A number of basic concepts related to the earth's orbit around the sun are required by the model to make solar radiation calculations. This section summarizes these concepts. Iqbal (1983) provides a detailed discussion of these and other topics related to solar radiation for users who require more information.

Once water is introduced to the system as precipitation, the available energy, specifically solar radiation, exerts a major control on the movement of water in the land phase of the hydrologic cycle. Processes that are greatly affected by temperature and solar radiation include snow fall, snow melt and evaporation. Since evaporation is the primary water removal mechanism in the watershed, the energy inputs become very important in reproducing or simulating an accurate water balance.

Temperature influences a number of physical, chemical and biological processes. Plant production is strongly temperature dependent, as are organic matter decomposition and mineralization. Daily air temperature may be input to the model or generated from average monthly values. Soil and water temperatures are derived from air temperature.

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.

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.

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.

SWAT+ requires daily values of precipitation, maximum and minimum temperature, solar radiation, relative humidity and wind speed. The user may choose to read these inputs from a file or generate the values using monthly average data summarized over a number of years.

SWAT+ includes the WXGEN weather generator model (Sharpley and Williams, 1990) to generate climatic data or to fill in gaps in measured records. This weather generator was developed for the contiguous U.S. If the user prefers a different weather generator, daily input values for the different weather parameters may be generated with an alternative model and formatted for input to SWAT+.

The occurrence of rain on a given day has a major impact on relative humidity, temperature and solar radiation for the day. The weather generator first independently generates precipitation for the day. Once the total amount of rainfall for the day is generated, the distribution of rainfall within the day is computed if the Green & Ampt method is used for infiltration. Maximum temperature, minimum temperature, solar radiation and relative humidity are then generated based on the presence or absence of rain for the day. Finally, wind speed is generated independently.

Maximum half-hour rainfall is required by SWAT+ to calculate the peak flow rate for runoff. When daily precipitation data are used by the model, the maximum half-hour rainfall may be calculated from a triangular distribution using monthly maximum half-hour rainfall data or the user may choose to use the monthly maximum half-hour rainfall for all days in the month. The maximum half-hour rainfall is calculated only on days where surface runoff has been generated.

The procedure used to generate daily values for maximum temperature, minimum temperature and solar radiation (Richardson, 1981; Richardson and Wright, 1984) is based on the weakly stationary generating process presented by Matalas (1967).

The climatic inputs to the model are reviewed first because it is these inputs that provide the moisture and energy that drive all other processes simulated in the watershed. The climatic processes modeled in SWAT+ consist of precipitation, air temperature, soil temperature, and solar radiation. Depending on the method used to calculate potential evapotranspiration, wind speed and relative humidity may also be modeled.

The extraterrestrial radiation falling on a horizontal surface during one hour is given by the equation:

$I_0=I_{SC}E_0(\sin\delta\sin\phi+\cos\delta\cos\phi\cos\omega*t)$ 1:1.2.8

where $I_0$ is the extraterrestrial radiation for 1 hour centered around the hour angle $t$.

An accurate calculation of the radiation for each hour of the day requires a knowledge of the difference between standard time and solar time for the location. SWAT+ simplifies the hourly solar radiation calculation by assuming that solar noon occurs at 12:00pm local standard time.

When the values of $I_0$ calculated for every hour between sunrise and sunset are summed, they will equal the value of $H_0$. Because of the relationship between $I_0$ and $H_0$, it is possible to calculate the hourly radiation values by multiplying $H_0$ by the fraction of radiation that falls within the different hours of the day. The benefit of this alternative method is that assumptions used to estimate the difference between maximum and actual solar radiation reaching the earth’s surface can be automatically incorporated in calculations of hourly solar radiation at the earth’s surface.

SWAT+ calculates hourly solar radiation at the earth’s surface with the equation:

$I_{hr}=I_{frac}*H_{day}$ 1:1.2.9

where$I_{hr}$ is the solar radiation reaching the earth’s surface during a specific hour of the day ($MJ m^{-2}hr^{-1}$), $I_{frac}$ is the fraction of total daily radiation falling during that hour, and $H_{day}$ is the total solar radiation reaching the earth’s surface on that day.

