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__Saturation
vapor pressure formulations__

Holger
Vömel

CIRES,
University of Colorado, Boulder

A large number of saturation vapor pressure equations exists to calculate the pressure of water vapor over a surface of liquid water or ice. This is a brief overview of the most important equations used. Several useful reviews of the existing vapor pressure curves are listed in the references. Please note the discussion of the WMO formulations.

Goff Gratch equation

*(Smithsonian Tables, 1984, after Goff and Gratch, 1946)*:

Log_{10} *e*_{w}
= -7.90298
(373.16/*T*-1)
[1]

+ 5.02808 Log_{10}(373.16/*T*)

- 1.3816 10^{-7 }(10^{11.344 (1-}^{T}^{/373.16)}
-1)

+ 8.1328 10^{-3 }(10^{-3.49149 (373.16/}^{T}^{-1) }
-1)

+ Log_{10}(1013.246)

with *T* in [K] and *e*_{w}
in [hPa]

Guide to Meteorological Instruments and Methods of Observation (CIMO Guide)

*(WMO, 2008)*

*e*_{w} =
6.112 e^{(17.62 t/(243.12 +
t))}
[2]

with *t* in [°C] and *e*_{w} in
[hPa]

WMO

*(Goff, 1957):*

Log_{10} e_{w}
= 10.79574
(1-273.16/*T*)
[3]

- 5.02800 Log_{10}(*T*/273.16)

+ 1.50475 10^{-4 }(1 - 10^{(-8.2969*(}^{T}^{/273.16-1))})

+ 0.42873 10^{-3 }(10^{(+4.76955*(1-273.16/}^{T}^{))
}- 1)

+ 0.78614

with *T* in [K] and *e*_{w}
in [hPa]

(Note: WMO based its
recommendation on a paper by Goff (1957), which is shown here. The
recommendation published by WMO (1988) has several typographical
errors and cannot be used. A corrigendum (WMO, 2000) shows the term
+0.42873 10^{-3 }(10^{(-4.76955*(1-273.16/}^{T}^{))
}- 1) in the fourth line compared to the original publication
by Goff (1957). Note the different sign of the exponent. The earlier
1984 edition shows the correct formula.)

Hyland and Wexler

*(Hyland and Wexler, 1983):*

Log *e*_{w} =
-0.58002206 10^{4} /
*T
*[4]

+ 0.13914993 10^{1}

- 0.48640239 10^{-1} *T*

+ 0.41764768 10^{-4} *T*^{2}

- 0.14452093 10^{-7} *T*^{3}

+ 0.65459673 10^{1} Log(*T*)

with *T* in [K]
and *e*_{w} in [Pa]

Buck

*(Buck Research Manual (1996); updated equation from Buck, A. L., New equations for computing vapor pressure and enhancement factor, J. Appl. Meteorol., 20, 1527-1532, 1981)*

e_{w}= 6.1121 e^{(18.678 - }^{t}^{ / 234.5) }^{t}^{ / (257.14 + }^{t}^{)}[1996] [5]e_{w}= 6.1121 e^{17.502 }^{t}^{ / (240.97 + }^{t}^{)}[1981] [6]

withtin [°C] ande_{w}in [hPa]

Sonntag

*(Sonntag, 1994)*

Log e_{w} =
-6096.9385 /
*T*
[7]

+ 16.635794

- 2.711193 10^{-2} * *T*

+ 1.673952 10^{-5} * *T*^{2}

+ 2.433502 * Log(*T*)

with *T* in [K] and e_{w}
in [hPa]

Magnus Tetens

*(Murray, 1967)*

*e*_{w} =
6.1078 e^{17.269388 * }^{(T-273.16)}^{ /
(}^{T –
35.86}^{)}
[8]

with *T* in [K] and e_{w} in [hPa]

Bolton

*(Bolton, 1980)*

*e*_{w} =
6.112 e^{17.67 * }^{t}^{ /
(}^{t}^{+243.5)}
[9]

with *t* in [°C] and e_{w} in [hPa]

