Relative humidity is required by SWAT+ when the Penman-Monteith equation is used to calculate potential evapotranspiration. It is also used to calculate the vapor pressure deficit on plant growth. Daily average relative humidity values are calculated from a triangular distribution using average monthly relative humidity. This method was developed by J.R. Williams for the EPIC model (Sharpley and Williams, 1990).

The triangular distribution used to generate daily relative humidity values requires four inputs: mean monthly relative humidity, maximum relative humidity value allowed in month, minimum relative humidity value allowed in month, and a random number between 0.0 and 1.0.

The maximum relative humidity value, or upper limit of the triangular distribution, is calculated from the mean monthly relative humidity with the equation:

$R_{hUmon}=R_{hmon}+(1-R_{hmon})*exp(R_{hmon}-1)$ 1:3.5.4

where $R_{hUmon}$ is the largest relative humidity value that can be generated on a given day in the month, and $R_{hmon}$ is the average relative humidity for the month.

The minimum relative humidity value, or lower limit of the triangular distribution, is calculated from the mean monthly relative humidity with the equation:

$R_{hLmon}=R_{hmon}*(1-exp(-R_{hmon}))$ 1:3.5.5

where $R_{hLmon}$ is the smallest relative humidity value that can be generated on a given day in the month, and $R_{hmon}$ is the average relative humidity for the month.

The triangular distribution uses one of two sets of equations to generate a relative humidity value for the day. If $rnd_1 \le (\frac{R_{hmon}-R_{hLmon}}{R_{hUmon}-R_{hLmon}})$ then

$R_h=R_{hmon}*\frac{R_{hLmon}+[rnd_1*(R_{hUmon}-R_{hLmon})*(R_{hmon}-R_{hLmon})]^{0.5}}{R_{hmon,mean}}$ 1:3.5.6

If $rnd_1>(\frac{R_{hmon}-R_{hLmon}}{R_{hUmon}-R_{hLmon}})$ then

$R_h=R_{hmon}*\frac{R_{hUmon}-(R_{hUmon}-R_{hmon})*[\frac{R_{hUmon}(1-rnd_1)-R_{hLmon}(1-rnd_1)}{R_{hUmon}-R_{hmon}}]^{0.5}}{R_{hmon,mean}}$ 1:3.5.7

where $R_h$ is the average relative humidity calculated for the day, $rnd_1$ is a random number generated by the model each day, $R_{hmon}$ is the average relative humidity for the month, $R_{hLmon}$ is the smallest relative humidity value that can be generated on a given day in the month, $R_{hUmon}$ is the largest relative humidity value that can be generated on a given day in the month, and $R_{hmon,mean}$ is the mean of $R_{hLmon},R_{hmon},$ and $R_{hUmon}$.

Relative humidity is defined as the ratio of the actual vapor pressure to the saturation vapor pressure at a given temperature:

$R_{hmon}=\frac{e_{mon}}{e^o_{mon}}$ 1:3.5.1

where $R_{hmon}$ is the average relative humidity for the month, $e_{mon}$ is the actual vapor pressure at the mean monthly temperature (kPa), and $e^o_{mon}$ is the saturation vapor pressure at the mean monthly temperature (kPa). The saturation vapor pressure, $e^o_{mon}$ , is related to the mean monthly air temperature with the equation:

$e^o_{mon}=exp[\frac{16.78*\mu tmp_{mon}-116.9}{\mu tmp_{mon}+237.3}]$ 1:3.5.2

where $e^o_{mon}$ is the saturation vapor pressure at the mean monthly temperature (kPa), and $\mu tmp_{mon}$ is the mean air temperature for the month (°C). The mean air temperature for the month is calculated by averaging the mean maximum monthly temperature, $\mu mx_{mon}$, and the mean minimum monthly temperature, $\mu mn_{mon}$.

The dew point temperature is the temperature at which the actual vapor pressure present in the atmosphere is equal to the saturation vapor pressure. Therefore, by substituting the dew point temperature in place of the average monthly temperature in equation 1:3.5.2, the actual vapor pressure may be calculated:

$e_{mon}=exp[\frac{16.78*\mu dew_{mon}-116.9}{\mu dew_{mon}+273.3}]$ 1:3.5.3

where $e_{mon}$ is the actual vapor pressure at the mean month temperature (kPa), and $\mu dew_{mon}$ is the average dew point temperature for the month (°C).

To incorporate the effect of clear and overcast weather on generated values of relative humidity, monthly average relative humidity values can be adjusted for wet or dry conditions.

The continuity equation relates average relative humidity adjusted for wet or dry conditions to the average relative humidity for the month:

$R_{hmon}*days_{tot}=R_{hWmon}*days_{wet}+R_{hDmon}*days_{dry}$ 1:3.5.8

where $R_{hmon}$ is the average relative humidity for the month, $days_{tot}$ are the total number of days in the month, $R_{hWmon}$ is the average relative humidity for the month on wet days, $days_{wet}$ are the number of wet days in the month, $R_{hDmon}$ is the average relative humidity of the month on dry days, and $days_{dry}$ are the number of dry days in the month.

The wet day average relative humidity is assumed to be greater than the dry day average relative humidity by some fraction:

$R_{hWmon}=R_{hDmon}+b_H*(1-R_{hDmon})$ 1:3.5.9

where $R_{hWmon}$ is the average relative humidity of the month on wet days, $R_{hDmon}$ is the average relative humidity of the month on dry days, and $b_H$ is a scaling factor that controls the degree of deviation in relative humidity caused by the presence or absence of precipitation. The scaling factor, $b_H$, is set to 0.9 in SWAT+.

To calculate the dry day relative humidity, equations 1:3.5.8 and 1:3.5.9 are combined and solved for $R_{hDmon}$:

$R_{hDmon}=(R_{hmon}-b_H*\frac{days_{wet}}{days_{tot}})*(1.0-b_H*\frac{days_{wet}}{days_{tot}})^{-1}$ 1:3.5.10

To reflect the impact of wet or dry conditions, SWAT+ will replace $R_{hmon}$ with $R_{hWmon}$ on wet days or $R_{hDmon}$ on dry days in equations 1:3.5.4 through 1:3.5.7.

Table 1:3-5: SWAT+ input variables that pertain to generation of relative humidity.