Returning to Fig. 3.18a, as light passes through this thin layer of air, it can be absorbed by the molecules of A. From the differential form of the Beer-Lambert law, Eq. (O), the change in light intensity, dl, on passing through this small volume of air containing the absorber A is given by dl/I a -(dl) = —kdl= -a[A]dl, where the negative sign indicates that the intensity decreases as the light passes through the sample.

It is important to note that the application of this form of the Beer-Lambert law, inherent in which are natural logarithms, means that the absorption cross section, cr, must be that to the base e.

The absolute change in intensity at a particular wavelength is thus given by dl = a(X)[A]I dl. This is just the number of photons absorbed as the light passes through the thin layer. I is the incident light, i.e., dP from Eq. (DD). If the quantum yield at this wavelength is 0(A), then using I = dP and substituting in for dP from Eq. (DD), the total number of molecules of A that photodissociate is given by

Number of A dissociating

= cf}( A) dl = 0(A)o-(A)[A] dPdl = <f)(k)cr(k)[A]dl

Since the total path length for light absorption is given by dl = dz/cos 9 (Fig. 3.f8b), this becomes

Number of A dissociating

= <HA)o-(A)[A] dz X {L(A, 9, 4>) dadoi dtdk}.

However, this represents the number of A molecules dissociating only due to light absorption in the wavelength interval d\ and only for incident light over the solid angle dco and surface area da. To obtain the total number of A dissociating, Eq. (LL) must be integrated over all wavelengths, solid angles, and surface areas

Total number of A dissociating f f f 4>(A)cr(A)[A]L(A,9,(l>)dzdadMdtdA

The first integration over the surface area a is just the volume. Rearranging, Eq. (MM) becomes

Total number of A dissociating/volume dt d[ A]

A oj

Quartz grains Coaxial quartz tubes

Quartz grains Coaxial quartz tubes

Outer dome Inner dome

[ | Interference filter

Outer dome Inner dome

Radiation shield

Radiation shield

[ | Interference filter

Photomultiplier

FIGURE 3.20 Schematic diagram of a 277 radiometer used to measure actinic fluxes (adapted from Junkermann et al., 1989).

Light intensities can be measured using flat-plate radiometers such as that in Fig. 3.17. As discussed earlier, this measures the flux through a horizontal plane, and as a result there is a difference of cos 6 between this measured irradiance and the actinic flux (Box 3.2). Another approach is to use what is sometimes referred to as a 2tt radiometer, which measures the light intensity striking half of a sphere. A typical example is shown in Fig. 3.20 (Junkermann et al., 1989). The light collector consists of coaxial quartz tubes located inside a quartz dome, such that light is collected from all directions in a hemisphere equally well (hence the "277" designation). Because light is collected from all directions within 2tt sr, the cos 9 factor does not apply. The space between the quartz tubes and the dome is filled with quartz grains to scatter the light. The scattered light is transmitted by the quartz tubes through a set of interference filters to a detector. The filter-detector combination is chosen for the particular measurement of interest, e.g., 03 or N02 photolysis. The shield is used to limit the field of view to exactly 2it sr. Two such detectors, one pointing up and one pointing down, can be used to cover the entire 477 sphere. A similar detector for aircraft use is described by Volz-Thomas et al. (1996).

b. Estimates of the Actinic Flux, F(A), at the Earth's Surface

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, f987, 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, HzO, 02, and C02, and scattering and absorption by particles are taken into account.

Madronich (1998) has calculated actinic fluxes using updated values of the extraterrestrial flux. In the 150-to 400-nm region, values from Atlas are used (see Web site in Appendix IV) whereas from 400 to 700 nm those of Neckel and Labs (1984) are used. In addition, the ozone absorption cross sections of Molina and Molina (1986) and the radiation scheme of Stamnes et al. (f988) were used. Other assumptions, e.g., the particle concentration and distribution, are the same as those of Demerjian et al. (1980).

