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

The radiant energy from the sun is practically the only source of energy that impacts climatic processes on earth. The solar constant, $I_{SC}$, 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,$I_0$,

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

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

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

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 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.

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 ($MJ m^{-2} d^{-1}$), $H_b$is the net incoming long-wave radiation ($MJ m^{-2} d^{-1}$) and the arrows indicate the direction of the radiation flux.

Net Short-Wave Radiation

Net short-wave radiation is defined as $(1-\alpha)*H_{day}$ . 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,

$\alpha=0.8$ 1:1.2.13

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

$\alpha=\alpha_{soil}$ 1:1.2.14

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

$\alpha=\alpha_{plant}*(1-cov_{sol})+\alpha_{soil}*cov_{sol}$ 1:1.2.15

where $_{plant}$ is the plant albedo (set at 0.23), and $cov_{sol}$ is the soil cover index. The soil cover index is calculated

$cov_{sol}=exp(-5.0X10^{-5}*CV)$ 1:1.2.16

where $CV$ is the aboveground biomass and residue ($kg ha^{-1}$).

Net Long-Wave Radiation

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

$H_R=\varepsilon*\sigma*T_K^{4}$ 1:1.2.17

where $H_R$ is the radiant energy ($MJ m^{-2} d^{-1})$, is the emissivity, is the Stefan-Boltzmann constant ($4.903 10^{-9} MJ m^{-2} K^{-4} d^{-1})$, and $T_K$ is the mean air temperature in Kelvin (273.15 + C). Net long-wave radiation is calculated using a modified form of equation 1:1.2.17 (Jensen et al., 1990):

$H_b=f_{cld}*(\varepsilon_a -\varepsilon_{vs})*\sigma*T_K^{4}$ 1:1.2.18

where $H_b$ is the net long-wave radiation ($MJ m^{-2} d^{-1}$), $f_{cld}$ is a factor to adjust for cloud cover, a is the atmospheric emittance, and vs is the vegetative or soil emittance.

Wright and Jensen (1972) developed the following expression for the cloud cover adjustment factor, $f_{cld}$:

$f_{cld}=a*\frac{H_{day}}{H_{MX}}-b$ 1:1.2.19

where a and b are constants, $H_{day}$ is the solar radiation reaching the ground surface on a given day ($MJ m^{-2}d^{-1}$), and $H_{MX}$ is the maximum possible solar radiation to reach the ground surface on a given day ($MJ m^{-2}d^{-1}$). 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):

$\varepsilon'=\varepsilon_a-\varepsilon_{vs}=-(a_1+b_1*\sqrt(e))$ 1:1.2.20

where $a_1$ and $b_1$ are constants and $e$ is the vapor pressure on a given day (kPa). The calculation of e 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:

$H_b=-[a*\frac{H_{day}}{H_{MX}}-b]*[a_1+b_1*\sqrt(e)]*\sigma*T_k^4$ 1:1.2.21

Experimental values for the coefficients $a,b,a_1$and $b_1$ are presented in Table 1:1.3. The default equation in SWAT+ uses coefficient values proposed by Doorenbos and Pruitt (1977):

$H_b=-[0.9*\frac{H_{day}}{H_{MX}}+0.1]*[0.34-0.139\sqrt(e)]*\sigma*T_k^4$ 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.