Question

4. [10 pointsl Consider a two-layer model of the Earth's atmosphere. A "main" atmospheric layer of temperature Tm that is transparent to solar radiation and absorbs a frac- tion ε-0.77 of terrestrial radiation A "thin layer with temperature Tt, above this main layer that is transparent to solar radiation and absorbs a small fraction ε « l of terrestrial radiation. This layer is often called the "planetary skin." Page 2 Calculate the temperature T,. This temperature represents the coldest temperature achievable in Earth's at mosphere in the absence of dynamical effects or absorption of solar radiation by gas molecules. Explain why briefly.

Answer 1

T_0 rightarrow surface temperature T_m rightarrow main layer temperature f rightarrow absorption coefficient alpha rightarrow constant F_s (1 - A)/4 = (1 - f) alpha T^4_0 + f alpha T^4_m -(i) incoming radiation = outgoing atmosphere + Earth radiation surface. Energy balance (T) we get f alpha T^4_0 = 2 f alpha T^4_m -(ii) T_0 = 2^1/4 T_m -(iii) from (ii) & (i) we get T_0 = [F_s (1 - A)/4 alpha (1 - f/2)]^1/4 -(iv) from (iv) we get T_0 = 288k T_0 sub in (iii) we get 288k = 2^1/4 T_m 288 times 1000/2^1/4 = T_m \r\n 288 times 10^3/2^1/4 = T_m T_m = 288 times 10^3/2^1/4 T_m = 288 times 10^3/2^0.5 T_m = 288 times 10^3/1.414 T_m = 288 times 10^3/1414 times 10^-3 T_m = 288/1414 T_m = 203k scale height of 7km similarly for skin temperature is T_s = 215k considering f < < 1