Introduction to climate dynamics and climate modelling - Daily insolation at the top of the atmosphere

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Technical Note

2.1.3.3 Daily insolation at the top of the atmosphere

If the Sun rises above the horizon on a particular day, we can compute the times of sunrise (HA_sr) and sunset (HA_ss) using Equation 2.21, since both events correspond to a solar zenith angle of 90°:

H A_{SR , SS} = \pm \arccos (- \tan φ \tan δ)

(2.24)

The length of the day (LOD) is then given by

L O D = \frac{24}{π} \arccos (- \tan φ \tan δ)

(2.25)

The coefficient 24/ $π$ is used to convert the result from radians for HA to hours for LOD.

At the equator, since $φ$ is equal to zero, LOD is always equal to 12 hours. At the equinoxes, $δ$ is equal to zero so LOD is equal to 12 hours everywhere. We can also estimate from Eq. 2.25 that in the polar regions of the Northern Hemisphere, where $φ$ + $δ$ $\geq$ 90^o, tan $φ$ tan $δ$ $\geq$ 1 in summer , the Sun is visible during the whole day (midnight sun) while where $φ$ - $δ$ $\geq$ 90^o, tan $φ$ tan $δ$ $\leq$ -1 in winter, the Sun is always below the horizon (polar night). Similar formulae can be obtained for the Southern Hemisphere.

Using Eqs 2.18, 2.20, and 2.21, 2.24, we can integrate S_h over time to compute S_{h, day} the daily insolation on a horizontal surface (in J m^-2). The mean insolation over one day (in W m^-2) is also often used. It is simply S_{h, day} divided by 24 hours

\begin{array}{rcl} S_{h, day} & = & S_{0} \frac{r_{m}^{2}}{r_{}^{2}} \int_{Sunrise}^{Sunset} (\sin φ \sin δ + \cos φ \cos δ \cos H A) d t \\ = & S_{0} \frac{r_{m}^{2}}{r_{}^{2}} \frac{86400}{2 π} \int_{H A_{S R}}^{H A_{S S}} (\sin φ \sin δ + \cos φ \cos δ \cos H A) d H A \\ = & S_{0} \frac{r_{m}^{2}}{r_{}^{2}} \frac{86400}{π} (H A_{S S} \sin φ \sin δ + \cos φ \cos δ \sin H A_{S S}) \end{array}

(2.26)

**Figure 2.11:** Mean daily insolation (in W m^-2) on a horizontal surface at the top of the atmosphere as a function of the day of the year and the latitude. White areas correspond to the polar night.

As expected, the daily insolation is higher in the summer hemisphere because of the lower zenith distance (i.e. a Sun higher above the horizon) and longer duration of the day. Averaged over one day, the maximum solar insolation occurs at the poles at the summer solstice (see Fig. 2.11), while averaged over one year the solar energy received at the top of the atmosphere at the equator is about twice that received at the poles.