Homework 6

Due Thursday, May 21, 2009.
A. Read Section 2.2, Section 3.1, Section 10.1
(whenever there is word “water” replace it with “cloud”)
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1. Recall that optical thickness tau= geometric depth times
extinction = c z. Discuss how optical thickness may depend on LWC or
IWC and wavelength (IR soloar) of the incoming light
2. In equation 3.1 assume that variable “l” is cloud optical thickness
(tau). Understand how we derive PDF (probability distribution
function) of optical thicknesses and its related CDF (cumulative
probability distribution function). Do not email equations to me.
3. Set detectors at different depths below the water surface (i.e.
cloud top) (say zd=0.01, 0.1, 1, 10). Plot intensity of photons
detected at these depths. Discuss the results and provide plot.

B. Read Section 4.1.4 on Henyey-Greenstein (HG) PDF for scattering.

1. Define what is PDF for HG. Is it normalized to 1. Why?
2. Derive all equations after 4.13 on page 13 (but do not include
derivations in homework answers). Done?
3. Plot CDF as a function of scattering angle. Also plot inverse of
this plot (i.e. Figure 4 on p. 14) – i.e. scattering angle as a
function of the CDF. Provide this plot. Discuss how results on
Figure 4 are directly applicable to Monte-Carlo simulations of light

C. Examine program on p. 41 and

1. Name 3 important physical processes are included in this code?
Discuss their PDFs and CDFs.

2. Plot several trajectories of photons (note that trajectories are up
or down along z-coordinate). Plot several trajectories which exit at
cloud top and track how many collisions (variable ns) contributed to
this trajectory.

3. Can you reproduce results in Table 7. Now run the code with mu0=1
but for several values of omega0. Discuss these results in terms of
reflectance (see equation 3 in this paper

4. (difficult) Modify the code to include Henyey-Greenstein
scattering instead of isotropic scattering. Set mu0=1, omega0=0.9 for
3 cases of the asymmetry parameter g=0.9, 0.5, g=0. What are the
results for these 3 cases? Discuss in terms of the upwelling
reflectance at the cloud top.

D. Derive equation for heating rates due to radiative divergence of
energy. Plot typical clear sky IR flux up, down, net, divergence, and
discuss associated heating rates. Why thin cirrus clouds are heated
in IR? Plot Sc IR fluxes and associated heating rates. Discuss
magnitude of heating at cloud top considering that it is 100m deep.

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