---
title: "Humidity Measures"
author: "Jun Cai <caijun@fudan.edu.cn>"
date: "`r Sys.Date()`"
output: bookdown::html_document2
bibliography: bibliography.bib
link-citations: yes
vignette: >
  %\VignetteIndexEntry{Humidity Measures}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

## Saturation Vapor Pressure $e_s$ {-}

[Saturation vapor pressure](https://glossary.ametsoc.org/wiki/saturation-vapor-pressure/) $e_s$ is calculated from a given temperature $T$ (in $K$) by using the [Clausius-Clapeyron equation](https://glossary.ametsoc.org/wiki/clausius-clapeyron-equation/).
\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)
  (\#eq: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](https://en.wikipedia.org/wiki/Latent_heat) of evaporation for water, and $R_w = \frac{1000R}{M_w} = 461.52 J/(kg K)$ is the [specific gas constant](https://en.wikipedia.org/wiki/Gas_constant#Specific_gas_constant) for water vapor (where $R = 8.3144621 J / (mol K)$ is the molar [gas constant](https://en.wikipedia.org/wiki/Gas_constant) and $M_w = 18.01528 g/mol$ is the [molar mass](https://en.wikipedia.org/wiki/Molar_mass) of water vapor). More details refer to @Shaman-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](https://en.wikipedia.org/wiki/Dew_point) $T_d$ (in $K$), the actual vapor pressure $e$ can be computed by plugging $T_d$ in place of $T$ into equation \@ref(eq:1). The resulting $e$ is in millibar ($mb$).

## Relative Humidity $\psi$ {-}

[Relative humidity](https://en.wikipedia.org/wiki/Humidity#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
  (\#eq: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 \@ref(eq:2).
$$
  e = \psi e_s
$$
The resulting $e$ is in $Pa$.

## Absolute Humidity $\rho_w$ {-}

[Absolute humidity](https://en.wikipedia.org/wiki/Humidity#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
  (\#eq:3)
\end{equation}

Absolute humidity $\rho_w$ is derived by solving equation \@ref(eq:3).
$$
  \rho_w = \frac{e}{R_w T}
$$
The resulting $\rho_w$ is in $kg/m^3$.

## Mixing Ratio $\omega$ {-}

[Mixing ratio](https://www.conservationphysics.org/teabag/ah_mr.php) $\omega$ 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](https://en.wikipedia.org/wiki/Humidity#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.
$$
\begin{equation}
  q = \frac{\frac{M_w}{M_d}e}{p - (1 - \frac{M_w}{M_d})e}
  (\#eq:4)
\end{equation}
$$
where $M_d = 28.9634 g/mol$ is the molar mass of dry air; $p$ represents [atmospheric pressure](https://en.wikipedia.org/wiki/Atmospheric_pressure) and the standard atmospheric pressure is equal to $101,325 Pa$. The details of formula derivation refer to [Wikipedia](https://zh.wikipedia.org/wiki/%E6%B9%BF%E5%BA%A6#.E6.AF.94.E6.BA.BC).

Substitute $\frac{M_w}{M_d} \approx 0.622$ into equation \@ref(eq:4) and simplify the formula.
$$
  q \approx \frac{0.622e}{p - 0.378e}
  (\#eq:5)
$$
The resulting $q$ is in $kg/kg$.

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

$$
  e \approx \frac{qp}{0.622 + 0.378q}
  (\#eq:6)
$$
Substituting equations \@ref(eq:1) and \@ref(eq:6) into equation \@ref(eq: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
