### ATMOSPHERIC SCIENCES 749 RADIATION TRANSFER HOMEWORK AND EXAMS

#### {HW1, HW2,HW3,HW4, HW5,HW6, HW7, HW8, HW9, HW10opt1, HW10opt2, HW11}

MAKE A SKETCH OF THE PROBLEM SET UP WHENEVER POSSIBLE! Homework problems are intended to help you master the course material. You are strongly encouraged to work with others, but make your submitted homework a unique expression of your knowledge of the material.
A more thorough discussion of this process is given in the course description.

Example 1 of an easy to read, clear homework page. (Nice handwriting with a pen, on one side of the page only).
Example 2 of a word-processed homework page.

ASSIGNMENT 11 due Tuesday December 9, during class. Start early, this is a fairly demanding problem set.

A. Do this multiple scattering problem involving cirrus clouds and small ice crystals.

B. Redo the problem in this presentation on the aerosol indirect effect.
However, this time use the form of the reflection coefficient where the single scatter albedo is not equal to 1.
Calculate the cloud albedo as a function of effective radius and liquid water path for single scattering albedo equal to 0.999, 0.98, and 0.95.
For each case, assume that the absorption is caused by black carbon aerosol embedded in the cloud.
Calculate the absorption coefficient necessary to give each value of the single scattering albedo as a function of the liquid water path.
(Here is a hint on how to do the absorption optical depth calculation).
Comment on the likelihood of observing these absorption coefficients.
Finally, comment on how aerosol light absorption impacts the aerosol indirect effect (i.e. the increased cloud albedo because of smaller more numerous droplets).

ASSIGNMENT 10 Option 2: due Tuesday, November 25th during class.

A. Problem 9.3

B. Calculate the absorption cross section per molecule for CO2 for wavenumbers between 667 cm-1 and 669 cm-1.
Do the calculation for the following 3 conditions: i. P=1000 mb, T=296 K, ii. P=100 mb, T=296 K, iii. P=100 mb, T=217 K.
Overlay all three curves on a single graph. Explain the differences among the curves.
Use spectralcalc for the calculation of absorption cross section per molecule. Use no instrument function spectral window.

C. Calculate the absorption cross section per molecule for H2O for wavenumbers the spectral range 800 to 1100 cm-1 in the window region.
Use spectralcalc for the calculation of absorption cross section per molecule. Be sure to use a very large gas cell length, like 1000000 cm, because absorption is weak.
Use a Gaussian window instrument function with a spectral bandwidth of 1 cm-1.
Do the calculation for the following 3 conditions: i. P=1000 mb, T=296 K, ii. P=100 mb, T=296 K, iii. P=100 mb, T=217 K.
Overlay all three curves on a single graph. Explain the differences among the curves. Water vapor continuum absorption?

ASSIGNMENT 10 Option 1: due Tuesday, November 25th during class.

The purpose of this problem is to cover all the topics in chapters 8 and 9 in a very concrete way, and to learn how to compute emissivity, transmission, and radiance.
You will learn the basics of computing line by line radiation transfer, like we have used from the calculator online.

Pick your favorite molecule, and your favorite transition for this molecule. For example, CO2, and a rotational-vibrational transition.

A. Calculate from the line strength of this transition, and the line shape, the absorption cross section per molecule as a function of wavenumber (units 1/cm) in the vicinity of this transition. Assume that your molecule is in a standard atmosphere at 0 meters and calculate the pressure and Doppler broadened absorption cross section per molecule (two separate calculations). Then calculate, at 50,000 meters, the pressure and Doppler broadened absorption cross section per molecule (two separate calculations). Which broadening mechanism dominates at each level for your molecule?

B. Assume a typical partial pressure for your gas, and a 1 km layer at the two altitudes, assuming a constant temperature at that of the standard atmosphere. Calculate the transmissivity of your layer, the emissivity, and the radiance it would have at the standard temperature, as a function of wavenumber in the vicinity of the transition wavenumber.

You can use the line list browser at the spectralcalc site, or the following, to get information about the lines.

Spectroscopic information is obtained from the HITRAN website.

HAWKS manual
HITRAN database
PC version of JAVA program.

B. You will need the JAVA version of the HAWKS program for PC, MAC, UNIX, or LINUX.
This program allows you to search the HITRAN database for your molecule and its spectroscopic properties.
C. You will need the HAWKS manual.
D. Also of course will need the HITRAN database containing a huge amount of spectroscopic information on molecules.

