Water Vapor - Other Formulas

My programs implement a number of formulas which claim to compute saturated water pressure at a given temperature. However, from time to time I run across new formulas - and occasionally, flat out wrong formulas. Since I don't want to be continuously updating my program, I will simply add these "new" formulas here.

The Engineering ToolBox

The Engineering ToolBox is a source of fairly reliable data. As a result, I was shocked (and disappointed) to find that their Saturation Pressure of Water Vapor formula is totally different than all the other formulas I've collected. In general, the results agree fairly well with GolfGratch up to about 0°C, then it begins to strongly diverge. For instance, (based on this table) Only GISS-AR4 (NASA) is worse.

Unfortunately, they do not reference their source for this equation - it is merely presented as "a statement of fact".

A Google search for that equation returned numerous hits. Unfortunately, all the examples I checked are obviously just quoting The Engineering ToolBox - either an explicit reference or the identical text and formatting.

In addition, on their Dry Bulb, Wet Bulb and Dew Point Temperatures page they say

which is, of course, wrong! It should be -273.15°C. (Yes - it does make a difference! Particularly in a reference book.)

NASA Tech Note D-8401

Equations for the Determination of Humidity from Dewpoint and Psychrometric Data - NASA Tech Note D-8401, Parish and Putnam, January 1977 - provides yet another set of equations with derivations. They claim that the following is the Magnus formula. which has the general form None of the other formulas I have seen to date have that form. The MagnusTetens formula is To be clear, several formulas contain Ta, but a is usually positive and there are always a lot more terms.

Their derivations of the constants - a, b, c - are provided in Appendix A. They always use 273 to convert between K and C (should be 273.15) and they use 2 vapor pressure callibration points - both are at 0°C (32°F).

Appendix A, Example 2 computes relative humidity from a dew point below freezing. Their value is 17.3%, my program, using MurphyKoop, computes 19.2% over water and 17.1% over ice. Example 3 uses 2 temperatures above freezing - they compute 84.6%, my program finds the same for most of the provided algorithms.

In addition, their definition of relative humidity (based on that giving in the Smithsonian Meteorological Tables, 1971) is a bit strange.

I have not seen any other papers specify that the denominator always be with respect to the saturation perssure over liquid water.

New Magnus Coefficients

Improved Magnus Form Approximation of Saturation Vapor Pressure, Alduchov (Ru) and Eskridge (US), April 1996, Journal of Applied Meteorology

This paper argues that the standard Magnus coefficients should be replaced so that the Magnus equation can be used instead of the much more complex equations of Wexler (1976), Sonntag (1990), and Goff and Gratch (1946).

As an aside, these dates do not match those provided by Vömel - Wexler (1983), and Sonntag (1994) - and the MurphyKoop (2005) formulation was published after this paper.

Part of the justification of this approach is that atmospheric temperature is reported to only 0.1°C accuracy world wide and only 0.5°C in the US. That, plus the general inaccuracies of the reported humidity proxies (such as dew point) means that the true accuracy of any computation is low enough that several formulations produce calculated saturation pressures that are "the same" within some margin of error.

This is the general form of the Magnus formula.

Alduchov provides 6 different sets of Magnus coefficients by others, and one "better formulation" suggested by himself. The following table only shows 3 - MagnusTetens from Vömel, one of the six "other" formulations, and the new "suggested" Alduchov formulation.

Full equations

The Alduchov paper discusses 4 "complex" equations by 3 authors - its purpose was to replace these "difficult to compute" formulations with a simpler Magnus form. (With modern computers, this goal is questionable.)


New Equations for Computing Vapor Pressure and Enhancement Factor, Arden L.Buck, 1981.

This paper contains a good overview of the progress toward developing equations to compute saturation vapor pressure. It includes a modification to the Magnus equation by Bögel and 2 equations by Wexler that are not included in Vömel.

washington.edu skewt.js

While developing a skew-T charting application, I found washington.edu skewt.js. This is their saturation vapor pressure subroutine (no source given) This is almost identical to the Goff Gratch equation (only one constant is different), but with the terms multiplied out (which presumably makes the computation faster) which, unfortunately, obscures the use of the triple point temperature. The following computations were performed using a spreadsheet. A few test computations show the same results as Goff Gratch to 2 decimal places - 4 significant figures.

Author: Robert Clemenzi
URL: http:// mc-computing.com / Science_Facts / Water_Vapor / Other_Formulas.html