Atmospheric Transmission Beer's Law

3/31/10Mass pathτ (z) = k aw 1ρ 0∞∫ze − z ʹ′Hdz ʹ′ = k au(z)€€u(z) =∞∫zρ 1( z ʹ′ )dzʹ′• Mass path of the constituentbetween altitude z and the top-ofthe-atmosphere(TOA)• Dimensions: mass per unit area(e.g., g m -2 )• Integral over all z for water vaporgives an important atmosphericquantity: total precipitable water,water vapor burden, water vaporpath, liquid water path, cloud waterpath….(also for ice)Ice Water Path (IWP)Transmittance in an exponential atmosphere⎡⎡t(z) = exp − τ (z) ⎤⎤ ⎡⎡⎢⎢ ⎥⎥ = exp − k w a 1ρ 0H⎢⎢⎣⎣ µ ⎦⎦ ⎣⎣ µe − z H⎤⎤⎥⎥⎦⎦€Where is the absorption occurring?low atmospheric densityNo penetration of radiationt€W(z) = dt(z)dz= β e(z)µ t(z)The local rate of absorption withinthe atmosphere (W(z)) equals thelocal rate of change of transmittancefrom level z to the TOA, or the localextinction coefficient at level z timesthe transmittance from z to the TOAW(z) is called the absorptionweighting function. Its peak dependson k e at the wavelength in question,but occurs where τ(z) = 126