A cloud layer has a vertical profile of ? that is quadratic in altitude z between cloud base zbase and cloud top ztop, with maximum ?mid at the midpoint of the cloud. At the base and top of the cloud, ? = 0. Let ? = C(ztop – z)(z – zbase).
a. Find the quadratic equation that describes ?(z) within the cloud layer.
b. Find an expression for total optical path ? measured vertically through the cloud layer.
c. Typical values for the above parameters for solar radiation incident on a thin cloud layer might be zbase = 1.0 km, ztop = 1.2 km, and? ?mid = 0.015 m^-1. Compute the total optical path for this case.
d. Based on your answer to (c), computer the vertical transmittance ? through the cloud.