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.

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:

$\delta = sin^{-1} \{0.4sin [ 2 \pi /365] (d_n - 82)\}$ 1:1.1.2

where $\delta$ is the solar declination reported in radians and $d_n$ is the day number of the year.

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 ($MJ m^{-2} d^{-1}$).

The mean distance between the earth and the sun is $1.496 X 10^8$ 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, $E_0$, of the earth's orbit:

$E_0 = (r_0/r)^2 = 1+ 0.033 cos [(2\pi d_n /365)]$ 1:1.1.1

where $r_0$is the mean earth-sun distance (1 AU), r is the earth-sun distance for any given day of the year (AU), and $d_n$ 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.

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.

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.

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 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 slr in the master weather station (weather-sta.cli) file identifies the method used to obtain solar radiation data. To read in daily solar radiation data, the slr variable is set to the name of the solar radiation data file. To generate daily solar radiation values, set the name of the solar radiation input file (slr) to sim. 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 the description for the .slr files on the page for input and format requirements if measured daily solar radiation data is being used.

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 $\omega t$.

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 , it is possible to calculate the hourly radiation values by multiplying 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:

1:1.2.9

where is the solar radiation reaching the earth’s surface during a specific hour of the day (), is the fraction of total daily radiation falling during that hour, and 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

1:1.2.10

where is the solar time at the midpoint of hour .

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:

$z_w=h_c+100$ 1:1.4.1

where $z_w$ is the height of the wind speed measurement (cm), and $h_c$ 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 wnd in the master weather () file identifies if there is available input wind speed data or if it will be simulated. The file includes a list of all available wind speed data stations, and can be empty if all locations are simulated. To read in daily wind speed data, the variable is set to the name of the wind speed data station. To generate daily wind speed values wnd is set to "sim". 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 on the page 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:

$H_{net}=H_{day}\downarrow-\alpha*H_{day}\uparrow+H_L\downarrow-H_L\uparrow$ 1:1.2.11

or

$H_{net} = (1-\alpha) H_{day} + H_b$ 1:1.2.12

where $H_{net}$ is the net radiation ($MJ m^{-2} d^{-1}$), $H_{day}$ is the short-wave solar radiation reaching the ground ($MJ m^{-2} d^{-1}$), is the short-wave reflectance or albedo, $H_L$ is the long-wave radiation ($MJ m^{-2} d^{-1}$), is the net incoming long-wave radiation () and the arrows indicate the direction of the radiation flux.

1:1.2.5.1 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 ().

1:1.2.5.2 Net Long-Wave Radiation

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

1:1.2.17

where is the radiant energy (, is the emissivity, is the Stefan-Boltzmann constant (, and is the mean air temperature in Kelvin (273.15 + ). Net long-wave radiation is calculated using a modified form of equation 1:1.2.17 (Jensen et al., 1990):

1:1.2.18

where is the net long-wave radiation (), is a factor to adjust for cloud cover, is the atmospheric emittance, and is the vegetative or soil emittance.

Wright and Jensen (1972) developed the following expression for the cloud cover adjustment factor, :

1:1.2.19

where and are constants, is the solar radiation reaching the ground surface on a given day (), and is the maximum possible solar radiation to reach the ground surface on a given day ().

The two emittances in equation 1:1.2.18 may be combined into a single term, the net emittance . The net emittance is calculated using an equation developed by Brunt (1932):

1:1.2.20

where and are constants and is the vapor pressure on a given day (). The calculation of is given in Chapter 1:2. Combining equations 1:1.2.18, 1:1.2.19, and 1:1.2.20 results in a general equation for net long-wave radiation:

1:1.2.21

Experimental values for the coefficients , and are presented in Table 1:1.3. The default equation in SWAT+ uses coefficient values proposed by Doorenbos and Pruitt (1977):

1:1.2.22

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 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 has the same units as the solar constant, . To calculate the irradiance on a horizontal surface, ,

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, is sunrise, and 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 ), is 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

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.

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:

$T_{soil}(z,d_n)=\overline T_{AA} +A_{surf}exp(-z/dd)sin(\omega_{tmp}d_n-z/dd)$ 1:1.3.2

where is the soil temperature () at depth (mm) and day of the year , is the average annual soil temperature (), is the amplitude of the surface fluctuations (), is the damping depth (mm) and is the angular frequency. When (soil surface), equation 1:1.3.2 reduces to As , equation 1:1.3.2 becomes .

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:

1:1.3.3

where is the soil temperature () at depth (mm) and day of the year , is the lag coefficient (ranging from 0.0 to 1.0) that controls the influence of the previous day's temperature on the current day's temperature , is the soil temperature () in the layer from the previous day, is the depth factor that quantifies the influence of depth below surface on soil temperature , is the average annual temperature (), and is the soil surface temperature on the day. SWAT+ sets the lag coefficient, to 0.80. The soil temperature from the previous day is known and the average annual air temperature is calculated from the long-term monthly maximum and minimum temperatures reported in the weather generator input () file. This leaves the depth factor, , and the soil surface temperature, , to be defined.

