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1:1.1.3 Solar Noon, Sunrise, Sunset, and Daylength

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.

1:1.1.3 Solar Noon, Sunrise, Sunset, and Daylength

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 $h^{-1}$ or 15˚ $h^{-1}$ ), and t is the solar hour. t equals zero at solar noon, is a positive value in the morning, and is a negative value in the evening. The combined term $\omega t$ is referred to as the hour angle.

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

$T_{SR} = +(\cos^{-1}[-\tan\delta \tan\phi]/\omega)$ 1:1.1.4

and

$T_{SS} = - (\cos^{-1}[-tan \delta tan\phi]/\omega)$ 1:1.1.5

Total daylength, $T_{DL}$ is calculated:

$T_{DL} = (2 \cos^ {-1}[-1\tan \delta \tan \phi]/\omega)$ 1:1.1.6

At latitudes above $66.5\degree$ or below $-66.5\degree$ , the absolute value of [ $\tan\delta \tan\phi$ ] can exceed 1 and the above equation cannot be used. When this happens, there is either no sunrise (winter) or no sunset (summer) and $T_{DL}$ 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 $-23.5\degree$ (-0.4102 radians) for the northern hemisphere or $+23.5\degree$ (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.

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