## Problems

6.1. To gain confidence that the N-stream equation [Eq. (6.2)] is correct (within the limits of the underlying approximations), show that it is correct for four streams. For simplicity take two directions in the downward hemisphere, two in the upward hemisphere, and the cosines in the downward direction equal in magnitude but opposite in sign to those in the upward hemisphere. Don't forget that attenuation is along a direction of propagation, which corresponds to the 2 direction only for light directed upward or downward.

6.2. Show that the Henyey-Greenstein phase function [Eq. (6.59)] is normalized.

HINT: This is perhaps done most easily by using the theorem for differentiation under the integral sign:

where f and its partial derivative are continuous and the limits of integration do not depend on a.

6.4. For the phase function for scattering by a spherical dipole (Prob. 7.18), sometimes called the Rayleigh phase function, derive in the same way that Eq. (6.64) is derived.

HINT: Cubic equations are exactly soluble.

6.5. Show that the vectors Eqs. (6.72)-(6.74) form an orthonormal, right-handed system.

6.6. Within the framework of the diffusion approximation, find the rate (volumetric) at which radiant energy is absorbed in an infinite, uniform, isotropic medium at any distance r from a point source (isotropic). A check on your solution is conservation of radiant energy: the rate of radiant energy from the source must be equal to the rate at which energy is absorbed in the entire medium.

HINT: The most difficult part of this problem is determining how to incorporate the point source as a boundary condition. Imagine a small spherical cavity to be carved out of the medium and determine the net rate at which radiant energy leaves the cavity in the limit as its radius approaches zero.

6.7. Use the result of the previous problem to determine the mean distance from the origin a photon travels (not the total path length of the photon) before it is absorbed. You can probably make a good guess at the answer without doing any calculations.

6.8. Derive the reflectivity of an infinite, plane-parallel, absorbing medium using diffusion theory and compare this reflectivity with Eq. (5.72) for the two-stream theory. By inspection of the diffusion theory reflectivity you should be able to give rough criteria (i.e., the range of medium properties) for when diffusion theory is definitely not a good approximation. After you have done so, try to obtain these criteria solely by physical arguments.

HINT: Reflectivities must be less than 1.

6.9. For what function, and only what function (of three space variables), is the directional derivative the same in all directions?

6.10. Equation (6.25) is a continuity equation for radiant energy. But it cannot be correct in general if the radiation field is explicitly time-dependent. Derive a more general form of this continuity equation. Why is it not, in general, identical in form to the continuity equation in fluid mechanics? When is it identical and why?

HINTS: Use the definition of the vector irradiance, the divergence theorem, and the same kinds of arguments used to derive the continuity equation of fluid mechanics.

6.11. The continuity equation derived in the previous problem is completely general. It does not depend on the equation of radiation transfer Eq. (6.15). To the contrary, this equation must be consistent with the continuity equation, and as it stands it is not. What time-dependent term must be added to the left side of Eq. (6.15) so that it is consistent with the general continuity equation?

6.12. Beginning with the general continuity equation obtained in Problem 6.10, derive a time-dependent diffusion equation for photons. Consider the special case of no absorption. Have you seen this equation before?

6.13. It is not necessary to solve a differential equation in order to obtain some insight from it. For example, beginning with the diffusion equation derived in Problem 6.12 for a nonabsorb-ing medium you should be able to answer the following question. Suppose that a plane-parallel medium is suddenly illuminated at its upper boundary. Everything takes time. Approximately how long after the illumination is turned on will the medium at a distance h from the boundary be illuminated? If your answer is not what might be expected at first glance, explain. HINT: This problem entails what the fluid mechanics folks call scale analysis.

6.14. Consider a cylindrical medium of radius a extending from z = 0 to to. The scattering and absorption properties of the medium are uniform and it is illuminated by a uniform and isotropic source at z = 0. The medium is surrounded by empty space, which means that no photons that leak out the sides can return. Using the diffusion approximation, find the rate at which irradiance is attenuated deep within the medium. By deep is meant that z is sufficiently large that attenuation is dominated by a single exponential term. Compare this attenuation rate with that for the same medium but infinite in lateral extent (a ^ to). Interpret your result physically. Also, consider the limiting case in which the medium is nonabsorbing.

