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.

1) Vapor pressure over liquid water below 0°C

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

Log₁₀ e_w =  -7.90298 (373.16/T-1)                                                              [1]
                    + 5.02808 Log₁₀(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)
                   + Log₁₀(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):

Log₁₀ e_w =  10.79574 (1-273.16/T)                                                          [3]
                    - 5.02800 Log₁₀(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 10⁴ / T                                                                  [4]
              + 0.13914993 10¹
              - 0.48640239 10^-1 T
              + 0.41764768 10^-4 T²
              - 0.14452093 10^-7 T³
              + 0.65459673 10¹ 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]
with t in [°C] and e_w in [hPa]

• Sonntag
  (Sonntag, 1994)

Log e_w =  -6096.9385 / T                                                                             [7]
                 + 16.635794
                 - 2.711193 10^-2 * T
                 + 1.673952 10^-5 * T²
                 + 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 ν³
                                                            + 22.6807411 ν^3.5
                                                             - 15.9618719 ν⁴
                                                            + 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.

2) Vapor pressure over ice

• Goff Gratch equation
  (Smithsonian Tables, 1984):

Log₁₀ e_i =  -9.09718 (273.16/T - 1)                                                             [12]
                   - 3.56654 Log₁₀(273.16/ T)
                   + 0.876793 (1 - T/ 273.16)
                   + Log₁₀(6.1071)
with T in [K] and e_i in [hPa]

• Hyland and Wexler
  (Hyland and Wexler, 1983.):

Log e_i =  -0.56745359 10⁴ / T                                                                     [13]
              + 0.63925247 10¹
              - 0.96778430 10^-2 T
             + 0.62215701 10^-6 T²
             + 0.20747825 10^-8 T³
             - 0.94840240 10^-12 T⁴
             + 0.41635019 10¹ 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]
with t in [°C] and e_i in [hPa]

• Marti Mauersberger
  (Marti and Mauersberger, 1993)

Log₁₀ e_i = -2663.5 / T + 12.537                                                                   [18]
with T in [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)
                - 0.00728332 T                                                                                 [19]
with T in [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.

3) References

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.