
T0 = 10°C & α = 6°C km-1 T0 = 20°C & α = 4.9°C km-1 T0 = 28°C & α = 4°C km-1
diagram Eq. (21) diagram Eq. (21) diagram Eq. (21)
10.34 0.86 0.862 0.2 2.07 2.069 0.0 4.06 4.095 0.9
8.53 0.85 0.855 0.6 2.06 2.024 1.8 3.95 3.922 0.7
7.32 0.85 0.845 0.6 2.02 1.967 2.7 3.80 3.744 1.5
6.1 0.84 0.823 2.1 1.94 1.876 3.4 3.55 3.490 1.7
4.88 0.80 0.783 2.2 1.79 1.732 3.3 3.18 3.140 1.3
3.66 0.73 0.709 3.0 1.57 1.513 3.8 2.72 2.659 2.3
2.44 0.60 0.579 3.6 1.22 1.183 3.1 2.05 2.011 1.9
1.22 0.37 0.360 2.8 0.71 0.699 1.6 1.17 1.144 2.3

A Formula for PWV in Atmosphere of Constant Lapse Rate 23
Acknowledgements The author would like to thank the
anonymous journal reviewers for their constructive re-
marks. This work was accomplished under partial support
of Iowa Flood Center. I also appreciate professors William
Eichinger and Allen Bradley for their guidance.
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