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.

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.

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 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:

$\cos\theta_z = \sin\delta\sin\phi + \cos\delta \cos\phi\cos\omega t$ 1:1.1.3

where $\delta$ is the solar declination in radians, $\phi$ is the geographic latitude in radians, $\omega$ is the angular velocity of the earth's rotation (0.2618 rad or 15˚ ), and t is the solar hour. equals zero at solar noon, is a positive value in the morning, and is a negative value in the evening. The combined term is referred to as the hour angle.

Sunrise, , and sunset, , occur at equal times before and after solar noon. These times can be determined by rearranging the above equation as:

1:1.1.4

and

1:1.1.5

Total daylength, is calculated:

1:1.1.6

At latitudes above or below , the absolute value of [ ] can exceed 1 and the above equation cannot be used. When this happens, there is either no sunrise (winter) or no sunset (summer) and must be assigned a value of 0 or 24 hours, respectively.

To determine the minimum daylength that will occur during the year, equation 1:1.1.6 is solved with the solar declination set to (-0.4102 radians) for the northern hemisphere or (0.4102 radians) for the southern hemisphere.

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.

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.

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.

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 .

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

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.

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.

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, , at a particular location on the earth's surface is calculated as:

1:1.2.7

where the maximum possible solar radiation, , is the amount of radiation reaching the earth's surface under a clear sky ().

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:

1:1.3.2

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.

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.

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.