1:2.3 Water Vapor

Relative humidity is required by SWAT+ if the Penman-Monteith or Priestley-Taylor equation is used to estimate potential evapotranspiration. It is also used to calculate the vapor pressure deficit on plant growth. The Penman-Monteith equation includes terms that quantify the effect of the amount of water vapor in the air near the evaporative surface on evaporation. Both Penman-Monteith and Priestley-Taylor require the actual vapor pressure, which is calculated from the relative humidity.

Relative humidity is the ratio of an air volume’s actual vapor pressure to its saturation vapor pressure:

$R_h=\frac{e}{e^o}$ 1:2.3.1

where $R_h$ is the relative humidity on a given day, $e^o$ is the actual vapor pressure on a given day ( $kPa$ ), and is the saturation vapor pressure on a given day ( $kPa$ ).

The saturation vapor pressure is the maximum vapor pressure that is thermodynamically stable and is a function of the air temperature. SWAT+ calculates saturation vapor pressure using an equation presented by Tetens (1930) and Murray (1967):

$e^o=exp[\frac{16.78*\overline T_{av}-116.9}{\overline T_{av}+237.3}]$ 1:2.3.2

where $e^o$ is the saturation vapor pressure on a given day ( $kPa$ ) and $\overline T_{av}$ is the mean daily air temperature ( $\degree C$ ). When relative humidity is known, the actual vapor pressure can be calculated by rearranging equation 1:2.3.1:

$e=R_h*e^o$ 1:2.3.3

The saturation vapor pressure curve is obtained by plotting equation 1:2.3.2. The slope of the saturation vapor pressure curve can be calculated by differentiating equation 1:2.3.2:

$\Delta=\frac{4098*e^o}{(\overline T_{av}+237.3)^2}$ 1:2.3.4

where is the slope of the saturation vapor pressure curve ( $kPa\degree C^{-1}$ $^{-1}$ ), $e^o$ is the saturation vapor pressure on a given day ( $kPa$ ) and $\overline T_{av}$ is the mean daily air temperature ( $\degree C$ ).

The rate of evaporation is proportional to the difference between the vapor pressure of the surface layer and the vapor pressure of the overlying air. This difference is termed the vapor pressure deficit:

$vpd=e^o-e$ 1:2.3.5

where $vpd$ is the vapor pressure deficit ( $kPa$ ), $e^o$ is the saturation vapor pressure on a given day ( $kPa$ ), and $e$ is the actual vapor pressure on a given day ( $kPa$ ). The greater the value of $vpd$ the higher the rate of evaporation.

The latent heat of vaporization, $\lambda$ , is the quantity of heat energy that must be absorbed to break the hydrogen bonds between water molecules in the liquid state to convert them to gas. The latent heat of vaporization is a function of temperature and can be calculated with the equation (Harrison, 1963):

$\lambda=2.501-2.361*10^{-3}*\overline T_{av}$ 1:2.3.6

where is the latent heat of vaporization ( $MJ\space kg^{-1}$ ) and $\overline T_{av}$ is the mean daily air temperature ( $\degree C$ ).

Evaporation involves the exchange of both latent heat and sensible heat between the evaporating body and the air. The psychrometric constant, $\gamma$ , represents a balance between the sensible heat gained from air flowing past a wet bulb thermometer and the sensible heat converted to latent heat (Brunt, 1952) and is calculated:

$\gamma=\frac{c_p*P}{0.622*\lambda}$ 1:2.3.7

where is the psychrometric constant ( $kPa\degree C^{-1}$ $^{-1}$ ), $c_p$ is the specific heat of moist air at constant pressure (1.013 x 10 $^{-3}$ $MJ\space kg^{-1}\degree C^{-1}$ $^{-1}$ ), P is the atmospheric pressure ( $kPa$ ), and is the latent heat of vaporization ( $MJ\space kg^{-1}$ ).

Calculation of the psychrometric constant requires a value for atmospheric pressure. SWAT+ estimates atmospheric pressure using an equation developed by Doorenbos and Pruitt (1977) from mean barometric pressure data at a number of East African sites:

$P=101.3-0.01152*EL+0.544*10^{-6}*EL^2$ 1:2.3.8

where $P$ is the atmospheric pressure ( $kPa$ ) and $EL$ is the elevation ( $m$ ).

The daily relative humidity data required by SWAT+ may be read from an input file or generated by the model. The variable hmd in the master weather file (weather-sta.cli) file identifies the method used to obtain relative humidity data. To read in daily relative humidity data, the variable is set to the name of the relative humidity data file(s). To generate daily relative humidity values, hmd is set to "sim". The equations used to generate relative humidity data in SWAT+ are reviewed in Chapter 1:3.

Table 1:2-2: SWAT+ input variables used in relative humidity calculations.

Definition

Source Name

Input Name

Input File

$R_h$ : daily average relative humidity