Murphy and Koop

*(Murphy and Koop, 2005)*

Log e_{w} =
54.842763

- 6763.22 / *T*

- 4.21 Log(*T)*

+ 0.000367 *T*

+ Tanh{0.0415 (*T* - 218.8)}

· (53.878 - 1331.22 / *T* - 9.44523 Log(*T*)
+ 0.014025 *T*)
[10]

with *T* in [K] and e_{w}
in [Pa]

International Association for the Properties of Water and Steam (IAPWS) Formulation 1995

*(Wagner and Pruß, 2002)*

Log (e_{w}*/22.064e6*)
= 647.096/T * ((-7.85951783 ν

+ 1.84408259 ν^{1.5}

- 11.7866497 ν^{3}

+ 22.6807411 ν^{3.5}

- 15.9618719 ν^{4}

+ 1.80122502 ν^{7.5}))
[11]

with *T* in [K] and e_{w}
in [Pa] and ν = 1 - T/647.096

At low temperatures most of these
are based on theoretical studies and only a small number are based on
actual measurements of the vapor pressure. The Goff Gratch equation
[1] for the vapor pressure over liquid water covers a region of -50°C
to 102°C [Gibbins 1990]. This work is generally considered the
reference equation but other equations are in use in the
meteorological community [Elliott and Gaffen, 1993]. There is a very
limited number of measurements of the vapor pressure of water over
supercooled liquid water at temperatures below °C. Detwiler
[1983] claims some indirect evidence to support the extrapolation of
the Goff-Gratch equation down to temperatures of -60°C. However,
this currently remains an open issue.

The WMO Guide to
Meteorological Instruments and Methods of Observation (CIMO Guide,
WMO No. 8) formulation [2] is widely used in Meteorology and appeals
for its simplicity. Together with the formulas by Bolton [9] and Buck
[6] it has the same mathematical form as older the Maguns Tetens [8]
formula and differs only in the value of the parameters.

The
Hyland and Wexler formulation is used by Vaisala and is very similar
to the formula by Sonntag [7]. The comparison for the liquid
saturation vapor pressure equations [2]-[11] with the Goff-Gratch
equation [1] in figure 1 shows that uncertainties at low temperatures
become increasingly large and reach the measurement uncertainty
claimed by some RH sensors. At -60°C the deviations range from
-6% to +3% and at -70°C the deviations range from -9% to +6%. For
RH values reported in the low and mid troposphere the influence of
the saturation vapor pressure formula used is small and only
significant for climatological studies [Elliott and Gaffen 1993].

The WMO (WMO No. 49, Technical Regulations) recommended
formula [3] is a derivative of the Goff-Gratch equation, originally
published by Goff (1957). The differences between Goff (1957) and
Goff-Gratch (1946) are less than 1% over the entire temperature
range. The formulation published by WMO (1988) cannot be used due to
several typographical errors. The corrected formulation WMO (2000)
differs in the sign of one exponent compared to Goff (1957). This
incorrect formulation is in closer agreement with the Hyland and
Wexler formulation; however, it is to be assumed that Goff (1957) was
to be recommended.

The review of vapor pressures of ice and supercooled water by Murphy and Kopp (2005) provides a formulation [10] based on recent data on the molar heat capacity of supercooled water. The comparison of the the vapor pressure equations with the formulation by Murphy and Koop is shown in figure 2.

The study by Fukuta and Gramada [2003] shows direct measurements of the vapor pressure over liquid water down to -38°C. Their result indicates that at the lowest temperatures the measured vapor pressure may be as much as 10% lower than the value given by the Smithsonian Tables [1], and as shown in figure 1 lower as any other vapor pressure formulation. However, these data are in conflict with measured molar heat capacity data (Muprhy and Koop, 2005), which have been measured both for bulk as for small water droplets.

Like most other formulations, the IAPWS formulation 1995 (Wagner and Pruß, 2002) are valid only above the triple point. The IAWPS formulation 1995 (Wagner and Pruß, 2002) is valid in the temperature range 273.16 K < T < 647.096 K.

