### Water Vapor / Relative Humidity

For a given number of water molecules in the atmosphere, the Relative Humidity varies depending on only the temperature. In principal, this is the Clausius-Clapeyron relation.

In general, meteorological instruments do not measure either the amount of water in the atmosphere or the relative humidity. Instead, they measure

• Dew point
• Frost point
• Temperature
• Pressure
depending on the sensors used. Then an equation is used to compute the Relative Humidity. According to Murphy & Koop, these techniques are at best 0.1% accurate, and in the stratosphere, perhaps only 10% accurate.

The odd thing is that relative humidity has little to do with cloud formation, and even less with rain. It seems intuitively obvious that clouds form when the relative humidity is 100% .. but they don't. In fact, they tend to evaporate at even higher water concentrations.

Equally odd is the fact that water won't freeze until about -38°C even though it melts at about 0.01°C.

### Clouds

The problem with clouds is simple - the partial pressure of water vapor at saturation is a function of the droplet size, dissolved gasses, dissolved minerals (salts), and the availability of inert matter (dust).

100% relative humidity is defined over a flat area of water where both the water and the air have the same temperature. In a cloud, the temperature of the water and the surrounding air are about the same - but the water is in small drops. As a result, at 100%RH it is easier for a small drop to lose a water molecule than it is for it to gain one. However, once the size crosses some threshold, the drop will suck up all the available water and make rain.

For clouds to form at 100%RH, there needs to be something that changes the balance. The "technical" term for this is "dust". Over oceans, this could be salt crystals; over a desert, silica. Air pollution may also help. In a cloud chamber, cosmic rays cause a liquid (usually alcohol or water) to condense.

Without getting into the details, the point is that clouds form in a range of relative humidities - not just when %RH = 100%.

### Ice

Just like clouds that don't form at 100%RH, ice tends to not form until the water temperature is around -38°C .. unless there is some help. In chemistry class, we would cool a beaker of water below 0°C. To produce ice, we hit the inside of the beaker with a glass rod.

When a liquid is cooled below its freezing point, it is referred to as super cooled. In the atmosphere, super cooled water is fairly common. (Think freezing rain. Sometimes this is just rain hitting a very cold surface. But it can also be super cooled rain that freezes on contact.)

Ice will form at a higher temperature if there is something to start the process - dust is the common suspect.

The relative humidity over ice is different than over liquid water. This is usually explained by observing that a certain amount of energy is required to evaporate water and that, for ice to sublime, it requires additional energy equivalent to the amount needed to raise the temperature to the triple point and the latent heat needed to convert it to the liquid state. However, reality is a bit more complicated, depending also on the crystal type (amorphous, cubic, or hexagonal) and the existence of sharp points (think snowflake vs sleet).

And of course, the computed RH values are over a large surface, and the stuff that falls has lots of small surfaces. Therefore, 100%RH is just a guideline for ice formation, not when it must occur.

### Calibration Standards

The Celsius temperature scale was originally calibrated using 2 points - the freezing and boiling points of water - defined as 0°C and 100°C respectively. Today there is only one calibration point, the triple point at 0.01°C. The second point at 100°C is no longer used. However, it is still necessary to use 2 points to define the temperature scale. Assuming the second point to be absolute zero (0.0 K), and that it can be determined by the gas law, places the boiling point at 99.974°C at 1 atm (1013.25 mbar).

From Review of the vapour pressures of ice and supercooled water for atmospheric applications by D. M. Murphy and T. Koop, 29 Dec 2006

 The vapour pressure of both ice and liquid water at the triple point is Pt = 611.657 ± 0.01 Pa at temperature Tt = 273.16 K (Guildner et al. 1976, Vapor pressure of water at its triple point. J. Res. Natl. Bur. Stand., 80A, 505–521)
Many of the formulas I've seen do not pass thru that point. Presumably, this is because other calibrations were used. This is not just because the quality of the work changed. In addition, many of the standards used to determine the partial pressure have changed over time. [Rossini 1963]
• In 1929, the calorie was defined to be 4.1868 joules exact. This definition is still in use.
• In 1930, the calorie was defined to be 4.1833 joules exact.
• In 1948, the calorie was redefined to be 4.184 joules exact because other units changed.
• In 1955, the temperature scale was changed - the triple point of water was changed from 0.0°C exact to 0.01°C exact.
• In 1960, C-12 became the standard for defining a mole, which affected Avogadro's number and the gas constant.
Note that there are still 2 different definitions of the calorie in use - 4.1868 and 4.184 joules - depending on the context.

