4.3: Humidity Variables

Vapor pressure is one way of defining humidity, but there are many others. Here is a non-comprehensive list of humidity variables and their typical units.

\(e\) = vapor pressure (kPa)
\(r\) = mixing ratio (g·kg^–1)
\(q\) = specific humidity (g·kg^–1)
\(\rho_V\) = absolute humidity (g·m^-3)
\(RH\) = relative humidity (%)
\(z_{LCL}\) = lifting condensation level (km)
\(T_d\) = dew point (temperature) (°C)
\(T_w\) = wet-bulb temperature (°C)

Vapor Pressure

We’ve already discussed saturation vapor pressure, \(e_s\), but you can also compute vapor pressure, \(e\). However, because \(T_d\) is often unknown, the easiest way is usually through relative humidity.

\[e=e_0 \cdot \exp \left[\frac{L_v}{R_v} \cdot\left(\frac{1}{T_0}-\frac{1}{T_d}\right)\right] \nonumber \]

Again, \(e_0\) is 0.6113 kPa, \(L_v\) is 2.5×10⁶ J·kg^–1, \(\Re_v\) is 461 J·K^–1·kg^–1, \(T_0\) is 273.15 K, and \(T_d\) is dew point temperature, which will be defined later.

Mixing Ratio

Mixing ratio, \(r\), is the ratio of the mass of water vapor to the mass of dry air. It is typically expressed as grams of water vapor per kilogram of air (g·kg^–1).

\[r=\frac{m_{\text {water vapor }}}{m_{\text {dry air }}} \nonumber \]

\[r=\frac{\epsilon \cdot e}{P-e} \nonumber \]

Pressure (\(P\)) should be in the same units as vapor pressure (\(e\)). The constant \(\varepsilon\) is 0.622, is the ratio between the gas constant for dry air and the gas constant for water vapor.

\[\epsilon=\frac{R_d}{R_v}=\frac{287 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}}{461 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}}=0.622 \nonumber \]

Saturation mixing ratio, \(r_s\), is computed the same way as the mixing ratio but with saturation vapor pressure, \(e_s\), instead of \(e\).

\[r_s=\frac{\epsilon \cdot e_s}{P-e_s} \nonumber \]

When calculating mixing ratio, the pressure units on the top of the fraction will cancel with the pressure units on the bottom of the fraction. While it appears unit-less, its technically not based on its definition of mass of water vapor as compared to mass of dry air. See the Pro Tip below for more information.

Pro Tip: Many units of moisture are given in g·kg^-1 or kg·kg^-1 so technically the units could cancel and it could be unitless! Don’t let this fool you. It is important to remember that the mass in the numerator and denominator are different. In the case of mixing ratio, the value is given in mass of water vapor proportional to the mass of dry air.

Specific Humidity

Specific humidity, \(q\), is the ratio of the mass of water vapor to the total mass of air (dry air and water vapor combined). It is expressed as grams of water vapor per kilogram of air (g·kg^–1).

\[q=\frac{m_{\text {water vapor }}}{m_{\text {total air }}} \nonumber \]

\[q=\frac{\epsilon \cdot e}{P-e \cdot(1-\epsilon)} \approx \frac{\epsilon \cdot e}{P} \nonumber \]

Again, saturation specific humidity, \(q_s\),, is computed with \(e_s\), instead of \(e\).

\[q_s=\frac{\epsilon \cdot e_s}{P-e_s \cdot(1-\epsilon)} \approx \frac{\epsilon \cdot e_s}{P} \nonumber \]

Absolute Humidity

Absolute humidity, \(\rho_v\), is the ratio of the mass of water vapor to the volume of air. It is expressed as grams of water vapor in a cubic meter of air (g·m^-3). It is effectively water vapor density.

\[\rho_v=\frac{m_{\text {water vapor }}}{\text { Volume }} \nonumber \]

\[\rho_v=\frac{e}{R_v \times T} \nonumber \]

Again, saturation absolute humidity, \(\rho_{vs}\), uses \(e_s\), instead of \(e\).

\[\rho_{v s}=\frac{e_s}{R_v \times T} \nonumber \]

Relative Humidity

Relative humidity, \(RH\), is the ratio of the amount of water vapor present in the air to the maximum amount of water vapor needed for saturation at a certain pressure and temperature. It is typically multiplied by 100 and expressed as a percent. Relative humidity shows how close the air is to being saturated, not how much water vapor the air contains. For this reason, \(RH\) is not a good indicator of the quantitative amount of water vapor in the air. It is only a relative measure that is highly dependent on the air temperature. Relative humidity greater than 100% is called supersaturation.

\[R H=\frac{e}{e_s}=\frac{R H \%}{100 \%} \nonumber \]

or

\[R H=\frac{q}{q_s}=\frac{\rho_v}{\rho_{v s}} \nonumber \]

Imagine two parcels of air with the same volume, pressure, and relative humidity. Parcel 1 has an air temperature of 20 °C while Parcel 2 has an air temperature of 30 °C. Which parcel contains more water vapor?

Dew Point Temperature

The dew point temperature, \(T_d\), is the temperature to which the air must be cooled to reach saturation, without changing the moisture or air pressure. It measures the actual moisture content of a parcel of air. Saturation occurs when the dew point temperature equals the air temperature.

\[T_d=\left[\frac{1}{T_0}-\frac{R_v}{L_v} \cdot \ln \left(\frac{e}{e_0}\right)\right]^{-1} \nonumber \]

\[T_d=\left[\frac{1}{T_0}-\frac{R_v}{L_v} \cdot \ln \left(\frac{r \cdot P}{e_0 \cdot(r+\epsilon)}\right)\right]^{-1} \nonumber \]

When the dew point temperature is lower than the freezing point of water, it is also called the frost point.

Wet-Bulb Temperature

Wet-bulb temperature, \(T_w\), is the lowest temperature that can be achieved if water evaporates within the air. When the relative humidity is 100%, the wet-bulb temperature is equal to the air temperature because there is no evaporation.

The wet-bulb temperature is difficult to calculate but easy to measure. To measure the wet-bulb temperature, all you need is a thermometer with a wet cloth wrapped around the bulb. Typically this thermometer is attached to an apparatus called a sling psychrometer to make it easy to spin around in the air to create lots of airflow over the wet cloth on the thermometer. The evaporation from the wet cloth cools the temperature measured, hence the wet-bulb temperature is always lower than the air temperature (or dry-bulb temperature) when relative humidity is less than 100%.

You can also estimate the wet bulb temperature using lines on a graph. Normand’s Rule is used to calculate the wet-bulb temperature from the air temperature and the dew point temperature. The wet-bulb temperature is always between the dew point and the dry-bulb temperature (\(T_d \leq T_w \leq T\)). This can be implemented on thermodynamic diagrams, such as the Skew-\(T \log P,\), which is discussed in more detail in the next chapter.

Take note of this description for later. To find the wet-bulb temperature on a Skew-\(T \log P,\) diagram, follow the dry adiabatic lapse rate line upward from the air temperature. Next, use the dew point temperature and follow an isohume (line of constant relative humidity) upward. The point where these two lines meet is called the lifting condensation level (LCL). From the meeting point, follow the moist (saturated) adiabatic lapse rate back down to obtain the wet-bulb temperature value. This is probably confusing at this point because we have not discussed the LCL or the moist adiabatic lapse rate, but don’t worry, we’ll repeat this logic again in the next chapter to make sure this is clear.