Solar zenith angles

The albedo depends on surface properties—whether ocean, land, or ice—on the presence or absence of clouds, and on the zenith angle of the sun. The formulation I use is based on a detailed study by Thompson and Barron (1981). I have fitted to the results of their theory the analytical expressions contained in subroutine SWALBEDO. Figures 7-2 and 7-3 illustrate the calculated albedos for various conditions Figure 7—2 shows the variation of albedo for clear and cloudy skies over land and ocean as a function of the daily average solar zenith angle, results that were calculated using subroutine SWALBEDO. The temperature was taken to be warm enough to eliminate ice and snow. The most important parameter is cloud cover, because the difference between land and ocean is most marked... 

Fig. 7-2. The albedo as a function of the daily average solar zenith angle comparing clear and cloudy land and ocean. The temperature for these calculations was taken as +15°C to suppress ice and snow. In this formulation, the albedo is...

Fig. 7-3. The albedo as a function of temperature at a solar zenith angle of 75°, comparing clear and cloudy ocean and land. The temperature effect results from the large albedo of ice and snow. The sensitivity is small for cloud-covered land and...

Also calculates daily average solar zenith angle for use in SWALBEDO day = tvar 365.2422... 

I modify the program to simulate the effect of permanent ice at high latitudes by setting the albedo for the two highest latitudes in each hemisphere equal to 0.7, independent of season, temperature, or solar zenith angle. These values are set at the end of subroutine SWALBEDO. The modified program is listed as DAV10. 

Of more direct interest for atmospheric photochemistry is the solar flux per unit interval of wavelength. Values up to approximately 400 nm are provided by Atlas 3 (see Web site in Appendix IV) and from 400 nm on by Neckel and Labs (1984). Figure 3.12 shows the solar flux as a function of wavelength outside the atmosphere and at sea level for a solar zenith angle of 0° (Howard et al., 1960). 

FIGURE 3.14 Definition of solar zenith angle 8 at a point on the earth s surface. 

The solar zenith angle can be calculated in the following manner for any particular location (i.e., latitude and longitude), day of the year (dn), and time of day as described by Spencer (1971) and Madronich (1993). First, one needs to calculate what is known as the local hour angle (th), which is defined as the angle (in radians) between the meridian of the observer and that of the sun ... 

The second derived parameter that is needed for calculating the solar zenith angle at a particular time... 

The solar zenith angle (9) for that particular time and place is then determined from ... 

There are a number of estimates of the actinic flux at various wavelengths and solar zenith angles in the literature (e.g., see references in Madronich, 1987, 1993). Clearly, these all involve certain assumptions about the amounts and distribution of 03 and the concentration and nature (e.g., size distribution and composition) of particles which determine their light scattering and absorption properties. Historically, one of the most widely used data sets for actinic fluxes at the earth s surface is that of Peterson (1976), who recalculated these solar fluxes from 290 to 700 nm using a radiative transfer model developed by Dave (1972). Demerjian et al. (1980) then applied them to the photolysis of some important atmospheric species. In this model, molecular scattering, absorption due to 03, H20, 02, and C02, and scattering and absorption by particles are taken into account. 

Table 3.9 summarizes the solar zenith angles at latitudes of 20, 30, 40, and 50°N as a function of month and true solar time. True solar time, also known as apparent solar time or apparent local solar time, is defined as the time scale referenced to the sun crossing the meridian at noon. For example, at a latitude of 50°N at the beginning of January, two hours before the sun crosses the meridian corresponds to a true solar time of 10 a.m. from Table 3.9, the solar zenith angle at this time is 77.7°. 

For other latitudes, dates, and times, the solar zenith angle can be calculated as described by Madronich (1993) and summarized earlier. 

Figure 3.22 shows the solar angle 0 as a function of true solar time for several latitudes and different times of the year. As expected, only for the lower latitudes at the summer solstice does the solar zenith angle approach 0° at noon. For a latitude of 50°N even at the summer solstice, 0 is 27°. 

