Snow melt is controlled by the air and snow pack temperature, the melting rate, and the areal coverage of snow. Snow melt is included with rainfall in the calculations of runoff and percolation. When SWAT+ calculates erosion, the rainfall energy of the snow melt fraction of the water is set to zero. The water released from snow melt is assumed to be evenly distributed over the 24 hours of the day.

Precipitation is the mechanism by which water enters the land phase of the hydrologic cycle. Because precipitation controls the water balance, it is critical that the amount and distribution of precipitation in space and time is accurately simulated by the model.

SWAT+ classifies precipitation as rain or freezing rain/snow by the mean daily air temperature. The boundary temperature, $T_{s-r}$, used to categorize precipitation as rain or snow is defined by the user. If the mean daily air temperature is less than the boundary temperature, then the precipitation within the HRU is classified as snow and the water equivalent of the snow precipitation is added to the snow pack.

Snowfall is stored at the ground surface in the form of a snow pack. The amount of water stored in the snow pack is reported as a snow water equivalent. The snow pack will increase with additional snowfall or decrease with snow melt or sublimation. The mass balance for the snow pack is:

$SNO=SNO+R_{day}-E_{sub}-SNO_{mlt}$ 1:2.4.1

where $SNO$ is the water content of the snow pack on a given day ($mm\space H_2O$), $R_{day}$ is the amount of precipitation on a given day (added only if ) ($mm\space H_2O$), $E_{sub}$ is the amount of sublimation on a given day ($mm\space H_2O$), and $SNO_{mlt}$ is the amount of snow melt on a given day (). The amount of snow is expressed as depth over the total HRU area.

Due to variables such as drifting, shading and topography, the snow pack in a subbasin will rarely be uniformly distributed over the total area. This results in a fraction of the subbasin area that is bare of snow. This fraction must be quantified to accurately compute snow melt in the subbasin.

The factors that contribute to variable snow coverage are usually similar from year to year, making it possible to correlate the areal coverage of snow with the amount of snow present in the subbasin at a given time. This correlation is expressed as an areal depletion curve, which is used to describe the seasonal growth and recession of the snow pack as a function of the amount of snow present in the subbasin (Anderson, 1976). The areal depletion curve requires a threshold depth of snow, , to be defined above which there will always be 100% cover. The threshold depth will depend on factors such as vegetation distribution, wind loading of snow, wind scouring of snow, interception and aspect, and will be unique to the watershed of interest. The areal depletion curve is based on a natural logarithm. The areal depletion curve equation is:

1:2.4.2

where is the fraction of the HRU area covered by snow, is the water content of the snow pack on a given day (), is the threshold depth of snow at 100% coverage (), and are coefficients that define the shape of the curve. The values used for and are determined by solving equation 1:2.4.2 using two known points: 95% coverage at 95% ; and 50% coverage at a user specified fraction of . Example areal depletion curves for various fractions of at 50% coverage are shown in the following figures.

It is important to remember that once the volume of water held in the snow pack exceeds the depth of snow over the HRU is assumed to be uniform, i.e. = 1.0. The areal depletion curve affects snow melt only when the snow pack water content is between 0.0 and . Consequently if is set to a very small value, the impact of the areal depletion curve on snow melt will be minimal. As the value for increases, the influence of the areal depletion curve will assume more importance in snow melt processes.

Table 1:2-3: SWAT+ input variables used in snow cover calculations.

The maximum half-hour rainfall is required by SWAT+ to calculate the peak runoff rate. The maximum half-hour rainfall is reported as a fraction of the total daily rainfall, 0.5. If sub-daily precipitation data is used in the model, SWAT+ will calculate the maximum half-hour rainfall fraction directly from the precipitation data. If daily precipitation data is used, SWAT+ generates a value for 0.5 using the equations summarized in Chapter 1:3.

The precipitation reaching the earth's surface on a given day, $R_{day}$, may be read from an input file or generated by the model. Users are strongly recommended to incorporate measured precipitation into their simulations any time the data is available. The ability of SWAT+ to reproduce observed stream hydrographs is greatly improved by the use of measured precipitation data.

Unfortunately, even with the use of measured precipitation the model user can expect some error due to inaccuracy in precipitation data. Measurement of precipitation at individual gages is subject to error from a number of causes and additional error is introduced when regional precipitation is estimated from point values. Typically, total or average areal precipitation estimates for periods of a year or longer have relative uncertainties of 10% (Winter, 1981).