The depth factor is calculated using the equation:

1:1.3.4

where is the ratio of the depth at the center of the soil layer to the damping depth:

1:1.3.5

where is the depth at the center of the soil layer (mm) and is the damping depth (mm).

From the previous three equations (1:1.3.3, 1:1.3.4 and 1:1.3.5) one can see that at depths close to the soil surface, the soil temperature is a function of the soil surface temperature. As the depth increases, soil temperature is increasingly influenced by the average annual air temperature, until at the damping depth, the soil temperature is within 5% of .

The damping depth, , is calculated daily and is a function of the maximum damping depth, bulk density and soil water. The maximum damping depth, , is calculated:

1:1.3.6

where is the maximum damping depth (mm), and is the soil bulk density (). The impact of soil water content on the damping depth is incorporated via a scaling factor,, that is calculated with the equation:

1:1.3.7

where is the amount of water in the soil profile expressed as depth of water in the profile (mm ), is the soil bulk density (), and is the depth from the soil surface to the bottom of the soil profile (mm).

The daily value for the damping depth, , is calculated:

1:1.3.8

where is the maximum damping depth (mm), and is the scaling factor for soil water. The soil surface temperature is a function of the previous day’s temperature, the amount of ground cover and the temperature of the surface when no cover is present. The temperature of a bare soil surface is calculated with the equation:

1:1.3.1.9

where is the temperature of the soil surface with no cover (), is the average temperature on the day (), is the daily maximum temperature (), is the daily minimum temperature (), and is a radiation term. The radiation term is calculated with the equation:

1:1.3.10

where is the solar radiation reaching the ground on the current day (), and is the albedo for the day. Any cover present will significantly impact the soil surface temperature. The influence of plant canopy or snow cover on soil temperature is incorporated with a weighting factor, , calculated as:

1:1.3.11

where is the total aboveground biomass and residue present on the current day (kg ha) and SNO is the water content of the snow cover on the current day (mm ). The weighting factor, , is 0.0 for a bare soil and approaches 1.0 as cover increases.

The equation used to calculate the soil surface temperature is:

1:1.3.12

where is the soil surface temperature for the current day (), is the weighting factor for soil cover impacts, is the soil temperature of the first soil layer on the previous day (), and is the temperature of the bare soil surface (). The influence of ground cover is to place more emphasis on the previous day’s temperature near the surface.

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 (snow_init in and rsd_init in ). The influence of these variables will be limited to the first few months of simulation. Finally, the timing of management operations in the file will affect ground cover and consequently soil temperature.

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

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:

1:1.3.13

where is the water temperature for the day (), and is the average air temperature on the day ().

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.

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 tmp in the master weather () 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 the name of the temperature data file(s). To generate daily air temperature values, tmp is set to "sim". 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.

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:

1:1.3.1

where is the air temperature during hour of the day (), is the average temperature on the day (), is the daily maximum temperature (), and is the daily minimum temperature ().

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

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 (), and is the saturation vapor pressure on a given 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.

The daily precipitation generator is a Markov chain-skewed (Nicks, 1974) or Markov chain-exponential model (Williams, 1995). A first-order Markov chain is used to define the day as wet or dry. When a wet day is generated, a skewed distribution or exponential distribution is used to generate the precipitation amount. Table 1:3-1 lists SWAT+ input variables that are used in the precipitation generator.

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).

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).

SWAT+ is capable of simulating a number of climate customization options. Orographic impacts on temperature and precipitation for watersheds in mountainous regions can be simulated. The model will also modify climate inputs for simulations that are looking at the impact of climatic change in a given watershed. Finally, SWAT+ allows a weather forecast period to be incorporated into a simulation to study the effects of predicted weather in a watershed.

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.

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 $i$ given a wet day on day $i-1,Pi-1(W/W)$, and the probability of a wet day on day $i$ given a dry day on day $i-1,P_i(W/D)$, for each month of the year. From these inputs the remaining transition probabilities can be derived:

$P_i(D/W)=1-P_i(W/W)$ 1:3.1.1

$P_i(W/W)=1-P_i(W/D)$ 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.

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.

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:

$R_{0.5sm(mon)}=\frac{R_{0.5x(mon-1)}+R_{0.5x(mon)}+R_{0.5x(mon+1)}}{3}$ 1:3.2.1

where $R_{0.5sm(mon)}$ is the smoothed maximum half-hour rainfall for a given month ($mm\space H_2O$) and $R_{0.5x}$ is the extreme maximum half-hour rainfall for the specified month ($mm\space H_2O$). Once the smoothed maximum half-hour rainfall is known, the representative half-hour rainfall fraction is calculated using the equation:

$\alpha_{0.5mon}=adj_{0.5\alpha}*[1-exp(\frac{R_{0.5sm(mon)}}{{\mu_{mon}}*ln*(\frac{0.5}{yrs*days_{wet}})})]$ 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 (), is the mean daily rainfall () 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.

