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, θz\theta_z (Figure 1:1-1). Solar noon occurs when this angle is at its minimum value for the day.

Figure 1:1-1 Diagram illustrating zenith angle

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

cosθz=sinδsinϕ+cosδcosϕcosωt\cos\theta_z = \sin\delta\sin\phi + \cos\delta \cos\phi\cos\omega t 1:1.1.3

where δ\delta is the solar declination in radians, ϕ\phi is the geographic latitude in radians, ω\omega is the angular velocity of the earth's rotation (0.2618 rad h1h^{-1} or 15˚ h1h^{-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 ωt\omega t is referred to as the hour angle.

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

TSR=+(cos1[tanδtanϕ]/ω)T_{SR} = +(\cos^{-1}[-\tan\delta \tan\phi]/\omega) 1:1.1.4

and

TSS=(cos1[tanδtanϕ]/ω)T_{SS} = - (\cos^{-1}[-tan \delta tan\phi]/\omega) 1:1.1.5

Total daylength, TDLT_{DL} is calculated:

TDL=(2cos1[1tanδtanϕ]/ω)T_{DL} = (2 \cos^ {-1}[-1\tan \delta \tan \phi]/\omega) 1:1.1.6

At latitudes above 66.5°66.5\degreeor below 66.5°-66.5\degree, the absolute value of [ tanδtanϕ\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 TDLT_{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°-23.5\degree (-0.4102 radians) for the northern hemisphere or +23.5°+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.

Variable Name

Definition

File Name

SUB_LAT

Latitude of the subbasin (degrees).

.sub

Last updated

#1315: katie.mendoza's Oct 3 ET chapter

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