The fraction of total daily radiation falling during an hour is calculated

$I_{frac}=$\frac{(\sin\delta\sin\phi+\cos\delta\cos\phi\cos\omega*t_i)}{$\displaystyle{\sum_{t=SR}^{SS}}(\sin\delta\sin\phi+\cos\delta\cos\phi\ cos\omega*t)}

1:1.2.10

where $t_i$ is the solar time at the midpoint of hour i.

The radiant energy from the sun is practically the only source of energy that impacts climatic processes on earth. The solar constant, ISC, is the rate of total solar energy at all wavelengths incident on a unit area exposed normally to rays of the sun at a distance of 1 AU from the sun. Quantifying this value has been the object of numerous studies through the years. The value officially adopted by the Commission for Instruments and Methods of Observation in October 1981 is

$I_{SC} = 1367 W m^{-2} = 4.921 MJm^{-2} h^{-1}$

On any given day, the extraterrestrial irradiance (rate of energy) on a surface normal to the rays of the sun, $I_{0n}$, is:

$I_{0n} = I_{SC}E_0$ 1:1.2.1

where $E_0$ is the eccentricity correction factor of the earth's orbit, and $I_{0n}$ has the same units as the solar constant, $I_{SC}$. To calculate the irradiance on a horizontal surface, $I_{SC}$,

To calculate the irradiance on a horizontal surface,$I_0$,

$I_0 = I_{0n} \cos\theta_z = I_{SC}E_0\cos\theta_z$ 1:1.2.2

where $cos\theta_z$, is defined in equation 1:1.1.3.

The amount of energy falling on a horizontal surface during a day is given by

$H_0 = \int_{SR}^{SS} I_0dt = 2 \int_0^{SS} I_0dt$ 1:1.2.3

where $H_0$ is the extraterrestrial daily irradiation$(MJ m^{-2} d^{-1})$, SR is sunrise, and SS is sunset. Assuming that $E_0$remains constant during the one day time step and converting the time $dt$ to the hour angle, the equation can be written

$H_0 = \frac{24}{\pi} I_{SC}E_0\int_0^{\omega.T_{SR} }(\sin\delta \sin\phi+\cos\delta\cos\phi\cos\omega.t)d\omega.t$ 1:1.2.4

or

$H_0 = \frac{24}{\pi} I_{SC}E_0[{\omega.T_{SR} }(\sin\delta \sin\phi+\cos\delta\cos\phi\sin(\omega.T_{SR}))]$ 1:1.2.5

where $I_{SC}$ is the solar constant (4.921 $MJ m^{-2} h^{-1}$), E$_0$is the eccentricity correction factor of the earth's orbit, is the angular velocity of the earth's rotation ($0.2618 rad h^{-1}$), the hour of sunrise, $T_{SR}$, is defined by equation 1:1.1.4, δ is the solar declination in radians, and is the geographic latitude in radians. Multiplying all the constants together gives

$H_0 = 37.59E_0[{\omega.T_{SR} }\sin\delta \sin\phi+\cos\delta\cos\phi\sin(\omega.T_{SR})]$ 1:1.2.6

The solar radiation reaching the earth's surface on a given day, $H_{day}$, may be less than $H_{mx}$ due to the presence of cloud cover. The daily solar radiation data required by SWAT+ may be read from an input file or generated by the model.

The variable SLRSIM in the master watershed (file.cio) file identifies the method used to obtain solar radiation data. To read in daily solar radiation data, the variable is set to 1 and the name of the solar radiation data file and the number of solar radiation records stored in the file are set. To generate daily solar radiation values, SLRSIM is set to 2. The equations used to generate solar radiation data in SWAT+ are reviewed in Chapter 1:3. SWAT+ input variables that pertain to solar radiation are summarized in Table 1:1-2.

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

see description of .slr file in the User’s Manual for input and format requirements if measured daily solar radiation data is being used.

When solar radiation enters the earth's atmosphere, a portion of the energy is removed by scattering and adsorption. The amount of energy lost is a function of the transmittance of the atmosphere, the composition and concentration of the constituents of air at the location, the path length the radiation travels through the air column, and the radiation wavelength.

Due to the complexity of the process and the detail of the information required to accurately predict the amount of radiant energy lost while passing through the atmosphere, SWAT+ makes a broad assumption that roughly 20% of the extraterrestrial radiation is lost while passing through the atmosphere under cloudless skies. Using this assumption, the maximum possible solar radiation, $H_{MX}$, at a particular location on the earth's surface is calculated as:

$H_{MX} = 30.0E_0[{\omega*T_{SR} }\sin\delta \sin\phi+\cos\delta\cos\phi\sin(\omega*T_{SR})]$ 1:1.2.7

where the maximum possible solar radiation, $H_{MX}$, is the amount of radiation reaching the earth's surface under a clear sky ($MJ m^{-2} d^{-1}$).

