Assignment # 2 Solutions

1. Compute the effective radiating temperature TE for Venus using the equation for planetary energy balance:   S(1-A) / 4 = sigma TE4.     On Venus the average solar flux over the surface is S=661 W/m2 and the albedo A = 0.8  (sigma = 5.67 X 10 -8 W/m2/K4  for all planets).  (Remember you can compute a number to the 1/4 power by taking its square root twice.) How does your answer for TE compare to the actual average surface temperature, which is 753 K?

ANSWER:  S(1-A) / 4 = sigma TE4
                33.05 W/m2  =  5.67 X 10 -8 W/m2/K4  X TE4
                TE4 = 5.83 X 10 8 K4
                TE=  155.4 K

             Unfortunately I gave the wrong value for S. On Venus it should be 4X larger, or S=2644W/m2, which gives TE=220 K. The effective radiating temperature of Venus is colder than Earth's because the albedo of Venus is so high. With either value for S, TE is much colder than the actual temperature of 753 K. The calculation is missing the greenhouse effect.

2. Use the inverse-square law (see Fig 3-5) to compute the solar energy flux at the distance RE from the sun. The solar flux at the surface of the sun is about 63,000,000 W/m2 and the radius of the sun is 696,000 km. The Earth-sun distance RE is 149,598,000 km.

ANSWER:  S =  So (ro/r)2 = 63,000,000 W/m2 (696,000 km / 149,598,000 km)2 =  1364 W/m2

3. Describe the greenhouse effect as if to a friend in about 50 words.

ANSWER:  The greenhouse effect occurs on Earth from the presence of greenhouse gases (GHGs) in the atmosphere, such as water vapor, carbon dioxide, ozone, nitrous oxide. All things emit radiation provided their temperature is above absolute zero. GHGs allow sunlight to mostly pass through the cloud-free atmosphere. Incoming sunlight is partly absorbed by Earth's surface, which results in heating the surface. The surface in turn emits IR radiation. GHGs absorb this and other IR radiation relatively well, which heats them. GHGs in turn emit IR radiation in all directions, some of which is returned to Earth and acts as another source of heat to the surface. Clouds also have a greenhouse effect, but it is more complex (see question 5 below).

4. If the albedo of a cloudy sky is Ac and the albedo of the surface is As,  the fraction of absorbed radiation by the planet is (1-A)=(1-As)(1-Ac).
a) Compute A if As = 0.2 and Ac = 0.7

    ANSWER: Using the equation, solve for A and insert As and Ac.
    (1-A)=(1-As)(1-Ac)       add A to both sides
    1=(1-As)(1-Ac) + A   subtract (1-As)(1-Ac) from both sides
    1-(1-As)(1-Ac) = A    substitute in for As and Ac
    1-(1-.2)(1-.7) = A       compute...
    1-(.8)(.3) = A
    1-(.24) = A
    76 = A

b) Compute A if As = 0.7 and Ac = 0.7 
            ANSWER: Repeat part a) with As = .7, result is A = .91

c) Use this calculation to explain that adding clouds over a darker surface affects the planetary albedo more than adding clouds over a lighter surface.

ANSWER: Any clouds added over a surface will reduce the amount of solar radiation available for the surface to absorb by a percentage equal to the cloud albedo, Ac. Because highly reflective surfaces absorb a lower percentage of solar radiation than less reflective surfaces, the reduction in absorbed solar flux by highly reflective surfaces is smaller than for darker, more absorbent surfaces. This can be seen in the results of the calculations in parts a) and b).

5. a) During the polar winter (total darkness for several months), what is the effect on surface temperature if a low cloud is added versus when a high cloud is added to an otherwise cloud-free atmosphere?

SHORT ANSWER: Because there is no solar radiation in polar winter, the albedo properties of the clouds has no effect on temperature. The longwave effects of clouds is all that is relevant. Low clouds have a modest longwave effect because their temperature is similar to the surface. In contrast, high clouds have a  considerable longwave  effect. Because they are cold and good absorbers/emitters, their addition reduces the radiation lost to space. The atmosphere and surface must warm to compensate, and the warmer atmosphere emits more radiation, some reaching the surface.

LONG ANSWER:  Clouds absorb some shortwave and most longwave radiation that reaches them and they in turn emit longwave radiation in all directions. On a cloud-free day, longwave radiation reaching the surface is mostly from gases near the surface anyway, except in the "atmospheric window". Thus adding a cloud with at first NO CHANGE TO THE  ATMOSPHERIC TEMPERATURE, only substantially increases longwave radiation reaching the surface in the atmospheric window.

If the cloud is added near the surface, it has a relatively modest effect because near surface clouds usually have about the same temperature as the surface. From the Top Of the Atmosphere (TOA) perspective, very little change is seen on the planet! Only IR radiation from near the surface that is not strongly absorbed by the atmosphere (ie wavelengths in the "atmospheric window") can escape the planet. A low cloud "looks" radiatively like the surface, so FOUT changes little. From the TOA, the planet is still in energy balance and there is no need for warming. From the surface perspective, as explained in the previous paragraph, the cloud only alters the incoming radiation in the atmospheric window. This has a modest effect on increasing the incident energy budget, so the surface warms modestly. 

Adding a high cloud is another matter. First (1) recall that gases are not very efficient IR radiators and (2) density decreases with height. Thus without any clouds the radiation leaving the planet is not necessarily emitted from molecules up high, which are very cold. Instead radiation leaving the planet on a cloud-free day comes from all levels in the atmosphere, much of it coming from molecules that are quite a bit warmer than molecules near the level of high clouds.  In contrast clouds are very good IR radiators. Adding a high cloud essentially absorbs (blocks) the longwave radiation emitted by the surface and lower gases from escaping.

(and if you want to know even more...) Adding a high cloud disrupts planetary energy balance, causing FIN > FOUT. The planet responds by warming up so that FOUT increases to regain a balance. THE ATMOSPHERE RESPONDS BY WARMING THROUGHOUT, so more IR radiation is emitted from greenhouse gases everywhere.  The puny little high cloud emits radiation towards Earth too, but it's the increased IR from the warmer greenhouse gases everywhere that really add up.

b) Repeat this question for polar summer (24 hour sunlight for several months). Explain if there are competing effects in this case.

ANSWER: Adding a cloud in polar summer will have competing effects: a cooling due to an increase in albedo that reflects solar radiation and a warming due to the longwave effects described in part (a). Because low clouds tend to be thicker, they have a high albedo that strongly reflects solar radiation. This leads to a strong cooling effect. As noted in part (a) low clouds have a modest longwave warming effect. The net result of adding a low cloud would be a cooling effect given the strength of these two competing effects. High clouds, on the other hand, are rather thin and reflect little solar radiation. Thus, they have a weak cooling effect. However, as noted in part (a), high clouds have a strong longwave waming effect.This combined with their relatively low albedo implies that high clouds would cause a warming.