Atmospheric Transmission Beer's Law

3/31/10Optical thickness and transmittance of clouds• Note that N and r are difficult to measure, but we can measure or estimatethe cloud water density ρ w (typically 0.1-1 g m -3 )• For a monodisperse cloud, the cloud water density is N times the mass ofwater in each droplet:ρ w= N 4 3 πr 3 ρ l• where ρ l is the density of pure water (~1000 kg m -3 )• This gives us a new expression for the volume extinction coefficient:€ β e= NQ eπr 2 = k eρ w= k eN 4 3 πr 3 ρ l€• And solving for k e gives:k e= 3Q e4ρ lrNote inversedependence on r!€Rain and fogr ≈ 1 mmk e = 1.5 m 2 kg -1ρ w = 0.1 g m -3Q e ≈ 2 at visible wavelengthsk e= 3Q e4ρ lrr ≈ 10 µmk e = 150 m 2 kg -1ρ w = 0.1 g m -3β e = 0.15 km -1β e= k eρ wβ e = 15 km -1€€28