Air temperature data are usually provided in the form of daily maximum and minimum temperature. A reasonable approximation for converting these to hourly temperatures is to assume a sinusoidal interpolation function between the minimum and maximum daily temperatures. The maximum daily temperature is assumed to occur at 1500 hours and the minimum daily temperature at 300 hours (Campbell, 1985). The temperature for the hour is then calculated with the equation:

$T_{hr} = \overline T_{av} + \frac{T_{mx}-T_{mn}}2 *cos(0.2618*(hr-15))$ 1:1.3.1

where $T_{hr}$ is the air temperature during hour $hr$of the day (C), $\overline T_{av}$is the average temperature on the day (C), $T_{mx}$ is the daily maximum temperature (C), and $T_{mn}$ is the daily minimum temperature (C).

Table 1:1-6: SWAT+ input variables that pertain to hourly air temperature.

The angle between the line from an observer on the earth to the sun and a vertical line extending upward from the observer is called the zenith angle, $\theta_z$ (Figure 1:1-1). Solar noon occurs when this angle is at its minimum value for the day.

For a given geographical position, the relationship between the sun and a horizontal surface on the earth's surface is:

and

The only SWAT+ input variable used in the calculations reviewed in Section 1:1.1 is given in Table 1:1-1.

Table 1:1-1: SWAT+ input variables that are used in earth-sun relationship calculations.

Water temperature is required to model in-stream biological and water quality processes. SWAT+ uses an equation developed by Stefan and Preud’homme (1993) to calculate average daily water temperature for a well-mixed stream:

$T_{water}=5.0+0.75\overline T_{av}$ 1:1.3.13

where $T_{water}$ is the water temperature for the day (C), and ${\overline T_{av}}$ is the average air temperature on the day (C).

Due to thermal inertia of the water, the response of water temperature to a change in air temperature is dampened and delayed. When water and air temperature are plotted for a stream or river, the peaks in the water temperature plots usually lag 3-7 hours behind the peaks in air temperature. As the depth of the river increases, the lag time can increase beyond this typical interval. For very large rivers, the lag time can extend up to a week. Equation 1:1.3.13 assumes that the lag time between air and water temperatures is less than 1 day.

In addition to air temperature, water temperature is influenced by solar radiation, relative humidity, wind speed, water depth, ground water inflow, artificial heat inputs, thermal conductivity of the sediments and the presence of impoundments along the stream network. SWAT+ assumes that the impact of these other variables on water temperature is not significant.

Table 1:1-8: SWAT+ input variables that pertain to water temperature.

Daily Air Temperature SWAT+ requires daily maximum and minimum air temperature. This data may be read from an input file or generated by the model. The user is strongly recommended to obtain measured daily temperature records from gages in or near the watershed if at all possible. The accuracy of model results is significantly improved by the use of measured temperature data.

The variable TMPSIM in the master watershed (file.cio) file identifies the method used to obtain air temperature data. To read in daily maximum and minimum air temperature data, the variable is set to 1 and the name of the temperature data file(s) and the number of temperature records stored in the file are set. To generate daily air temperature values, TMPSIM is set to 2. The equations used to generate air temperature data in SWAT+ are reviewed in Chapter 1:3. SWAT+ input variables that pertain to air temperature are summarized in Table 1:1-5.

Table 1:1-5: SWAT+ input variables that pertain to daily air temperature.

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

Wind speed is required by SWAT+ if the Penman-Monteith equation is used to estimate potential evapotranspiration and transpiration. SWAT+ assumes wind speed information is collected from gages positioned 1.7 meters above the ground surface.

When using the Penman-Monteith equation to estimate transpiration, the wind measurement used in the equation must be above the canopy. In SWAT+, a minimum difference of 1 meter is specified for canopy height and wind speed measurements. When the canopy height exceeds 1 meter, the original wind measurements is adjusted to:

1:1.4.1

where is the height of the wind speed measurement (cm), and is the canopy height (cm).

