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homework5_answers - Homework Problem Set #5 ATOC/ASEN 5235,...

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1 Homework Problem Set #5 ATOC/ASEN 5235, Fall 2010 Due Thursday, 28 October 1. 10 points. Petty 7.5. At a certain wavelength in the visible band, the optical thickness of the cloud- free atmosphere is τ * = 0.2. Determine the transmittance of sunlight at this wavelength when the sun is 10° above the horizon. Answer : t* = exp(-m τ *), m = 1/cos(80°) t* = exp(-0.2/cos(80°)) = 0.32 = 32% 2. 20 points. Petty 7.8. A ground-based radiometer operating at λ = 0.45 μ m is used to measure the solar intensity, I λ (0). For a solar zenith angle of θ =30°, I λ (0)=1.74 × 10 7 W m -2 μ m -1 sr -1 . For θ =60°, I λ (0)=1.14 × 10 7 W m -2 μ m -1 sr -1 . From this information, determine the top-of-the-atmosphere solar intensity S λ and the atmospheric optical thickness τ λ . Answer : From the chapter 7 lecture (and Petty Eq. 7.39) we know: ln(I λ ) = -m τ λ + ln(S λ ), where m is 1/cos( θ ). This is the equation for a line where ln(I λ ) is plotted vs. 1/cos( θ ); the slope is – τ λ and the intercept is ln(S λ ). τ λ = [ln(1.14 × 10 7 ) – ln(1.74 × 10 7 )] / [1/cos(30) – 1/cos(60)] = 0.5 {i.e., –slope = (y 1 -y 2 )/(x 2 -x 1 )} S λ = exp[ln(I λ )+ τ λ /cos θ ] = I λ × exp( τ λ /cos θ ) = 1.74 × 10 7 exp[0.5/cos(30)] S λ = 3.1 × 10 7 W m -2 μ m -1 sr -1 . 3. 20 points. Petty 7.9. At three different wavelengths, λ 1 , λ 2 , and λ 3 , the profile of absorption coefficient due to a certain atmospheric constituent is given by: β e (z) = k n ρ o exp[-z/H] where ρ o = 4 g/m 3 is the density of the constituent at sea level, and H = 8 km is the scale height. The wavelength-dependent mass extinction coefficients k 1 , k 2 , and k 3 , respectively, are 0.05, 0.10, and 0.15 m 2 kg -1 . Find the altitudes z n of the corresponding peaks of the absorption weighting functions W(z) for radiation incident at the top of the atmosphere with θ = 60°. Answer: Petty equation 7.60 tells us that the weighting function is maximized when: () 1 * / = H z e z μ τ For the problem here, H = 8 km and μ = cos(60°) = 0.5. The equation for τ * is 7.45: H k o ωρ = * where ω is the mixing ratio and k is the mass extinction coefficient (k
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homework5_answers - Homework Problem Set #5 ATOC/ASEN 5235,...

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