Saturation Vapor Pressure \(e_s\)

Saturation vapor pressure \(e_s\) is calculated from a given temperature \(T\) (in \(K\)) by using the Clausius-Clapeyron relation. \[\begin{equation} e_s(T) = e_s(T_0)\times \exp \left(\frac{L}{R_w}\left(\frac{1}{T_0} - \frac{1}{T}\right)\right) \tag{1} \end{equation}\] where \(e_s(T_0) = 6.11 hPa\) is the saturation vapor pressure at a reference temperature \(T_0 = 273.15 K\), \(L = 2.5 \times 10^6 J/kg\) is the latent heat of evaporation for water, and \(R_w = \frac{1000R}{M_w} = 461.52 J/(kg K)\) is the specific gas constant for water vapor (where \(R = 8.3144621 J / (mol K)\) is the molar gas constant and \(M_w = 18.01528 g/mol\) is the molar mass of water vapor). More details refer to Shaman and Kohn (2009).

An alternative way to calculate saturation vapor pressure \(e_s\) is per the equation proposed by Murray (1967). \[\begin{equation} e_s = 6.1078\exp{\left[\frac{a(T - 273.16)}{T - b}\right]} \end{equation}\] where \(\begin{cases} a = 21.8745584 \\ b = 7.66 \end{cases}\) over ice; \(\begin{cases} a = 17.2693882 \\ b = 35.86 \end{cases}\) over water.

The resulting \(e_s\) is in hectopascal (\(hPa\)) or millibar (\(mb\)).

Vapor Pressure \(e\)

When given dew point \(T_d\) (in \(K\)), the actual vapor pressure \(e\) can be computed by plugging \(T_d\) in place of \(T\) into equation (1). The resulting \(e\) is in millibar (\(mb\)).

Relative Humidity \(\psi\)

Relative humidity \(\psi\) is defined as the ratio of the partial water vapor pressure \(e\) to the saturation vapor pressure \(e_s\) at a given temperature \(T\), which is usually expressed in \(\%\) as follows \[\begin{equation} \psi = \frac{e}{e_s}\times 100 \tag{2} \end{equation}\]

Therefore, when given the saturation vapor pressure \(e_s\) and relative humidity \(\psi\), the partial water vapor pressure \(e\) can also be easily calculated per equation (2). \[ e = \psi e_s \] The resulting \(e\) is in \(Pa\).

Absolute Humidity \(\rho_w\)

Absolute humidity \(\rho_w\) is the total amount of water vapor \(m_w\) present in a given volume of air \(V\). The definition of absolute humidity can be described as follows \[ \rho_w = \frac{m_w}{V} \]

Water vapor can be regarded as ideal gas in the normal atmospheric temperature and atmospheric pressure. Its equation of state is \[\begin{equation} e = \rho_w R_w T \tag{3} \end{equation}\]

Absolute humidity \(\rho_w\) is derived by solving equation (3). \[ \rho_w = \frac{e}{R_w T} \] The resulting \(\rho_w\) is in \(kg/m^3\).

Mixing Ratio \(\omega\)

Mixing ratio \(\omega\) is the ratio of water vapor mass \(m_w\) to dry air mass \(m_d\), expressed in equation as follows \[ \omega = \frac{m_w}{m_d} \]

The resulting \(\omega\) is in \(kg/kg\).

Specific Humidity \(q\)

Specific humidity \(q\) is the ratio of water vapor mass \(m_w\) to the total (i.e., including dry) air mass \(m\) (namely, \(m = m_w + m_d\)). The definition is described as \[ q = \frac{m_w}{m} = \frac{m_w}{m_w + m_d} = \frac{\omega}{\omega + 1} \]

Specific humidity can also be expressed in following way. \[ \begin{equation} q = \frac{\frac{M_w}{M_d}e}{p - (1 - \frac{M_w}{M_d})e} \tag{4} \end{equation} \] where \(M_d = 28.9634 g/mol\) is the molar mass of dry air; \(p\) represents atmospheric pressure and the standard atmospheric pressure is equal to \(101,325 Pa\). The details of formula derivation refer to Wikipedia.

Substitute \(\frac{M_w}{M_d} \approx 0.622\) into equation (4) and simplify the formula. \[ q \approx \frac{0.622e}{p - 0.378e} \tag{5} \] The resulting \(q\) is in \(kg/kg\).

Hence, by solving equation (5) we can obtain the equation for calculating the partial water vapor pressure \(e\) given the specific humidity \(q\) and atmospheric pressure \(p\).

\[ e \approx \frac{qp}{0.622 + 0.378q} \tag{6} \] Substituting equations (1) and (6) into equation (2), we can get the equation for converting specific humidity \(q\) into relative humidity \(\psi\) at a given temperature \(T\) and under atmospheric pressure \(p\).

References

Murray, F. W. 1967. “On the Computation of Saturation Vapor Pressure.” J. Appl. Meteor. 6 (1). American Meteorological Society: 203–4. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2.

Shaman, J., and M. Kohn. 2009. “Absolute Humidity Modulates Influenza Survival, Transmission, and Seasonality.” PNAS 106 (9). Natl Acad Sciences: 3243–8. https://doi.org/10.1073/pnas.0806852106.