The variation of wind speed with elevation near the ground surface is estimated with the equation (Haltiner and Martin, 1957):

1:1.4.2

where is the wind speed (m s) at height (cm), is the wind speed (m s) at height (cm), and is an exponent between 0 and 1 that varies with atmospheric stability and surface roughness. Jensen (1974) recommended a value of 0.2 for and this is the value used in SWAT+.

The daily wind speed data required by SWAT+ may be read from an input file or generated by the model. The variable WNDSIM in the master watershed (file.cio) file identifies the method used to obtain wind speed data. To read in daily wind speed data, the variable is set to 1 and the name of the wind speed data file and the number of different records stored in the file are set. To generate daily wind speed values, WNDSIM is set to 2. The equations used to generate wind speed data in SWAT+ are reviewed in Chapter 1:3.

Table 1:1-9: SWAT+ input variables used in wind speed calculations.

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

Net radiation requires the determination of both incoming and reflected short-wave radiation and net long-wave or thermal radiation. Expressing net radiation in terms of the net short-wave and long-wave components gives:

1:1.2.11

or

1:1.2.12

where is the net radiation (), is the short-wave solar radiation reaching the ground (), is the short-wave reflectance or albedo, is the long-wave radiation (), is the net incoming long-wave radiation () and the arrows indicate the direction of the radiation flux.

Net Short-Wave Radiation

Net short-wave radiation is defined as . SWAT+ calculates a daily value for albedo as a function of the soil type, plant cover, and snow cover. When the snow water equivalent is greater than 0.5 mm,

1:1.2.13

When the snow water equivalent is less than 0.5 mm and no plants are growing in the HRU,

1:1.2.14

where is the soil albedo. When plants are growing and the snow water equivalent is less than 0.5 mm,

1:1.2.15

where is the plant albedo (set at 0.23), and is the soil cover index. The soil cover index is calculated

1:1.2.16

where is the aboveground biomass and residue ().

Net Long-Wave Radiation

Long-wave radiation is emitted from an object according to the radiation law:

Table 1:1-3: Experimental coefficients for net long-wave radiation equations (from Jensen et al., 1990).

Table 1:1-4: SWAT+ input variables used in net radiation calculations.

The precipitation reaching the earth's surface on a given 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 PCPSIM in the master watershed file (file.cio) identifies the method used to obtain precipitation data. To read in precipitation data, the variable is set to 1 and the names of the precipitation data files and the number of precipitation records stored in the files are defined. To generate precipitation values, PCPSIM is set to 2. 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 .pcp file in the User’s Manual for input and format requirements if measured daily precipitation data is being used.

Soil temperature will fluctuate due to seasonal and diurnal variations in temperature at the surface. Figure 1:1-2 plots air temperature and soil temperature at 5 cm and 300 cm below bare soil at College Station, Texas. Figure 1:1-2: Four-year average air and soil temperature at College Station, Texas.

This figure illustrates several important attributes of temperature variation in the soil. First, the annual variation in soil temperature follows a sinusoidal function. Second, the fluctuation in temperature during the year (the amplitude of the sine wave) decreases with depth until, at some depth in the soil, the temperature remains constant throughout the year. Finally, the timing of maximum and minimum temperatures varies with depth. Note in the above graph that there is a three month difference between the recording of the minimum temperature at the surface (January) and the minimum temperature at 300 cm (March).

Carslaw and Jaeger (1959) developed an equation to quantify the seasonal variation in temperature:

In order to calculate values for some of the variables in this equation, the heat capacity and thermal conductivity of the soil must be known. These are properties not commonly measured in soils and attempts at estimating values from other soil properties have not proven very effective. Consequently, an equation has been adopted in SWAT+ that calculates the temperature in the soil as a function of the previous day’s soil temperature, the average annual air temperature, the current day’s soil surface temperature, and the depth in the profile.

The equation used to calculate daily average soil temperature at the center of each layer is:

The depth factor is calculated using the equation:

The equation used to calculate the soil surface temperature is:

SWAT+ input variables that directly impact soil temperature calculations are listed in Table 1:1-7. There are several other variables that initialize residue and snow cover in the subbasins or HRUs (SNO_SUB and SNOEB in .sub; RSDIN in .hru). The influence of these variables will be limited to the first few months of simulation. Finally, the timing of management operations in the .mgt file will affect ground cover and consequently soil temperature.

Table 1:1-7: SWAT+ input variables that pertain to soil temperature.