According to the International Union of Pure and Applied Chemistry, Subcommittee on Calibration and Test Materials, 1975, p 1449 (pdf p 12), the saturation partial pressure of water is (was) 0.6111 kPa at 273.16 K. The notes for the related section indicate that

• The National Bureau of Standards new value was 0.61164 +/- 0.00006 kPa at 273.16 K
• The 1968 temperature scale was currently being revised and the table values would be changing
• The Steam point would become a measured point, not a defined point
In my programs, note that the WMO_Goff value is 0.611139 kPa - GISS (NASA) used 0.61080 KPa for its 2007 climate models.

The other major source of uncertainty is the fact that the equations model pure water with no dissolved salts or gases. Since that is not representative of the atmosphere, there is an "Enhancement Factor" to produce actual values - the error is about 0.5% near sea level and decreases with decreasing total pressure. For example, at 0°C, Buck gives the saturation partial pressure over liquid Water as 6.1121 mbar, corrected to 6.1360 mbar at 1,000 mbar total pressure - 6.1202 mbar at 250 mbar total pressure.

At any rate, these are some of the reasons it is difficult to find a "correct" equation or to evaluate any work (think climate model) that uses relative humidity.

### The Equations

In order to determine the relative humidity, you need a way to compute the saturation partial pressure at a given temperature. Unfortunately, there is no one correct way to compute that value. For instance, the Voemel text provides 20 different equations and the associated source code provides 5 more - each with a different answer.

My programs implement

• 25 algorithms from Voemel
• 1 algorithm from a Seinfeld & Pandis textbook
• 2 algorithms used in NASA's GISS Model-E for IPCC AR4

It is difficult to calibrate equations that fit a bunch of data points.

In the original definition - water freezes at 0°C and boils at 100°C.

However, both of those values depend on the total pressure. In general, both should be defined at 1013.25 hPa (101,325 Pa).

Also, the "freezing" calibration point has been replaced with the triple point - 0.01°C (273.16 K).

Of the 20 formulas I've used for liquid water, only one has a partial pressure of 1013.25 hPa at 100°C and 4 have a partial pressure of 611.657 Pa at 273.16 K. (7 at 611.657 ± 0.01 Pa). Of these, the "best" appear to be the ones from HylandWexler and Sonntag because they nearly intersect both calibration points. The Seinfeld and Pandis formula is also pretty good since it nearly intersects the old calibration points.

Below freezing, there are 4 calibration points, but these are only used by Fukuta.

Over ice, the curve is different and there is only one calibration point - 611.657 ± 0.01 Pa at 273.16 K - and only 4 of 11 formulas intersect it.