Point measurements of precipitation generally capture only a fraction of the true precipitation. The inability of a gage to capture a true reading is primarily caused by wind eddies created by the gage. These wind eddies reduce the catch of the smaller raindrops and snowflakes. Larson and Peck (1974) found that deficiencies of 10% for rain and 30% for snow are common for gages projecting above the ground surface that are not designed to shield wind effects. Even when the gage is designed to shield for wind effects, this source of error will not be eliminated. For an in-depth discussion of this and other sources of error as well as methods for dealing with the error, please refer to Dingman (1994).

The variable pcp in the master weather file () identifies the method used to obtain precipitation data. To read in precipitation data, the variable is set to the names of the precipitation data file(s). To generate precipitation values, pcp is set to "sim". The equations used to generate precipitation data in SWAT+ are reviewed in Chapter 1:3. SWAT+ input variables that pertain to precipitation are summarized in Table 1:2-1.

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

See description of . file in the User’s Manual for input and format requirements if measured daily precipitation data is being used.

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):

1:2.3.2

where is the saturation vapor pressure on a given day () and is the mean daily air temperature (). When relative humidity is known, the actual vapor pressure can be calculated by rearranging equation 1:2.3.1:

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:

1:2.3.4

where is the slope of the saturation vapor pressure curve (), is the saturation vapor pressure on a given day () and is the mean daily air temperature ().

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:

1:2.3.5

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

The latent heat of vaporization, , 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):

1:2.3.6

where is the latent heat of vaporization () and is the mean daily air temperature ().

Evaporation involves the exchange of both latent heat and sensible heat between the evaporating body and the air. The psychrometric constant, , 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:

1:2.3.7

where is the psychrometric constant (), is the specific heat of moist air at constant pressure (1.013 x 10 ), P is the atmospheric pressure (), and is the latent heat of vaporization ().

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:

1:2.3.8

where is the atmospheric pressure () and is the elevation ().

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 () 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.

See description of file in the User’s Manual for input and format requirements if measured relative humidity data is being used.

The snow pack temperature is a function of the mean daily temperature during the preceding days and varies as a dampened function of air temperature (Anderson, 1976). The influence of the previous day’s snow pack temperature on the current day’s snow pack temperature is controlled by a lagging factor,$\ell_{sno}$ . The lagging factor inherently accounts for snow pack density, snow pack depth, exposure and other factors affecting snow pack temperature. The equation used to calculate the snow pack temperature is:

$T_{snow(d_n)}=T_{snow(d_n-1)}*(1-\ell_{sno})+\overline T_{av}*\ell_{sno}$ 1:2.5.1

where $T_{snow(d_n)}$ is the snow pack temperature on a given day ($\degree C$), $T_{snow(d_n-1)}$ is the snow pack temperature on the previous day ($\degree C$), $\ell_{sno}$ is the snow temperature lag factor, and $\overline T_{av}$ is the mean air temperature on the current day ($\degree C$). As $\ell_{sno}$ approaches 1.0, the mean air temperature on the current day exerts an increasingly greater influence on the snow pack temperature and the snow pack temperature from the previous day exerts less and less influence.

The snow pack will not melt until the snow pack temperature exceeds a threshold value, . This threshold value is specified by the user.

The snow melt in SWAT+ is calculated as a linear function of the difference between the average snow pack-maximum air temperature and the base or threshold temperature for snow melt:

$SNO_{mlt}=b_{mlt}*sno_{cov}*[\frac{T_{snow}+T_{mx}}{2}-T_{mlt}]$ 1:2.5.2

where $SNO_{mlt}$ is the amount of snow melt on a given day (mm H$_2$O), $b_{mlt}$ is the melt factor for the day ($mm\space H_2O/day \degree C$), $sno_{cov}$ is the fraction of the HRU area covered by snow, $T_{snow}$ is the snow pack temperature on a given day ($\degree C$), $T_{mx}$ is the maximum air temperature on a given day ($\degree C$), and $T_{mlt}$ is the base temperature above which snow melt is allowed ($\degree C$).

The melt factor is allowed a seasonal variation with maximum and minimum values occurring on summer and winter solstices:

1:2.5.3

where is the melt factor for the day (), is the melt factor for June 21 (), is the melt factor for December 21 (), and is the day number of the year.

In rural areas, the melt factor will vary from 1.4 to 6.9 (Huber and Dickinson, 1988). In urban areas, values will fall in the higher end of the range due to compression of the snow pack by vehicles, pedestrians, etc. Urban snow melt studies in Sweden (Bengston, 1981; Westerstrom, 1981) reported melt factors ranging from 3.0 to 8.0 . Studies of snow melt on asphalt (Westerstrom, 1984) gave melt factors of 1.7 to 6.5 .

Table 1:2-4: SWAT+ input variables used in snow melt calculations.