The solar declination is the earth's latitude at which incoming solar rays are normal to the earth's surface. The solar declination is zero at the spring and fall equinoxes, approximately +23½° at the summer solstice, and approximately -23½° at the winter solstice. A simple formula to calculate solar declination from Perrin de Brichambaut (1975) is:

1:1.1.2

where is the solar declination reported in radians and is the day number of the year.

The mean distance between the earth and the sun is km and is called one astronomical unit (AU). The earth revolves around the sun in an elliptical orbit and the distance from the earth to the sun on a given day will vary from a maximum of 1.017 AU to a minimum of 0.983 AU. An accurate value of the earth-sun distance is important because the solar radiation reaching the earth is inversely proportional to the square of its distance from the sun. The distance is traditionally expressed in mathematical form as a Fourier series type of expansion with a number of coefficients. For most engineering applications a simple expression used by Duffie and Beckman (1980) is adequate for calculating the reciprocal of the square of the radius vector of the earth, also called the eccentricity correction factor, , of the earth's orbit:

1:1.1.1

where is the mean earth-sun distance (1 AU), r is the earth-sun distance for any given day of the year (AU), and is the day number of the year, ranging from 1 on January 1 to 365 on December 31. February is always assumed to have 28 days, making the accuracy of the equation vary due to the leap year cycle.

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:

1:2.3.1

where is the relative humidity on a given day, 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 (kPa) and is the mean daily air temperature (C). 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 (kPa C), is the saturation vapor pressure on a given day (kPa) and is the mean daily air temperature (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:

1:2.3.5

where is the vapor pressure deficit (kPa), is the saturation vapor pressure on a given day (kPa), and is the actual vapor pressure on a given day (kPa). 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 (MJ kg) and is the mean daily air temperature (C).

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:

The daily relative humidity data required by SWAT+ may be read from an input file or generated by the model. The variable RHSIM in the master watershed (file.cio) file identifies the method used to obtain relative humidity data. To read in daily relative humidity data, the variable is set to 1 and the name of the relative humidity data file and the number of different records stored in the file are set. To generate daily relative humidity values, RHSIM is set to 2. 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 .hmd file in the User’s Manual for input and format requirements if measured relative humidity data is being used.

The radiant energy from the sun is practically the only source of energy that impacts climatic processes on earth. The solar constant, , is the rate of total solar energy at all wavelengths incident on a unit area exposed normally to rays of the sun at a distance of 1 AU from the sun. Quantifying this value has been the object of numerous studies through the years. The value officially adopted by the Commission for Instruments and Methods of Observation in October 1981 is

On any given day, the extraterrestrial irradiance (rate of energy) on a surface normal to the rays of the sun, , is:

1:1.2.1

where is the eccentricity correction factor of the earth's orbit, and has the same units as the solar constant,.

To calculate the irradiance on a horizontal surface,,

1:1.2.2

where , is defined in equation 1:1.1.3.

The amount of energy falling on a horizontal surface during a day is given by

1:1.2.3

where is the extraterrestrial daily irradiation , SR is sunrise, and SS is sunset. Assuming that remains constant during the one day time step and converting the time to the hour angle, the equation can be written

1:1.2.4

or

1:1.2.5

where is the solar constant (4.921 ), Eis the eccentricity correction factor of the earth's orbit, is the angular velocity of the earth's rotation (), the hour of sunrise, , is defined by equation 1:1.1.4, δ is the solar declination in radians, and is the geographic latitude in radians. Multiplying all the constants together gives

1:1.2.6

see description of .tmp file in the User’s Manual for input and format requirements if measured temperature 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, . 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:

1:2.5.1

where is the snow pack temperature on a given day (C), is the snow pack temperature on the previous day (C), is the snow temperature lag factor, and is the mean air temperature on the current day (C). As 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.

Numerous probability distribution functions have been used to describe the distribution of rainfall amounts. SWAT+ provides the user with two options: a skewed distribution and an exponential distribution.

The skewed distribution was proposed by Nicks (1974) and is based on a skewed distribution used by Fiering (1967) to generate representative streamflow. The equation used to calculate the amount of precipitation on a wet day is:

1:3.1.3

where is the amount of rainfall on a given day (mm HO), is the mean daily rainfall (mm HO) for the month, is the standard deviation of daily rainfall (mm HO) for the month, is the standard normal deviate calculated for the day, and is the skew coefficient for daily precipitation in the month.