Figure 3.23 shows the diurnal variation of the solar zenith angle as a function of season for Los Angeles, which is located at a latitude of 34.1°N. Clearly, the peak solar zenith angle varies dramatically with season. 

TABLE 3.7 Actinic Flux Values F( A) at the Earth s Surface as a Function of Wavelength Interval and Solar Zenith Angle within Specific Wavelength Intervals for Best Estimate Surface Albedo Calculated by Madronich (1998) ... 

FIGURE 3.21 Calculated actinic flux centered on the indicated wavelengths at the earth s surface using best estimate albedos as a function of solar zenith angle (from Madronich, 1998). 

Figure 3.24 shows the relative changes in the total actinic flux as a function of altitude from 0 to 15 km at solar zenith angles of 20, 50, and 78° and at wavelengths of 332.5 (part a), 412.5 (part b), and 575 nm (part c), respectively. Again, since these are relative changes, these results of Peterson (1976) and Demerjian et al. (1980) are not expected to be significantly different from those that would be obtained with the Madronich (1998) actinic flux estimates. 

TABLE 3.9 Tabulation of Solar Zenith Angles (deg) as a Function of True Solar Time and Month... 

FIGURE 3.22 Effect of latitude on solar zenith angle. On the scale of true solar time, also called apparent solar time and apparent local solar time, the sun crosses the meridian at noon. The latitudes and seasons represented are as follows I, 20°N latitude, summer solstice II, 35°N latitude, summer solstice III, 50°N latitude, summer solstice IV, 20°N latitude, winter solstice V, 35°N latitude, winter solstice VI, 50°N latitude, winter solstice (from Leighton, 1961). 

FIGURE 3.23 Relation between solar zenith angle and time of day at Los Angeles, California (from Leighton, 1961). 

TABLE 3.10 Percentage Increase in the Calculated Actinic Flux at a Surface Elevation of 1.5 km Using Best Estimate Albedos as a Function of Solar Zenith Angle and Selected Wavelengths"... 

From Peterson (1976) and Demerjian et al. (1980) although these are based on actinic fluxes different from those in Table 3.7, the relative changes calculated here should be similar to those that would be derived using the model from which the data in Table 3.7 were derived the changes in the particle concentration are relative to the base case shown in Table 3.7. h Solar zenith angle. 

Calculated for typical winter conditions of a solar zenith angle of 70° and a surface reflectivity of 80% for a collimated incident light beam. 

The cloudless case shown first for a small solar zenith angle and typical summertime conditions shows an enhancement due to reflections from the surface. The cloud with an optical depth of 8 corresponds to a total of 67% transmission of the light through the cloud, but essentially all of it is diffused by the cloud and is therefore not directly transmitted light. The cloud with an optical depth of 128 only transmits a total of 9% of the light, essentially all of which is again diffuse. 

Under the typical summertime conditions, the thinner cloud shows an increase of 65% in the actinic flux above the cloud whereas the thicker cloud shows an increase of almost a factor of three, the maximum theoretically possible. This is due to scattering of diffuse light from the top of the cloud, as well as from the ground. As expected, below the thicker cloud, the total actinic flux is reduced, in this calculation, to 19% of the clear-sky value. However, for the thinner cloud of optical density 8, the actinic flux below the cloud is actually calculated to be greater than for the cloudless case. This occurs in the case of a small solar zenith angle and direct (rather than diffuse) incident light because the direct incident light is diffused as it traverses the cloud as discussed earlier for the case of the actinic flux above a Lambertian surface, conversion of a direct to diffuse source leads to an enhancement in the actinic flux. 

Figure 3.28 gives a typical measurement of the upward component of 7(01D) as well as the total value during one flight made at a solar zenith angle of 62° (Junkerman et al., 1994). The values of /(O D) are... 

---