Humidity variables
In atmospheric physics several different variables are used to describe the amount of water vapour in the air. These tend to be used in different contexts and here we put the most commonly encountered variables together.
The first variable is the concentration by mass of water vapour, usually called the specific humidity q,
Pd + Pv with pv and pd the local densities of the vapour and the dry air respectively. It was introduced in section 1.3 to define the virtual temperature. The specific humidity is dimensionless but it is often given 'units' of kg kg1 or, because there is usually much less water vapour than dry air, units of g kg1. A related quantity is the mass mixing ratio rv,
The mass mixing ratio is also dimensionless but is often given units of kg kg1 or g kg1. From their definitions it follows that the two variables can be transformed by q rv rv = and q = (5.21)
Because generally q < 1 we find that rv & q (or more precisely: r = q + q2 + q3 + q4 + ...). In the Earth's atmosphere the two are the same to within one part in a hundred.
From the ideal gas law, p = pRT, we can relate the partial densities pv and pd to the partial pressures of the water vapour e and the dry air p  e, with p the total pressure. It then follows that e e (5 22)
with id and iv the effective molar masses of dry air and water respectively, so id/iv = Rv/Rd = 1.61. (5.23)
The relative humidity (RH) is defined as the ratio of the actual vapour pressure  to the saturated vapour pressure vapour s at the given temperature,
The relative humidity is dimensionless but it is most commonly expressed as a percentage. Because s increases with temperature, the relative humidity at constant specific humidity will decrease at increasing temperature. Relative humidity indicates how far we are away from saturation. For example, just above the sea surface the relative humidity is usually very close to 100% while the specific humidity varies strongly with temperature.
Another commonly used humidity variable is the dewpoint temperature. It is defined as follows:
Dewpoint temperature Td is that temperature to which moist air has to be cooled isobarically to achieve saturation.
Following Eq. 5.22, the vapour pressure also remains constant in such a process, because rv remains the same. In equations, the dewpoint temperature Td is defined implicitly by
that is, the vapour pressure equals the saturated vapour pressure at the dewpoint temperature. The relative humidity can now be expressed as
The difference between the actual temperature and the dewpoint temperature, T — Td, is called the dewpoint depression. Air at low relative humidity has a large dewpoint depression; air at 100% relative humidity has a dewpoint depression of 0oC.
The definition of dewpoint temperature suggests a way to measure the humidity in air: cool it isobarically until saturation  the saturated vapour pressure at that temperature is the actual vapour pressure. This is the principle behind accurate measurements of humidity. However, it does require a fairly complex apparatus such as, for example, a chilled mirror hygrometer which works by cooling a mirror until condensation is optically detected.
A simpler way is to put water at the same temperature in contact with the air parcel and to cool the air parcel isobarically by evaporating the water into the parcel. Doing so will change the actual water vapour content of the parcel but this can be taken into account. It forms the basis of a classic humidity measurement using the socalled wetbulb temperature. It is defined as follows:
Wetbulb temperature Tw is that temperature to which air can be cooled isobarically by evaporating water into it.
At Tw the air is saturated. If it were not we could evaporate more water into it and cool the parcel further. Because in the process the specific humidity of the parcel has increased, the wetbulb temperature is higher than the dewpoint temperature.
Here we present a quick calculation to indicate how the wetbulb temperature relates to the vapour mixing ratio; a more accurate calculation is presented in the next section. To evaporate a unit mass of water we need energy L, the latent heat of evaporation. Let us assume that this energy is provided by the internal energy of the dry air, which forms the bulk of an air parcel. Further assume the dry air is an ideal gas. Because the process is isobaric we have
with Mv the mass of the vapour in the parcel and Md the mass of the dry air in the parcel. Divide this equation by Md to get an equation in terms of vapour mixing ratio rv,
Further assuming L and Cp to be constant, this equation can be integrated from the initial, drybulb temperature T to the final, wetbulb temperature Tw, when the air is saturated. This gives the socalled psychrometric equation,
with the additional subscript s indicating saturated values of the vapour mixing ratio. The difference T  Tw is called the wetbulb depression. The factor cp/L is called the psychrometric constant.