The standard normal deviate for the day is calculated:

1:3.1.4

where and are random numbers between 0.0 and 1.0.

The exponential distribution is provided as an alternative to the skewed distribution. This distribution requires fewer inputs and is most commonly used in areas where limited data on precipitation events is available. Daily precipitation is calculated with the exponential distribution using the equation:

1:3.1.5

where is the amount of rainfall on a given day (mm HO), is the mean daily rainfall (mm HO) for the month, is a random number between 0.0 and 1.0, and is an exponent that should be set between 1.0 and 2.0. As the value of is increased, the number of extreme rainfall events during the year will increase. Testing of this equation at locations across the U.S. have shown that a value of 1.3 gives satisfactory results.

Table 1:3-1: SWAT+ input variables that pertain to generation of precipitation.

For each month, users provide the maximum half-hour rain observed over the entire period of record. These extreme values are used to calculate representative monthly maximum half-hour rainfall fractions.

Prior to calculating the representative maximum half-hour rainfall fraction for each month, the extreme half-hour rainfall values are smoothed by calculating three month average values:

1:3.2.1

where is the smoothed maximum half-hour rainfall for a given month (mm HO) and is the extreme maximum half-hour rainfall for the specified month (mm HO). Once the smoothed maximum half-hour rainfall is known, the representative half-hour rainfall fraction is calculated using the equation:

1:3.2.2

where is the average half-hour rainfall fraction for the month, is an adjustment factor, is the smoothed half-hour rainfall amount for the month (mm HO), is the mean daily rainfall (mm HO) for the month, is the number of years of rainfall data used to obtain values for monthly extreme half-hour rainfalls, and are the number of wet days in the month. The adjustment factor is included to allow users to modify estimations of half-hour rainfall fractions and peak flow rates for runoff.

With the first-order Markov-chain model, the probability of rain on a given day is conditioned on the wet or dry status of the previous day. A wet day is defined as a day with 0.1 mm of rain or more.

The user is required to input the probability of a wet day on day given a wet day on day , and the probability of a wet day on day given a dry day on day , for each month of the year. From these inputs the remaining transition probabilities can be derived:

1:3.1.1

1:3.1.2

where is the probability of a dry day on day given a wet day on day and is the probability of a dry day on day given a dry day on day .

To define a day as wet or dry, SWAT+ generates a random number between 0.0 and 1.0. This random number is compared to the appropriate wet-dry probability, or . If the random number is equal to or less than the wet-dry probability, the day is defined as wet. If the random number is greater than the wet-dry probability, the day is defined as dry.

For simulations where the timing of rainfall within the day is required, the daily rainfall value must be partitioned into shorter time increments. The method used in SWAT+ to disaggregate storm data was taken from CLIGEN (Nicks et al., 1995).

A double exponential function is used to represent the intensity patterns within a storm. With the double exponential distribution, rainfall intensity exponentially increases with time to a maximum, or peak, intensity. Once the peak intensity is reached, the rainfall intensity exponentially decreases with time until the end of the storm.

The exponential equations governing rainfall intensity during a storm event are:

1:3.3.1

,

where is the rainfall intensity at time (mm/hr), is the maximum or peak rainfall intensity during the storm (mm/hr), is the time since the beginning of the storm (hr), is the time from the beginning of the storm till the peak rainfall intensity occurs (hr), is the duration of the storm (hr), and and are equation coefficients (hr).

The maximum or peak rainfall intensity during the storm is calculated assuming the peak rainfall intensity is equivalent to the rainfall intensity used to calculate the peak runoff rate. The equations used to calculate the intensity are reviewed in Chapter 2:1 (section 2:1.3.3).

The normalized time to peak intensity is calculated by SWAT+ using a triangular distribution. The triangular distribution used to generate the normalized time to peak intensity requires four inputs: average time to peak intensity expressed as a fraction of total storm duration , maximum time to peak intensity expressed as a fraction of total storm duration , minimum time to peak intensity expressed as a fraction of total storm duration and a random number between 0.0 and 1.0.

The maximum time to peak intensity, or upper limit of the triangular distribution, is set at 0.95. The minimum time to peak intensity, or lower limit of the triangular distribution is set at 0.05. The mean time to peak intensity is set at 0.25.

