The definition is simple - Relative Humidity is the ratio of the actual partial pressure of water vapor to the maximum partial pressure at saturation, expressed as a percent.
Of course, there is no way to measure either the relative humidity or the actual partial pressure. The tools normally used to determine these values - known as hygrometers (water measuring devices) .. such as psychrometers (2 thermometers, one bulb is wet and the other is dry), dew point mirrors, and frost point devices - do not actually measure Relative Humidity. Instead they measure a temperature that must be entered into some algorithm (model) that allows some parameter to be determined. In general, these techniques also rely on knowing what the saturation humidity is at a given temperature - that requires yet another algorithm (model). (Newer electronic devices determine relative humidity by measuring how much some physical property changes and computing an approximate value based on some calibration.)
Well, since we all know that "the science is done", I was very surprised to discover that there is no equation to determine what the saturation partial pressure is - instead, I found more than 28 different equations! 25 are provided by Holger Voemel. To these I added one from a textbook and two used in IPCC AR4. I have several more, but I decided to investigate just these 28. In version 1.2, I increased this to 30.
At saturation, a quantity of water will neither gain nor loose mass. The rate of evaporation and the rate of condensation are, by definition, equal. At a given temperature, the saturation partial pressure is strongly dependent on the shape of the surface. The equations model the saturation vapor pressure over a flat sheet of water. For droplets of water, the saturation pressure increases as the radius decreases. As a result, in clean air, the relative humidity can be larger than 100% without producing clouds. In addition to the surface shape, many substances - inert dust, dissolved salts, various gases - also affect the saturation partial pressure. In fact, saturated salt solutions are used to calibrate humidity sensors. In some cases, the references indicate that the equations fit data collected over water with all dissolved gasses removed. However, this is such an important consideration, that that condition may be true for all of them. On the other hand, when studying the atmosphere, there are always dissolved gasses. Ignoring the fact that CO2 is increasing, the amount of CO2 and various exhaust and industrial gases found near population centers could affect the probability of rain simply because they affect the saturation partial pressure of water vapor. In addition, the presence of dust (solid material that does not dissolve in water) helps water to condense at lower values. As a result, in an actual weather setting, the results from all of the equations contain a significant amount of uncertainty.
Besides being something that water can stick to, dust may actually be cooler than the surrounding air. This is because the air emits IR in a limited range of frequencies and is nearly IR opaque in those frequencies but solid matter (dust) emits with a blackbody (continuous) spectrum. As a result, more radiation from dust will escape the local environment and cause local cooling. Because the radiation spectrum of greenhouse gases decreases with altitude (lower pressure and lower temperature), dust should have a greater temperature effect at higher altitudes.
In an attempt to understand the relationships between Relative Humidity, Dew Point, Temperature, Pressure, and the Water Vapor Mixing Ratio, and to investigate various models, I wrote 2 programs - Saturation_vapor_pressure.exe and Water_Vapor.exe.
Saturation_vapor_pressure.exe shows the results of these 30 algorithms. Some are used by weather forecasters, others are used in radiosondes. The fact that there is no standard should be enough to make every scientist skeptical of climate models.
The second program - Water_Vapor.exe - allows you to select two of the 30 methods of computing saturation (one over water and another over ice) and computes the water vapor mixing ratio at the specified temperature, pressure, and relative humidity. Notice that the saturated partial pressures are independent of atmospheric pressure, but that the saturated mixing ratios are dependent on it. This is because the number air molecules per unit volume is related to the total pressure, but the saturated partial pressure of water is related to only the temperature. (Outside the temperature and pressure ranges found in the atmosphere, total pressure will have an effect. However, my programs consider that effect to be negligible. In a scuba tank - high pressure - total pressure is important.)
The program computes the dew point and the frost point. Obviously, frost points above freezing are of no value. However, dew points below freezing are useful - they could indicate the possibility of a freezing rain event, a condition where super cooled liquid water freezes on contact. The Cloud Base approximations assume that the temperature decreases from the surface to altitude at a rate equivalent to one of the 2 common lapse rates. Typically, the actual lapse rate is unknown. The 2 main exceptions are the DALR over land on a hot summer afternoon, and the ELR over the oceans.
I hope you have fun with these. I learned an awful lot trying to write them.
I was surprised to learn that the (in)famous GISS Model-E used in IPCC AR4 (Hansen, Schmidt, et al.) uses its own algorithms! (It has the same form as the August equation, but with different constants.) This looks like a major problem, but an example shows it probably does not matter.
The following tables compare the GoffGratch algorithm with GISS_AR4. They assume 15°C, sea level pressure (1013.25 mbar), and average relative humidities of 35% and 60%. (I don't know what good values are, just making this up.) The energy absorbed was computed with my program for a column of atmosphere one kilometer thick with those conditions held constant. The H2O column ignores the overlap with CO2.