It is important to note that in the upper troposphere, water vapor measurements reported in the WMO convention as relative humidity with respect to liquid water depend critically on the saturation vapor pressure equation that was used to calculate the RH value.

Figure 1: Comparison of equations
[2]-[11] with the Goff Gratch equation [1] for the saturation
pressure of water vapor over liquid water. The measurements by Fukuta
et al. [2003] are shown as well. ^{(*)}WMO
(2000) is also shown. This is based on Goff (1957) with the different
sign of one exponent, likely due to a typographical error.

Figure 2: Comparison of several equations with the equation by Sonntag [7] for the saturation pressure of water vapor over liquid water.

The equations by Hyland and Wexler [4], the nearly identical equation by Wexler (1976, see reference below) and the equation by Sonntag [7] are the most commonly
used equations among radiosonde manufacturers and should be used in upper air applications to avoid inconsistencies.

Goff Gratch equation

*(Smithsonian Tables, 1984)*:

Log_{10} e_{i}
= -9.09718 (273.16/*T* -
1)
[12]

- 3.56654 Log_{10}(273.16/ *T*)

+ 0.876793 (1 - *T*/ 273.16)

+ Log_{10}(6.1071)

with *T* in [K] and e_{i}
in [hPa]

Hyland and Wexler

*(Hyland and Wexler, 1983.):*

Log e_{i} =
-0.56745359 10^{4} /
*T*
[13]

+ 0.63925247 10^{1}

- 0.96778430 10^{-2} *T*

+ 0.62215701 10^{-6} *T*^{2}

+ 0.20747825 10^{-8} *T*^{3}

- 0.94840240 10^{-12} *T*^{4}

+ 0.41635019 10^{1} Log(*T*)

with *T* in [K]
and e_{i} in [Pa]

Guide to Meteorological Instruments and Methods of Observation (CIMO Guide)

*(WMO, 2008)*

*e*_{i} =
6.112 e^{(}^{22.46 t/(272.62 +
t}^{))}
[14]

with *t* in [°C] and e_{i} in [hPa]

Magnus Teten

*(Murray, 1967)*

*e*_{i} =
6.1078 e^{21.8745584 * }^{(T-273.16)}^{ /
(}^{T –
7.66}^{)}
[15]

with *T* in [K] and e_{w} in [hPa]

Buck

*(Buck Research Manual, 1996)*

e_{i}= 6.1115 e^{(23.036 - }^{t}^{ / 333.7) }^{t}^{ / (279.82 + }^{t}^{)}[1996] [16]e_{i}= 6.1115 e^{22.452 }^{t}^{ / (272.55+}^{t}^{)}[1981] [17]

withtin [°C] and e_{i}in [hPa]

Marti Mauersberger

*(Marti and Mauersberger, 1993)*

Log_{10}e_{i}= -2663.5 /T+ 12.537 [18]

withTin [K] and e_{i}in [Pa]

Murphy and Koop

*(Murphy and Koop, 2005)*

Log e_{i}= 9.550426

- 5723.265/T

+ 3.53068 Log(T)

- 0.00728332T[19]

withTin [K] and e_{i}in [Pa]

The Goff Gratch equation [11] for
the vapor pressure over ice covers a region of -100°C to 0°C.
It is generally considered the reference equation; however, other
equations have also been widely used. The equations discussed here
are mostly of interest for frost-point measurements using chilled
mirror hygrometers, since these instruments directly measure the
temperature at which a frost layer and the overlying vapor are in
equilibrium. In meteorological practice, relative humidity is given
over liquid water (see section 1) and care needs to be taken to
consider this difference.

Buck Research, which manufactures
frost-point hygrometers, uses the Buck formulations in their
instruments. These formulations include an enhancement factor, which
corrects for the differences between pure vapor and moist air. This
enhancement factor is a weak function of temperature and pressure and
corrects about 0.5% at sea level. For the current discussion it has
been omitted.

The Marti Mauersberger equation is the only
equation based on direct measurements of the vapor pressure down to
temperatures of 170 K.