The triangular distribution uses one of two sets of equations to generate a normalized peak intensity for the day. If then

1:3.3.9

If then

1:3.3.10

where is the time from the beginning of the storm till the peak intensity expressed as a fraction of the total storm duration (0.0-1.0), is the average time to peak intensity expressed as a fraction of storm duration, is a random number generated by the model each day, is the minimum time to peak intensity that can be generated, is the maximum time to peak intensity that can be generated, and is the mean of and .

SWAT+ classifies precipitation as rain or freezing rain/snow by the mean daily air temperature. The boundary temperature, , 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:

1:2.4.1

where is the water content of the snow pack on a given day (mm HO), is the amount of precipitation on a given day (added only if ) (mm HO), is the amount of sublimation on a given day (mm HO), and is the amount of snow melt on a given day (mm HO). 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 (mm HO), is the threshold depth of snow at 100% coverage (mm HO), 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.

Figure 1:2-1:10% = 50% coverage

Figure 1:2-2: 30% = 50% coverage

Figure 1:2-3: 50% = 50% coverage

Figure 1:2-4: 70% = 50% coverage

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

Residuals for maximum temperature, minimum temperature and solar radiation are required for calculation of daily values. The residuals must be serially correlated and cross-correlated with the correlations being constant at all locations. The equation used to calculate residuals is:

1:3.4.1

where is a 3 × 1 matrix for day whose elements are residuals of maximum temperature (), minimum temperature () and solar radiation (), ) is a 3 × 1 matrix of the previous day’s residuals, is a 3 × 1 matrix of independent random components, and and are 3 × 3 matrices whose elements are defined such that the new sequences have the desired serial-correlation and cross-correlation coefficients. The and matrices are given by

1:3.4.2

1:3.4.3

where the superscript denotes the inverse of the matrix and the superscript T denotes the transpose of the matrix. and are defined as

1:3.4.4

1:3.4.5

is the correlation coefficient between variables and on the same day where and may be set to 1 (maximum temperature), 2 (minimum temperature) or 3 (solar radiation) and is the correlation coefficient between variable and with variable lagged one day with respect to variable . Correlation coefficients were determined for 31 locations in the United States using 20 years of temperature and solar radiation data (Richardson, 1982). Using the average values of these coefficients, the and matrices become

1:3.4.6

1:3.4.7

Using equations 1:3.4.2 and 1:3.4.3, the A and B matrices become

1:3.4.8

1:3.4.9

The A and B matrices defined in equations 1:3.4.8 and 1:3.4.9 are used in conjunction with equation 1:3.4.1 to generate daily sequences of residuals of maximum temperature, minimum temperature and solar radiation.

The daily generated values are determined by multiplying the residual elements generated with equation 1:3.4.1 by the monthly standard deviation and adding the monthly average value.

1:3.4.10

1:3.4.11

1:3.4.12

where is the maximum temperature for the day (°C), is the average daily maximum temperature for the month (°C), is the residual for maximum temperature on the given day, is the standard deviation for daily maximum temperature during the month (°C), is the minimum temperature for the day (°C), is the average daily minimum temperature for the month (°C), is the residual for minimum temperature on the given day, is the standard deviation for daily minimum temperature during the month (°C), is the solar radiation for the day (MJ m), is the average daily solar radiation for the month (MJ m), is the residual for solar radiation on the given day, and is the standard deviation for daily solar radiation during the month (MJ m).

The user is required to input standard deviation for maximum and minimum temperature. For solar radiation the standard deviation is estimated as ¼ of the difference between the extreme and mean value for each month.

1:3.4.13

where is the standard deviation for daily solar radiation during the month (MJ m), is the maximum solar radiation that can reach the earth’s surface on a given day (MJ m), and is the average daily solar radiation for the month (MJ m).

The rainfall intensity distribution given in equation 1:3.3.1 can be normalized to eliminate units. To do this, all time values are divided, or normalized, by the storm duration and all intensity values are normalized by the average storm intensity. For example,

1:3.3.2

1:3.3.3

where the normalized rainfall intensity at time , is the rainfall intensity at time T (mm/hr), is the average storm rainfall intensity (mm/hr), is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), is the time since the beginning of the storm (hr), and is the duration of the storm (hr).

