HINT: You can do this proof by induction. Assume that the truth of Eq. (5.12) for arbitrary N implies the truth of it for 2N using Eq. (5.9), the general rule for finding the reflectivity of 2N plates given that for N plates. We know that Eq. (5.12) is true for N = 2 and N = 4.

5.2. In Section 5.1.3 we determine the mean free path for absorption (scattering). What is the root-mean-square free path? Determine the nth-root-mean free path, that is, (xn)1/n. What is the median free path? What is the most probable free path?

5.3. To convince yourself that the denominator in Eq. (5.67) is a consequence of multiple reflections between cloud and ground, derive this equation by summing the infinite series of such reflections. Treat the cloud as a slab with reflectivity R and transmissivity T overlying a surface with reflectivity Rs. You can check this solution by solving the equations of radiative transfer subject to suitable boundary conditions.

5.4. A magazine advertisement for Bermuda once caught our eye. The ad showed a couple in bathing suits sitting together romantically on a beach. A distinct line running parallel to the water separated bright white sand from darker pink sand (at water's edge), and the couple was sitting on the pink sand. The caption read "Does pink sand feel softer?" Don't bother answering this question but instead explain the abrupt change in color (and brightness) of the beach.

HINT: It might help to learn a bit about coral.

5.5. Estimate the wavelength of incident radiation such that a snowpack can be considered optically homogeneous. Take the snow volume fraction f to be 0.3, a typical value.

5.6. We once were asked the following question by a professor at another university: "The bottoms of thick clouds often are dark or gray. Yet scattering by cloud droplets is strongly peaked in the forward direction. Doesn't this imply that the bottoms of clouds should be bright?" Answer this question.

5.7. We once received an anguished message from a scientist interested in transmission of visible light by suspensions of particles. He stated that "as the beam marches through the sample it is assumed to be extinguished (scattering + absorption) exponentially as exp(-aexth)," where h is the sample thickness and ^ext — fCext/v, f is the volume fraction of particles in the suspension, Cext is the extinction cross section, and v is the volume of a single particle.

This scientist stated that using values of extinction for a (nonabsorbing) sphere of diameter 1.05 |m, he computed a value for aext of 0.443 mm-1 (f — 10~4). He further noted that "for our path length of 23 mm, the incident beam [according to calculations] is attenuated to about 3.8 x 10~5 of its original value (i.e., almost totally extinguished). This attenuation is far too great. We know [from laboratory observations] that a 10~4 suspension is very transparent [i.e., visible light is transmitted by it]. What are we doing wrong?"

Respond to this message. If necessary, draw upon examples in the atmosphere to help this person.

5.8. Within the simple two-stream model of multiple scattering, it is not possible to determine how the reflectivity of a cloud depends on the angle of incidence of the illumination (angle between the normal to the cloud and the incident beam). Nevertheless, on the basis of physical intuition acquired from the two-stream model you should be able to guess how the reflectivity of a cloud depends on this angle. Sketch this dependence and briefly explain your reasoning.

5.9. This problem is related to the previous one. Using a simple extension of the two-stream theory of reflection by clouds, and guided by physical intuition, you should be able do more than just make a crude sketch of cloud reflectivity versus solar zenith angle. You should be able to determine the slope of this curve and its limiting value.

5.10. Find the most general expression for the rate (per unit volume) at which radiant energy (of a given frequency) is deposited (transformed) within a medium according to the two-stream theory.

HINT: It might help to contemplate Fig. 5.10 and review the section on flux divergence (2.3).

5.11. Given the result of Problem 5.10, find the rate of transformation of radiant energy at each point of an infinite, absorbing medium according to the two-stream theory. The medium is illuminated from above. As a way of checking your result integrate this local energy transformation rate over the entire medium. You should obtain an expression you could have written down immediately without doing any integration.

5.12. This problem is related to the previous two. Pure lake or seawater (water containing no solid particle or biological organisms) is heated by absorption of solar radiation through a depth of tens of meters (see Figs. 2.2 and 5.12). Suppose that particles are added to the water. What happens to the radiative heating rate just below the surface (assuming no change in the incident solar radiation)? You may assume that the particles are nonabsorbing at solar wavelengths and that the reflectivity of the water is zero. Your first reaction to this question might be that because reflection by the water is increased by the particles, the heating rate decreases. Although this is true globally (for the medium as a whole) show either by simple mathematical or physical arguments (or some combination of both) that the local heating rate may not necessarily decrease. How might the particles change the temperature and its gradient in the water? Devise a simple experiment for testing your conclusions.

5.13. If the atmosphere were isothermal but not uniform in composition (i.e., the concentration of infrared active gases varies with height), would emission by the atmosphere to space be the same as, greater than or less than emission by the atmosphere to the ground? Here emission to space does not include radiation from the ground transmitted by the atmosphere. You may assume that the atmosphere emits only up or down.

5.14. On a sunny day we once noticed small clouds scattered over the horizon in all directions. The sun was perhaps 30 - 40° above the horizon. Clouds toward the sun were bright, but those away from the sun were darker. Moreover, it was possible to find pairs of clouds of apparent similar size, although in different directions, one of the pair dark, the other bright. We also observed that among the dark clouds (those opposite the sun) the brighter ones were also the largest. Explain. A diagram here will be helpful.

5.15. You occasionally come across the assertion that it is easier to get a sunburn on a cloudy day than on a clear day. The explanation sometimes offered for this is that clouds "transmit more ultraviolet radiation" (by "more" here presumably is meant more than on a clear day). Discuss this explanation. If you don't believe it, and can back this up with physical arguments, put forward an alternative explanation (under the assumption that the assertion is true).

5.16. Einstein's formula (see references at the end of Ch. 3) for the scattering coefficient (which he called an absorption coefficient because it determines the apparent absorption of a beam of light) of a nonabsorbing, one-phase, homogeneous medium can be written kBT (n2 - 1)2(n2 +2)2 4

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