View source for Goff–Gratch equation

The '''Goff–Gratch equation''' is one (arguably the first reliable in history) amongst many experimental correlation proposed to estimate the [[Vapour pressure of water|saturation water vapor pressure]] at a given temperature.

Another similar equation based on more recent data is the [[Arden Buck equation]].

==Historical note==

This equation is named after the authors of the original scientific article who described how to calculate the saturation water vapor pressure above a flat free water surface as a function of temperature (Goff and Gratch, 1946). Goff (1957) later revised his formula, and the latter was recommended for use by the [[World Meteorological Organization]] in 1988, with further corrections in 2000. {{Citation needed|date=October 2011}}

The current 2015 edition of the WMO Technical Regulations (WMO-No. 49) however states in Volume 1, Part III, Section 1.2.1, that any formula or constant given in the Guide to Meteorological Instruments and Methods of Observation a.k.a. CIMO-Guide (WMO-No. 8) shall be used, and this document only contains the much simpler Magnus formula (Annex 4.B. – Formulae for the computation of measures of humidity). Regarding the measurement of upper-air humidity, this publication also reads (in Section 12.5.1):

''The saturation with respect to water cannot be measured much below –50 °C, so manufacturers should use one of the following expressions for calculating saturation vapour pressure relative to water at the lowest temperatures – Wexler (1976, 1977),<ref name=Wexler1976>{{cite journal |first=A. |last=Wexler |title=Vapor pressure formulation for water in range 0 to 100°C. A revision |journal= Journal of Research of the National Bureau of Standards Section A |volume=80A |issue=5–6 |pages=775–785 |year=1976 |doi=10.6028/jres.080a.071|pmid=32196299 |pmc=5312760 |doi-access=free }}</ref><ref name=Wexler1977>{{cite journal |first=A. |last=Wexler |title=Vapor pressure formulation for ice |journal= Journal of Research of the National Bureau of Standards Section A |volume=81A |issue=1 |pages=5–20 |year=1977 |doi=10.6028/jres.081a.003|doi-access=free |pmc=5295832 }}</ref> reported by Flatau et al. (1992).,<ref name=Flatau1992>{{cite journal |last1=Flatau |first1=P.J. |last2=Walko |first2=R.L. |last3=Cotton |first3=W.R. |title=Polynomial fits to saturation vapor pressure |journal=Journal of Applied Meteorology |volume=31 |issue=12 |pages=1507–13 |year=1992 |doi=10.1175/1520-0450(1992)031<1507:PFTSVP>2.0.CO;2 |bibcode=1992JApMe..31.1507F |doi-access=free }}</ref> Hyland and Wexler (1983) or Sonntag (1994) – and not the Goff-Gratch equation recommended in earlier WMO publications.''

==Experimental correlation==

The original Goff–Gratch (1945) experimental correlation reads as follows:

:{|
|-
|<math>\log\ e^*\ = </math>
|<math>-7.90298(T_\mathrm{st}/T-1)\ +\ 5.02808\ \log(T_\mathrm{st}/T)</math>
|-
|
|<math>-\ 1.3816\times10^{-7}(10^{11.344(1-T/T_\mathrm{st})}-1)</math>
|
|-
|
|<math>+\ 8.1328\times10^{-3}(10^{-3.49149(T_\mathrm{st}/T-1)}-1)\ +\  \log\ e^*_\mathrm{st}</math>
|}

where:
:log refers to the logarithm in base 10
:''e''<sup>*</sup> is the saturation water vapor pressure ([[Pascal (unit)|hPa]])
:''T'' is the [[absolute temperature|absolute air temperature]] in [[kelvin]]s
:''T''<sub>st</sub> is the steam-point (i.e. [[boiling point]] at 1 atm.) temperature (373.15 K) 
:''e''<sup>*</sup><sub>st</sub> is ''e''<sup>*</sup> at the steam-point pressure (1 atm = 1013.25 hPa)

Similarly, the correlation for the saturation water vapor pressure over ice is:

:{|
|-
|<math>\log\ e^*_i\ = </math>
|<math>-9.09718(T_0/T-1)\ -\ 3.56654\ \log(T_0/T)</math>
|-
|
|<math>+\ 0.876793(1-T/T_0) +\ \log\ e^*_{i0}</math>
|}

where:
:log stands for the logarithm in base 10
:''e''<sup>*</sup><sub>i</sub> is the saturation water vapor pressure over ice (hPa)
:''T'' is the air temperature (K)
:''T''<sub>0</sub> is the ice-point ([[triple point]]) temperature (273.16 K) 
:''e''<sup>*</sup><sub>i0</sub> is e<sup>*</sup> at the ice-point pressure (6.1173 hPa)

==See also==
*[[Vapour pressure of water]]
*[[Antoine equation]]
*[[Arden Buck equation]]
*[[Tetens equation]]
*[[Lee–Kesler method]]

==References==
{{refbegin}}
* Goff, J. A., and Gratch, S. (1946) Low-pressure properties of water from &minus;160 to 212&nbsp;°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. (1957) 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.
* World Meteorological Organization (1988) General meteorological standards and recommended practices, Appendix A, WMO Technical Regulations, WMO-No. 49.
* World Meteorological Organization (2000) General meteorological standards and recommended practices, Appendix A, WMO Technical Regulations, WMO-No. 49, corrigendum.
*{{cite web |title=WMO Guide To Meteorological Instruments and Methods of Observation (the CIMO Guide) |year=2014 |id=WMO-No. 8. |url=http://www.wmo.int/pages/prog/www/IMOP/CIMO-Guide.html}}
*{{cite journal |last1=Murphy |first1=D.M. |last2=Koop |first2=T. |title=Review of the vapour pressures of ice and supercooled water for atmospheric applications |journal=Quarterly Journal of the Royal Meteorological Society |volume=131 |issue=608 |pages=1539–65 |year=2005 |doi=10.1256/qj.04.94 |bibcode=2005QJRMS.131.1539M |s2cid=122365938 |url=https://zenodo.org/record/1236243 |doi-access=free }}
;Notes
<references />
{{refend}}
==External links==
*{{cite web |first=Holger |last=Vömel |title=Saturation vapor pressure formulations |date=2016 |publisher=Earth Observing Laboratory, National Center for Atmospheric Research |location=Boulder CO |url=http://cires.colorado.edu/~voemel/vp.html}}
* [http://www.phymetrix.com/Software.htm Free Windows Program, Moisture Units Conversion Calculator w/Goff-Gratch equation] — PhyMetrix

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[[Category:Atmospheric thermodynamics]]