The normalized storm intensity distribution is:

1:3.3.4

,

where the normalized rainfall intensity at time , is the normalized maximum or peak rainfall intensity during the storm, is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), is the time from the beginning of the storm till the peak intensity expressed as a fraction of the total storm duration (0.0-1.0), and are equation coefficients.

The relationship between the original equation coefficients and the normalized equation coefficients is:

1:3.3.5

1:3.3.6

where is the equation coefficient for rainfall intensity before peak intensity is reached (hr), is the normalized equation coefficient for rainfall intensity before peak intensity is reached, is the equation coefficient for rainfall intensity after peak intensity is reached (hr), is the normalized equation coefficient for rainfall intensity after peak intensity is reached, and is the storm duration (hr).

Values for the equation coefficients, and , can be determined by isolating the coefficients in equation 1:3.3.4. At = 0.0 and at = 1.0,

1:3.3.7

1:3.3.8

where is the normalized equation coefficient for rainfall intensity before peak intensity is reached, is the normalized equation coefficient for rainfall intensity after peak intensity is reached, is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), is the time from the beginning of the storm till the peak intensity expressed as a fraction of the total storm duration (0.0-1.0), is the normalized rainfall intensity at time , and is the normalized maximum or peak rainfall intensity during the storm.

The user has the option of using the monthly maximum half-hour rainfall for all days in the month or generating a daily value. The variable sed_det in the basin input file (codes.bsn) defines which option the user prefers. The randomness of the triangular distribution used to generated daily values can cause the maximum half-hour rainfall value to jump around. For small plots or microwatersheds in particular, the variability of the triangular distribution is unrealistic.

The triangular distribution used to generate the maximum half-hour rainfall fraction requires four inputs: average monthly half-hour rainfall fraction, maximum value for half-hour rainfall fraction allowed in month, minimum value for half-hour rainfall fraction allowed in month, and a random number between 0.0 and 1.0.

The maximum half-hour rainfall fraction, or upper limit of the triangular distribution, is calculated from the daily amount of rainfall with the equation:

1:3.2.3

where is the largest half-hour fraction that can be generated on a given day, and is the precipitation on a given day (mm HO). The minimum half-hour fraction, or lower limit of the triangular distribution, , is set at 0.02083.

The triangular distribution uses one of two sets of equations to generate a maximum half-hour rainfall fraction for the day. If then

1:3.2.4

If then

1:3.2.5

where is the maximum half-hour rainfall fraction for the day, is the average maximum half-hour rainfall fraction for the month, is a random number generated by the model each day, is the smallest half-hour rainfall fraction that can be generated, is the largest half-hour fraction that can be generated, and is the average of , , and .

Table 1:3-2: SWAT+ input variables that pertain to generation of maximum half-hour rainfall.

The continuity equation relates average daily maximum temperature adjusted for wet or dry conditions to the average daily maximum temperature for the month:

1:3.4.14

where is the average daily maximum temperature for the month (°C), are the total number of days in the month, is the average daily maximum temperature of the month on wet days (°C), are the number of wet days in the month, is the average daily maximum temperature of the month on dry days (°C), and are the number of dry days in the month.

The wet day average maximum temperature is assumed to be less than the dry day average maximum temperature by some fraction of ():

1:3.4.15

where is the average daily maximum temperature of the month on wet days (°C), is the average daily maximum temperature of the month on dry days (°C), is a scaling factor that controls the degree of deviation in temperature caused by the presence or absence of precipitation, is the average daily maximum temperature for the month (°C), and is the average daily minimum temperature for the month (°C). The scaling factor, , is set to 0.5 in SWAT+.

To calculate the dry day average maximum temperature, equations 1:3.4.14 and 1:3.4.15 are combined and solved for :

1:3.4.16

Incorporating the modified values into equation 1:3.4.10, SWAT+ calculates the maximum temperature for a wet day using the equation:

1:3.4.17

and the maximum temperature for a dry day using the equation:

1:3.4.18

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

1:3.5.1

where is the average relative humidity for the month, is the actual vapor pressure at the mean monthly temperature (kPa), and is the saturation vapor pressure at the mean monthly temperature (kPa). The saturation vapor pressure, , is related to the mean monthly air temperature with the equation:

1:3.5.2

where is the saturation vapor pressure at the mean monthly temperature (kPa), and 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, , and the mean minimum monthly temperature, .

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:

1:3.5.3

where is the actual vapor pressure at the mean month temperature (kPa), and is the average dew point temperature for the month (°C).