Physics 137 - Introduction to the Atmosphere and Weather                                                                Homework Key - Assignment #2
Revised - Fall 2002

2-QR18.  Explain how the atmosphere near the earth's surface is warmed from below.

    First, incoming radiation from the sun, the ultimate source of atmospheric energy, passes largely through the visually transparent atmosphere, either directly or by atmospheric scattering, without being absorbed. When it reaches the surface beneath it is mostly absorbed, warming the surface. This warmed surface heats the atmosphere above it by several means: (1) It radiates infrared radiation back into the atmosphere where much of it is absorbed, mostly by clouds, by water vapor and by carbon dioxide. (2) It conductively heats the air just above it which then forms thermals and heats the overlying air by convection. (3) Water which is evaporated at the surface condenses in the atmosphere to form clouds, thus releasing latent heat to the atmosphere.

2-QT1.  Explain why the bridge in the diagram is the first to become icy.

    Other than on the bridge the road is warmed by conductive heat flow from the ground beneath, making it somewhat slow to freeze even when the surrounding air drops below freezing.  The road on the bridge lacks this heat source so, because of its low heat capacity, drops to freezing almost as quickly as the air around it does.  Also the bridge cools radiatively from its underside while the roadbed does not.

2-QT10.  Which will show the greatest increase in temperature when illuminated with direct sunlight:  a plowed field or a blanket of snow?  Explain.

    A plowed field. Because of its relatively low albedo, compared with the high albedo of snow, the plowed field will absorb most of the energy in the incident sunlight while the snow will reflect most of the incident energy. Thus the plowed field will warm more quickly.

2-QT11. Why does the surface temperature often increase on a clear, calm night as a low cloud moves overhead?

    The cloud acts as a source of heat because of the infrared radiation it emits. If it transfers heat to the ground faster than it permits heat to escape the ground, then the temperature at ground level increases.

2-PE1.  Suppose that 500 grams (g) of water vapor condense to make a cloud about the size of an average room.  If we assume that the latent heat of condensation is 600 cal/g, how much heat would be released to the air?   If the total mass of air before condensation is 100 kg, how much warmer would the air be after condensation?  Assume that the air is not undergoing any pressure changes.  (Hint: Use the specific heat of air in Table 2.1, p.30.)

    Let Q_air be the heat released to the air. Then Q_air= mL = (500 g)(600 cal/g) = 300,000 cal. Hence the temperature increase of the air is DT = Q_air / (mc) = 300,000 cal / (10⁵g × 0.24 cal/g°C) = 12.5°C.

2-PE3.  Suppose, in question 2, the temperature of planet B doubles.  (a) What would be its wavelength of maximum energy emission?  (b) In what region of the electromagnetic spectrum would this wavelength be found?  (c) If the temperature of planet A remained the same, determine which planet (A or B) would now be emitting more radiation (use the Stefan-Boltzmann relationship).  Explain your answer.

    (a) l_max = 2897 µm K / T[K] = 2897 µm K / 3000 K = 0.966 µm.

    (c) Since E = (constant) × (Area) × T⁴, E_B/E_A = (Area_B /Area_A) × (T_B/T_A)⁴ = (1/2) × (2/1)⁴ = 8, i.e., planet B would now be
         emitting 8 times as much radiant energy as planet A.