Figure 5.3 illustrates the construction of the wetbulb temperature on a graph of vapour mixing ratio versus temperature. It also illustrates the construction of the dewpoint temperature.
Dewpoint temperature is usually measured with a whirling psychrometer (or sling psychrometer). A whirling psychrometer has two thermometers, one of which has a wet wick (piece of fabric) around its bulb. The thermometers are whirled round so as to ventilate air through the wet wick. The air in the wick cools down and saturates, and the temperature of the air in the wick now equals the wetbulb temperature. The difference between the temperatures of the two thermometers is the wetbulb depression. The psychrometric equation is then solved to find the mixing ratio from the wetbulb depression and the temperature. There are tables, charts, slide rules, or computer programs to solve the psychrometric equation; Table 5.2 and Figure 5.4 are examples. If Tw < o°C there is a chance the wet bulb might freeze, so these tables and charts have limited validity in that regime although they typically continue
Figure 5.3 Construction of dewpoint temperature Td and wetbulb temperature Tw. The dewpoint temperature is found by cooling a parcel at fixed vapour mixing ratio rv until saturation is achieved, rv = rvs(Td). The wetbulb temperature is found by cooling the parcel by evaporating water into it until saturation is achieved. In this process the vapour mixing ratio changes according to drv/dT = —cp/L; for constant cp/L we can infer from the geometry in the figure below that (T — Tw) (cp/L) = rvs (Tw)rv, which is the psychrometric equation.
Figure 5.3 Construction of dewpoint temperature Td and wetbulb temperature Tw. The dewpoint temperature is found by cooling a parcel at fixed vapour mixing ratio rv until saturation is achieved, rv = rvs(Td). The wetbulb temperature is found by cooling the parcel by evaporating water into it until saturation is achieved. In this process the vapour mixing ratio changes according to drv/dT = —cp/L; for constant cp/L we can infer from the geometry in the figure below that (T — Tw) (cp/L) = rvs (Tw)rv, which is the psychrometric equation.
working down to Tw 2°C. For below freezing Tw we need to use an icebulb thermometer.
Modern radiosondes use a capacitor with a porous dielectric material inside. The capacitance changes according to the amount of vapour the dielectric absorbs. An electric circuit is then used to determine the capacitance and from this any humidity variable can be determined. Other instruments rely
1 1 w 
(°C)  
0.5 
1 
1.5 
2 
2.5 
3 
3.5 
4 
5 
6 
8 
10 
12 
14  
0 
91 
82 
74 
65 
57 
49 
40 
32 
16 
1  
2 
92 
84 
76 
68 
61 
53 
46 
38 
24 
10  
4 
93 
85 
78 
71 
64 
57 
50 
43 
30 
17 
RH (%)  
6 
93 
86 
80 
73 
67 
60 
54 
48 
36 
24 
1  
8 
94 
87 
81 
75 
69 
63 
57 
52 
40 
30 
9  
10 
94 
88 
83 
77 
71 
66 
60 
55 
45 
35 
15  
12 
94 
89 
84 
78 
73 
68 
63 
58 
48 
39 
21 
4  
15 
95 
90 
85 
80 
76 
71 
66 
62 
53 
44 
28 
13  
18 
95 
91 
86 
82 
77 
73 
69 
65 
57 
49 
34 
20 
7  
22 
96 
92 
87 
84 
80 
76 
72 
68 
61 
54 
41 
28 
17 
6 
26 
96 
92 
89 
85 
81 
78 
74 
71 
64 
58 
46 
35 
24 
14 
32 
97 
93 
90 
87 
83 
80 
77 
74 
68 
63 
52 
42 
33 
Figure 5.4 Psychrometric chart giving the water vapour mixing ratio r (in gkg1, solid lines) at 1000 hPa and relative humidity (in %, dashed lines) as a function of drybulb temperature and wetbulb depression, based on the psychrometric equation, Eq. 5.41. Figure 5.4 Psychrometric chart giving the water vapour mixing ratio r (in gkg1, solid lines) at 1000 hPa and relative humidity (in %, dashed lines) as a function of drybulb temperature and wetbulb depression, based on the psychrometric equation, Eq. 5.41. on the change of resistance of a material with humidity and use measured resistance to determine the humidity variables. 
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