The comparison of equations 12-17 with the
Goff Gratch equation (figure 3) shows, that with the exception of the
Magnus Teten formula, the deviations in the typical meteorological
range of -100°C to 0°C are less than 2.5%, and smaller than
typical instrumental errors of frost-point hygrometers of 5-10%.

Not
shown is the WMO recommended equation for vapor pressure over ice,
since it is nearly identical with the Goff-Gratch equation [12].

Figure 3: Comparison of equations
[13]-[18] with the Goff Gratch equation [12] for the saturation
pressure of water vapor over ice.

Bolton, D., The
computation of equivalent potential temperature, Monthly Weather
Review, 108, 1046-1053, 1980..

Buck, A. L.,
New equations for computing vapor pressure and enhancement factor, J.
Appl. Meteorol., 20, 1527-1532, 1981.

Buck
Research Manuals, 1996

Detwiler, A.,
Extrapolation of the Goff-Gratch formula for vapor pressure over
liquid water at temperatures below 0°C, J. Appl. Meteorol., 22,
503, 1983.

Elliott, W. P. and D. J. Gaffen,
On the utility of radiosonde humidity archives for climate studies,
Bull. Am. Meteorol. Soc., 72, 1507-1520, 1991.

Elliott,
W. P. and D. J. Gaffen, Effects of conversion algorithms on reported
upper air dewpoint depressions, Bull. Am. Meteorol. Soc., 74,
1323-1325, 1993.

Fukuta, N. and C. M.
Gramada, Vapor pressure measurement of supercooled water, J. Atmos.
Sci., 60, 1871-1875, 2003.

Gibbins, C. J., A
survey and comparison of relationships for the determination of the
saturation vapour pressure over plane surfaces of pure water and of
pure ice, Annales Geophys., 8, 859-886, 1990.

Goff,
J. A., and S. Gratch, Low-pressure properties of water from -160 to
212 F, in Transactions of the American society of heating and
ventilating engineers, pp 95-122, presented at the 52nd annual
meeting of the American society of heating and ventilating engineers,
New York, 1946.

Goff, J. A. Saturation
pressure of water on the new Kelvin temperature scale, Transactions
of the American society of heating and ventilating engineers, pp
347-354, presented at the semi-annual meeting of the American society
of heating and ventilating engineers, Murray Bay, Que. Canada, 1957.

Hyland, R. W. and A. Wexler, Formulations for the
Thermodynamic Properties of the saturated Phases of H2O from 173.15K
to 473.15K, ASHRAE Trans, 89(2A), 500-519, 1983.

Marti,
J. and K Mauersberger, A survey and new measurements of ice vapor
pressure at temperatures between 170 and 250 K, GRL 20, 363-366, 1993

Murphy, D. M. and T. Koop, Review of the vapour
pressures of ice and supercooled water for atmospheric applications,
Quart. J. Royal Met. Soc, 131, 1539-1565, 2005.

Murray,
F. W., On the computation of saturation vapor pressure, J. Appl.
Meteorol., 6, 203-204, 1967.

Smithsonian Met.
Tables, 5th ed., pp. 350, 1984.

Sonntag,
D., Advancements in the field of hygrometry, Meteorol. Z., N. F., 3,
51-66, 1994.

Wagner W. and A. Pruß, The
IAPWS formulation 1995 for the thermodynamic properties of ordinary
water substance for general and scientific use, J. Phys. Chem. Ref.
Data, 31, 387-535, 2002.

Wexler, A., Vapor Pressure Formulation for Water in Range 0 to 100°C.
A Revision, Journal of Research of the National Bureau of Standards, 80A, 775-785, 1976.

World Meteorological
Organization, General meteorological standards and recommended
practices, Appendix A, WMO Technical Regulations, WMO-No. 49, Geneva
1988.

World Meteorological Organization,
General meteorological standards and recommended practices, Appendix
A, WMO Technical Regulations, WMO-No. 49, corrigendum, Geneva August
2000.

World Meteorological Organization, Guide to Meteorological
Instruments and Methods of Observation, Appendix 4B, WMO-No. 8 (CIMO
Guide), Geneva 2008.

1 December 2011

Holger.Voemel@Colorado.edu