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.

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:

$R_{day}=\mu_{mon}+2*\sigma_{mon}*(\frac{[(SND_{day}-\frac{g_{mon}}{{6}})*\frac{g_{mon}}{{6}}+1]^3-1}{g_{mon}})$ 1:3.1.3

where $R_{day}$ is the amount of rainfall on a given day ($mm\space H_2O$), $\mu_{mon}$ is the mean daily rainfall ($mm\space H_2O$) for the month, $\sigma_{mon}$ is the standard deviation of daily rainfall ($mm\space H_2O$) for the month, $SND_{day}$ is the standard normal deviate calculated for the day, and $g_{mon}$ 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 (), is the mean daily rainfall () 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.

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:

$\alpha_{0.5U}=1-exp(\frac{-125}{R_{day}+5})$ 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 (). 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.

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:

$i(T)={i_{mx}*exp[\frac{T-T_{peak}}{\delta_{1}}], i_{mx}*exp[\frac{T_{peak}-T}{\delta_2}}]$ 1:3.3.1

$0\le T \le T_{peak}$ , $T_{peak} < T <T_{dur}$

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

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).

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 (), and is the saturation vapor pressure at the mean monthly temperature (). The saturation vapor pressure, , is related to the mean monthly air temperature with the equation:

1:3.5.2

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

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

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 (), is the average daily maximum temperature for the month (), is the residual for maximum temperature on the given day,

Maximum temperature and solar radiation will be lower on overcast days than on clear days. To incorporate the influence of wet/dry days on generated values of maximum temperature and solar radiation, the average daily maximum temperature, , and average daily solar radiation, , in equations 1:3.4.10 and 1:3.4.12 are adjusted for wet or dry conditions.

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:

1:3.5.8

where is the average relative humidity for the month, are the total number of days in the month, is the average relative humidity for the month on wet days, are the number of wet days in the month, is the average relative humidity of the month on dry days, and are the number of dry days in the month.

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

1:3.4.19

where is the average daily solar radiation for the month (MJ m), are the total number of days in the month, is the average daily solar radiation of the month on wet days (MJ m), are the number of wet days in the month, is the average daily solar radiation of the month on dry days (MJ m), and are the number of dry days in the month.

The wet day average solar radiation is assumed to be less than the dry day average solar radiation by some fraction:

The volume of rain is related to rainfall intensity by:

1:3.3.11

where is the amount of rain that has fallen at time () and is the rainfall intensity at time ().

Using the definition for rainfall intensity given in equation 1:3.3.1, equation 1:3.3.11 can be integrated to get:

1:3.3.12

Wind speed is required by SWAT+ when the Penman-Monteith equation is used to calculate potential evapotranspiration. Mean daily wind speed is generated in SWAT+ using a modified exponential equation:

1.3.6.1

where is the mean wind speed for the day (), is the average wind speed for the month (), and is a random number between 0.0 and 1.0.

Table 1:3-6: SWAT+ input variables that pertain to generation of wind speed.

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:

1:3.5.4

where is the largest relative humidity value that can be generated on a given day in the month, and 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:

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 (), are the total number of days in the month, is the average daily maximum temperature of the month on wet days (), are the number of wet days in the month, is the average daily maximum temperature of the month on dry days (), 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 ():

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(), is the average storm rainfall intensity (), 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 (

The impact of global climate change on water supply is a major area of research. Climate change can be simulated with SWAT+ by manipulating the climatic input that is read into the model (precipitation, temperature, solar radiation, relative humidity, wind speed, potential evapotranspiration and weather generator parameters). A less time-consuming method is to set adjustment factors for the various climatic inputs.

SWAT+ will allow users to adjust precipitation, temperature, solar radiation, relative humidity, and carbon dioxide levels in each subbasin. The alteration of precipitation, temperature, solar radiation and relative humidity are straightforward:

1:4.2.1

where is the precipitation falling in the subbasin on a given day (mm HO), and is the % change in rainfall.

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 (), is the amount of precipitation on a given day (added only if ) (), is the amount of sublimation on a given day (), and

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, (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:

1:1.1.3

where is the solar declination in radians, is the geographic latitude in radians,

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).

Orographic precipitation is a significant phenomenon in certain areas of the world. To account for orographic effects on both precipitation and temperature, SWAT+ allows up to 10 elevation bands to be defined in each subbasin. Precipitation and maximum and minimum temperatures are calculated for each band as a function of the respective lapse rate and the difference between the gage elevation and the average elevation specified for the band. For precipitation,

when 1:4.1.1

where is the precipitation falling in the elevation band (mm HO), is the precipitation recorded at the gage or generated from gage data (mm HO), is the mean elevation in the elevation band (m), is the elevation at the recording gage (m), is the precipitation lapse rate (mm HO/km), is the average number of days of precipitation in the subbasin in a year, and 1000 is a factor needed to convert meters to kilometers. For temperature,

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.