Due Thursday, May 21, 2009.

A. Read Section 2.2, Section 3.1, Section 10.1

http://clouds.wikidot.com/local--files/files/Ocean-optics_RT.pdf

(whenever there is word “water” replace it with “cloud”)

Download the

http://clouds.wikidot.com/local--files/files/ez_monte.m

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

scattering.

C. Examine program on p. 41 and

http://clouds.wikidot.com/local--files/file/iso_monte.m

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

http://clouds.wdfiles.com/local--files/files/chylek95.pdf)

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.