HINT: This is an advanced problem. To solve it requires knowing how to solve partial differential equations in more than one variable using the method of separation of variables. An outline of the solution is given by Craig F. Bohren and Bruce R. Barkstrom, 1974: Theory of the optical properties of snow. Journal ofGeophysical Research, Vol. 79, pp. 4527-35.

6.15. If you assume that the area of a circle is proportional to the area of the square that circumscribes it, you can estimate the value of n by generating many points randomly in a square with side one unit long and counting the fraction of points that lie within the circle. Try it.

6.16. By computations similar to those used to obtain n in Problem 6.15 you can estimate the circumference of a circle. Does an additional source of error enter into the estimate for this problem that is absent from Problem 6.15?

6.17. To do Problems 6.20, 6.21, and 6.22 you need a correct algorithm for choosing source points at random on a circular disc. You might think that the way to do this is to choose cylindrical polar coordinates (r, p) at random. That is, choose r by picking a random number between 0 and 1 (the disc has radius 1) and p by choosing a random number between 0 and 1 and multiplying it by 2n. Try this. Plot enough points to see a pattern. It is not likely to look random. Why? Can you come up with an algorithm (or even two) that does result in a distribution of points on the disc that at least looks random?

6.18. Use the expression for a small solid angle in spherical coordinates to derive Eq. (6.80). It may be easier to derive this result by considering isotropic emission by a surface.

6.19. Derive equations Eqs. (6.82) and (6.83). Write a Monte Carlo code that demonstrates that these equations do indeed produce the correct distributions.

6.20. Redo Problem 4.63 using the Monte Carlo method to determine the (average) irradiance at any depth z in the black tube. Compare your computational results with the (approximate) analytical expression obtained in that problem.

6.21. As a variation on Problem 6.20, suppose that the walls of the tube are specularly reflecting with a reflectivity less than 100%. For simplicity take this reflectivity to be independent of direction of incidence. Again, determine the average irradiance at a depth z in the tube as a function of reflectivity.

6.22. As a variation on Problem 6.20, suppose that the walls of the tube are diffusely reflecting with a reflectivity less than 100%. For simplicity take this reflectivity to be independent of direction of incidence. Again, determine the average irradiance at a depth z in the tube as a function of reflectivity.

6.23. Within the framework of diffusion theory, find the angular dependence of the radiance reflected by an infinite, absorbing, plane-parallel medium illuminated by irradiance F0.

6.24. Within the framework of diffusion theory, find the angular dependence of the radiance reflected by a finite, nonabsorbing, plane-parallel medium overlying a medium with an absorptivity of 1 and illuminated by irradiance F0.

6.25. Suppose that you propose to measure absorption by a medium by measuring the angular dependence of the reflected radiance from it. You argue that the greater the slope of the radiance (relative to the radiance at = n/22) versus cos the more absorbing the medium. Based on inspection of the solution to the two previous diffusion theory problems, what is one of the major drawbacks to your idea?

6.26. The two-stream theory is fundamentally incapable of yielding reflected radiances. What we often do, therefore, is assume that the radiance is isotropic and hence can be obtained from the irradiance. The diffusion approximation is not quite so limited, but can at best yield only an approximate radiance of simple form. Because of these limitations of the two (similar) theories, explain why they are not likely to give good results for optically thin, negligibly absorbing media and for strongly absorbing media.

6.27. Show that within the framework of the diffusion approximation it would be possible to infer ground albedo under a completely overcast sky by measuring the (relative) slope of the curve of radiance versus cosine of the radiance zenith angle (equivalent to the angle between the downward directed normal at cloud base and the direction of the radiance as it leaves the bottom of the cloud). Assume that the clouds are nonabsorbing. This problem was inspired by a discussion on pages 653-4 of Hendrik C. van de Hulst, 1980: Multiple Light Scattering: Tables, Formulas, and Applications. Vol. 2, Academic, who in turn was inspired by work (in Russian) by K. S. Shifrin and D. A. Kozhaev.

