HW4soln

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Apr 25, 2024

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SIO 172 Physics of the Atmosphere Homework #4 1. A stratocumulus cloud layer is 300 m thick, has liquid water content equal to 0.7 g m −3 , and has droplets with radius equal to 14 μm. The liquid water content and droplet radius are the same everywhere in the cloud. The scattering efficiency of the droplets at visible wavelengths is equal to 2. a) What is the optical thickness of the cloud layer at visible wavelengths? τ = (3/2) w l ∆z / ( r e ρ l ) = ( 1.5) (0.0007 kg m −3 ) (300 m) / ((14×10 −6 m) (1000 kg m −3 )) = 22.5 b) What is the number concentration of droplets in units of cm −3 ? N (4/3) π r e 3 ρ l = w l N = w l / ((4/3) π r e 3 ρ l ) = (0.0007 kg m −3 ) / ((4/3) π (14×10 −6 m) 3 (1000 kg m −3 )) N = 60.9×10 6 m −3 = 60.9 cm −3 c) Drizzle then forms in the stratocumulus cloud layer and precipitates out, which reduces the liquid water content to a value equal to 0.6 g m −3 . If the cloud droplet number concentration stays the same (new droplets are activated to replace those collected by the drizzle drops), what is the new cloud droplet radius in units of 14 μm? N = w l_new / ((4/3) π r new 3 ρ l ) = w l_old / ((4/3) π r old 3 ρ l ) w l_new / r new 3 = w l_old / r old 3 r new = r old ( w l_new / w l_old ) 1/3 = 14 μm (0.6 g m −3 / 0.7 g m −3 ) 1/3 = 13.3 μm d) What is the new optical thickness of the cloud layer at visible wavelengths? τ = (1.5) (0.0006 kg m −3 ) (300 m) / ((13.3×10 −6 m) (1000 kg m −3 )) = 20.3

2. A larger drop falls through a cloud while collecting smaller droplets along the way. The fall speed of the drop v is linearly proportional to its radius r according to the relationship below , where C = 6000 s −1 and r is in units of meters. v = C r Answer the following questions using the following assumptions: • The temperature of the cloud layer is above 0 °C everywhere • The liquid water content w l of the cloud is 1 g m −3 everywhere and at all times • The collection efficiency E c of the drop is 0.8 and does not change over time • Condensation or evaporation of water vapor on the drop is negligible • The cloud droplets are much smaller than the larger drop • The smaller cloud droplets have negligible fall speed a) Provide an equation that expresses the rate of growth of the radius of the drop in terms of the fall speed of the drop, the liquid water content of the smaller droplets, the collection efficiency, and any other necessary parameters. Use the above relationship for velocity in the equation. dr / dt = C r w l E c / (4 ρ l ) b ) Derive an equation that expresses the radius of the drop r in terms of the length of time t it has been growing through collection from an initial time when it had radius r 0 , along with any other necessary parameters. dr / dt = C r w l E c / (4 ρ l ) dr / r = C w l E c dt / (4 ρ l ) The above equation can be integrated from an initial time = 0 and radius = r 0 to a later time = t and radius = r . log( r ) − log( r 0 ) = C w l E c t / (4 ρ l ) r = r 0 exp[ C w l E c t / (4 ρ l )] c) If the initial radius of a drop is 20 μm, how much time does it take to grow to a radius of 400 μm? t = 4 ρ l log( r / r 0 ) / ( C w l E c ) t = (4) (1000 kg m −3 ) log(400 μ m / 20 μm) / ((6000 s −1 ) (0.001 kg m −3 ) (0.8)) = 2496 s t = 41.6 min

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