Computed Saturation Pressure (mbar) Formula Ice Water 0.01°C 0.01°C ne - no equation     Some of the sources only provide over ice or over water Calibration Values 6.11657 6.11657 1013.25 1013.25 The boiling point values are defined by the calibration method GoffGratch 6.1071 6.1077976 1011.9447 1012.8844 -50°C to 102°C over water-100°C to 0°C over ice CIMO 6.1170373 6.1164311 1037.4672 1038.4492 Widely used in Meteorology HylandWexler 6.1165702 6.1165703 1013.2464 1014.1872 Commonly used among radiosonde manufacturers and should be used in upper air applications to avoid inconsistencies. MartiMauersberger 6.1137044 MagnusTetens 6.1078 6.1078 1020.6138 1021.5702 Buck_original 6.1165364 6.1165407 1035.1794 1036.1563 Buck_manual 6.1165331 6.1165411 1012.1404 1013.0778 Buck Research Manual (1996) - They make frost-point hygrometers WMO_Goff 6.11139 6.11139 1012.3113 1013.2513 WMO2000 6.11139 993.59275 994.50492 Identical to WMO_Goff except this contains an exponent sign error Wexler 6.0863961 995.03074 995.94972 Sonntag 6.1171458 6.1165707 1013.2497 1014.1904 Bolton 6.1164367 1046.7137 1047.7066 Fukuta 6.1025599 -378825.41 -379605.02 -20°C to -35°C -- Goes negative at +21°C IAPWS 6.1165707 1013.2393 1014.1799 International Association for the Properties of Water and Steam (IAPWS) Formulation 1995 Valid only above the triple point - 273.16 K to 647.095 K Preining 6.1074255 1011.9379 1012.8777 MurphyKoop 6.1165707 6.1165704 1019.2698 1020.2225 Seinfeld_Pandis 6.116602 1012.3101 1013.25 From textbook - intersects both calibration points GISS_AR4 6.1080103 6.1080091 1239.6974 1240.9521 GISS Model-E for IPCC AR4 and AR5 Antoine equation 6.0602333 1012.4211 1013.3651 Based on Clausius-Clapeyron relation for 0 to 100°C August equation 6.5934454 1011.7726 1012.7427
Saturation_vapor_pressure.exe normally limits the temperature precision to only 2 decimal places. To use additional precision, type the value into the Kelvin field.
 ```99.974°C = 373.124 K ```

Seinfeld and Pandis claim their formula produces a value of 31.387 mbar (30,984 ppm) when T = 298 K. My program produces 31.398836062 mbar, 30,996 ppm. To produce a value of 31.387 mbar, my program needs a temperature of 297.9937 K. This small difference implies that the difference is due to round off / precision at each step of the calculation.

"The Hyland and Wexler (1983) parameterization is used for calibration of the widely used Vaisala radiosondes (Miloshevich et al. 2001), [but] is only valid above 0°C (273.15 K)." Murphy and Koop, 2006

The Fukuta algorithm, calibrated for -20°C to -35°C, is worthless above 0°C - it goes negative around +21°C.

Water becomes a super critical fluid above 373.946°C (647.10 K) - above that temperature, there is no distinction between liquid and gas.

Over liquid water

 The Goff Gratch equation for the vapor pressure over liquid water covers a region of -50°C to 102°C. This work is generally considered the reference equation but other equations are in use in the meteorological community. The equations by Hyland and Wexler, the nearly identical equation by Wexler, and the equation by Sonntag are the most commonly used equations among radiosonde manufacturers and should be used in upper air applications to avoid inconsistencies. [ref]
Over ice
 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 ... and care needs to be taken to consider this difference. [ref]
Rather than use a standard, NASA's GISS Model-E (used for IPCC AR4) has its own equations
 ```Psat := 6.108 * exp(2.5E6 * (7.93252E-6 - 2.166847E-3/T_K)); // Latent_heat water = 2.5E6 Psat := 6.108 * exp(2.834E6*(7.93252E-6 - 2.166847E-3/T_K)); // Latent_heat ice = 2.834E6 ```
I have found their value of 6.108 mbar for the triple point in a number of references - but I haven't determined what study this comes from.

Well, maybe it doesn't matter. For my use, it is possible that one model (algorithm) is as good as any other. With a few exceptions (such as Fukuta which gives ridiculous results outside its calibration range), the values are pretty close. Keep in mind - I don't care what the relative humidity is. I simply need to use it to determine how many water molecules are actually in a unit volume of atmosphere - expressed in parts-per-million (ppm) - which is easy to compute from the water vapor mixing ratio. My line-by-line program (not ready for distribution) uses the ppm values to compute the amount of IR radiation absorbed (or emitted) by a given parcel. I have an analysis showing the difference at 15°C.

### Software

At any rate, I have written two programs to help understand the information presented here. These programs allow you to enter temperatures on any of 3 different scales .. and automatically converts to the other 2. It also allows you to either see (or select) all 30 algorithms.

Below 180 K (-93°C), the saturation partial pressure is so low that computations have little value. Most of them are worthless below 150 K, some even produce negative results. Because of this, and because none of the algorithms are calibrated below 238 K (-35°C), the program simply returns zero below 150 K.

### References

Author: Robert Clemenzi