6.28. The equation of transfer in Section 6.1.2 is for incoherent scattering. We pointed out in Section 3.4.8 that scattering in the exact forward direction by two (or more) particles is in phase regardless of their separation. Thus the theory in this chapter is not applicable to this direction. And the same is true, although perhaps not as obvious, for the exact backward direction. Even multiple scattering by a suspension of randomly distributed particles can give rise to coherent backscattering. This has essentially nothing to do with the glory (Sec. 8.4.3) because coherent backscattering can be obtained with particles too small to yield the glory. What is special about the backward direction such that multiply scattered waves can interfere constructively for this direction regardless of the number of scatterings and the separation of the scatterers?

HINT: A sketch of various rays scattered by two or more scatterers is essential as is invoking time reversal (Sec. 1.3). A simple figure is all that is needed. For a good expository article see R. Corey, M. Kissner, and P. Saulnier, 1995: Coherent backscattering of light. American Journal ofPhysics, Vol. 63, pp. 560-4.

6.29. In Problem 6.28, what is the ratio of the radiance in the exact backscattering direction taking into account coherent backscattering relative to the radiance in this direction assuming complete incoherence? What is the error in the reflected irradiance as a result of ignoring coherent backscattering? Is this error of any consequence for irradiances given that the angular region of coherent backscattering is a few mrad.

HINT: For the second part of this problem assume that the incoherent reflected radiance is constant and that the coherent reflected radiance is constant everywhere and equal to the incoherent value except within an angle S of the backward direction, where it is equal to the value determined in the first part of this problem.

6.30. According to the two-stream equations in Section 5.2 for a nonabsorbing medium, the net irradiance (difference between upward and downward irradiances) is the same for every altitude z. Show that this is a general result for any plane-parallel, nonabsorbing medium. HINT: Use the equation satisfied by the vector irradiance and the divergence theorem.

6.31. Figure 6.15 shows Monte Carlo calculations of the relative difference in radiances at two closely-spaced frequencies in the near infrared (around 770 nm). The absorption optical thickness is 0.11 at one frequency (moderate absorption), but much smaller at the other frequency (weak absorption). Show that this relative difference for both the upward radiance at the top of the clear atmosphere and the downward radiance at the bottom is approximately equal to the negative of the absorption optical thickness. Does this result square with the detailed Monte Carlo calculations?

HINTS: No elaborate derivation is necessary. Assume negligible multiple scattering. You can derive the radiances by evaluating a path integral of radiance similar to Eq. (8.3) with the addition of absorption. Assume the uniform atmosphere approximation of Section 8.1.1. Both the scattering optical thickness and absorption optical thickness are C 1.

6.32. According to Fig. 6.20 the upward irradiance for a clear sky is approximately constant with altitude for the infrared part of the solar spectrum but increases slightly with altitude for the combined visible and ultraviolet. Try to explain this by physical arguments. You can check your intuition with the two-stream theory of Section 5.2.

6.33. Show, using the two-stream theory, that the relative values of the upward and downward radiances in Fig. 6.19 for the cloud with the smallest aspect ratio are plausible.

6.34. Explain the sudden rise in the heating rate by solar radiation at altitudes greater than about 20 km in Fig. 6.24.

6.35. We state in Section 6.3.5 that at terrestrial temperatures emission at visible and near-visible wavelengths is exceedingly small. To support this statement determine the ratio of spectral emission at the long-wavelength end of the solar spectrum (often taken to be 2.5 pm) to emission at the middle of the terrestrial spectrum (often taken to be 10 pm) at terrestrial temperatures.

6.36. Write a Monte Carlo program for investigating absorption by a cloudy sphere, large compared with the wavelength. See Problem 5.57.

6.37. Why are the Monte Carlo method upward and downward irradiances near cloud top [see Fig. 6.10 (bottom) and Fig. 6.11] greater than the incident irradiance and why does the two-stream theory not predict this?