In the "average" case for which the model calculations were carried out, absorption and scattering as the light traveled from the top of the atmosphere to the earth's surface were assumed to be due to 03 (UV light absorption with column 03 of 300 Dobson units), air molecules (scattering), and particles (scattering and absorption). The "best estimate" surface albedo varied from 0.05 in the 290- to 400-nm region to 0.15 in the 660- to 700-nm region. The surface was assumed to be what is known as an ideal "Lambert surface," meaning that it diffuses the incident light sufficiently well that it is reradiated equally in all upward directions, i.e., isotropically.

Table 3.7 gives the calculated actinic fluxes at the earth's surface as a function of zenith angle assuming the "best estimate" surface albedo. These data are plotted for six wavelength intervals as a function of zenith angle in Fig. 3.21. The initially small change in actinic flux with zenith angle as it increases from 0 to ~ 50° at a given wavelength followed by the rapid drop of intensity from 50 to 90° is due to the fact that the air mass m changes only gradually to ~ 50° but then increases much more rapidly to 9 = 90° [see Table 3.5 and Eq. (X)]. At the shorter wavelengths at a fixed zenith angle, the rapid increase in actinic flux with wavelength is primarily due to the strongly decreasing 03 absorption in this region.

The actinic fluxes calculated by Madronich (1998) for altitudes of f5, 25, and 40 km are collected in Tables 3.15 to 3.17.

c. Effects of Latitude, Season, and Time of Day on F(\)

To estimate photolysis rates for a given geographical location, one must take into account the latitude and season, as well as the time of day.

The data in Table 3.7 are representative for the average earth-sun distance characteristic of early April and October. The orbit of the earth is slightly elliptical, so that there is a small change in the earth-sun distance, which causes a small change (< 3%) in the solar flux with season. Correction factors for this seasonal variation for some dates from Demerjian et al. (1980) are given in Table 3.8. As discussed by Madronich (1993), the correction factors for solar intensity can be calculated for any other date using the following

(R()/R„ f = 1.000110 + 0.034221 cos N + 1.280 X 10"3 sin N + 7.19 X 10~4 cos 2N + 7.7 X 10"5 sin 2N, where R{) is the average earth-sun distance, Rn is the earth-sun distance on day dn as defined earlier, N = 2irdn/365 radians, and (R{)/R„)2 represents the correction factor.

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 fO a.m.; from Table 3.9, the solar zenith angle at this time is 11.7°.

To obtain the actinic flux at this time at any wavelength, one takes the fluxes in Table 3.7 listed under 78°; thus the flux in the 400- to 405-nm wavelength interval at 10 a.m. at 50°N latitude is 0.48 X 1015 photons cm-2 s-1.

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

Afternoon values of 9 are not given in Table 3.9 as the data are symmetrical about noon. Thus at a time of 2 p.m. at 50°N latitude, the flux would be the same as calculated for 10 a.m.

Figure 3.22 shows the solar angle 9 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, 9 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.

These differences in light intensity, and in its diurnal variation at different latitudes and seasons, are critical because they alter the atmospheric chemistry at various geographical locations due to the fact that photochemistry is the major source of the free radicals such as OH that drive the chemistry.

d. Effect of Surface Elevation on F( k)

The variation in the actinic flux with surface elevation is important because some of the world's major cities are located substantially above sea level. For example, Mexico City and Denver, Colorado, are at elevations of 2.2 and f .6 km, respectively.

Table 3.10 shows the calculated percentage increase in the actinic flux at the earth's surface for an elevation of f.5 km and atmospheric pressure of 0.84 atm (corresponding approximately to Denver) as a function of zenith angle for four wavelength intervals. In this calculation, it was assumed that the vertical 03 and particle

TABLE 3.7 Actinic Flux Values F(\) 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)"

Wavelength _

interval (nm) Exponent6 0 10 20

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