Read the chapter with these things in mind. We will again share homework discussions afterwards.

C. Estimate the downwelling IR in the atmospheric window region using the RH and temperature in the first few kilometers. The wavenumber range is 750 cm-1 to 1300 cm-1.
First calculate the absorption cross section for water vapor using the spectralcalc site. The calculate the absorption coefficient and optical depth; the transmission and absorption,
and finally the emissivity and blackbody radiance for the average temperature. The product of the emissivity and blackbody radiance is the approximate downwilling IR in this range.

ASSIGNMENT 9 due Tuesday, November 18th during class.

Join either the FTIR or Solar Groups; Analyze the data from measurements on 11/6/2008. Describe what the data shows.
For example, how did the temperature of the boundary layer develop? How and why did the spectrum of sunlight change from early morning to afternoon?
How did the FTIR spectrum evolve under clear and cloudy sky conditions? Be creative; don't limit yourself to just these suggestions. Each group will make a presenation
on 18 Nov.

RESULTS: IR, SOLAR, WEATHER SUMMARY.

ASSIGNMENT 8 due Tuesday, November 4th before class.

1. Do the problem discussed in this presentation (calculation of weighting functions for ground-based FTIR measurements in the CO2 and O3 regions).
This is a deceptively simple problem; one where you should start early, and be creative in your solution.

2. Problems 8.8, 8.9, and 8.10.

MIDTERM: Learn a lot about the aerosol indirect effect by doing the problem in this presentation.

ASSIGNMENT 7 due October 14th before class.

1. Do the problem discussed at the end of this presentation.

2. Problems 7.1, 7.2 (note: this is only the direct beam transmission; no diffuse radiation is considered), 7.10, 7.13.

ASSIGNMENT 6 due October 7th before class.

1. Students must go through these online tutorials on microwave active and passive remote sensing:

a). Microwave remote sensing Introduction, b). Applications, c). Advanced Applications.

2. Problems 6.12, 6.15, 6.28.

3. Climate sensitivity calculation. Use Eq. 6.35 for the radiative equilibrium surface temperature for this problem. Calculate the derivatives dTs/dasw and dTs/dalw and estimate their values. Comment on the relative impacts of changes in the short wave and long wave absorption in the atmosphere to surface temperature changes. What does this model predict for the surface temperature when the atmosphere becomes totally opaque with respect to IR radiation (alw=1)? Interpret these results.

ASSIGNMENT 5 due September 30th before class.

Problems 5.2, 5.5, 6.1, 6.2, 6.5

ASSIGNMENT 4 due September 23rd before class.

1. Download the latest compilation of the complex refractive index for ice from here, or locally from here. Then calculate and interpret the following:
a) Absorption coefficient.
b) Penetration depth.
c) Reflection coefficient for normal incidence.
d) Reflection coefficient for 45 degree incidence (both polarizations and the average of them).
Hint: You might wish to solve this problem first and use its results, or just use complex variables in Excel or Fortran, etc.
e) Brewster angle.

Scholars can study these papers related to wave propagation in complex media:
Ray tracing, general theory, total internal reflection, quick discussion.

2. Derive an expression for the rainbow angle of a spherical particle as a function of the refractive index (assumed real only). You can do this by solving the condition db/dTheta = 0 (see figures 4.7 and 4.8.) Here b is the impact parameter equal to the sphere radius * sin(thetai). Use your solution to discuss features of the rainbow including for water (rain drops or cloud droplets) why we see color, and why the rainbow is bright. What condition on the refractive index must happen so that the rainbow angle is associated with exact backscattering?

Extra Credit: (due any time during the semester).
3. Repeat problem 1 for water (refractive index table , and fortran code for the same).

4. Repeat problem 2, though for ice and atmospheric halos and sun dogs.

ASSIGNMENT 3 due September 16th before class.

1. Do this Sunphotometer related problem.

2. Sunphotometer measurements.

You will also set up an account so you can more effectively use this atmospheric radiation transfer site.
To subscribe for this site, do the following:
1. Go to the webpage http://www.spectralcalc.com/spectralcalc.php.
2. Click on the subscribe link in the upper left corner, and then subscribe as a new user. Save your login name and password.
3. Email the company at this address, spectralcalc@gats-inc.com, and tell them the following in your message:
My name is _____.
I am a student at the University of Nevada Reno.
We have a university wide license for this site.
Thank you.

Reference Notes:

Solar Zenith Angle: You will need this for the Langley plot as discussed below.
(see this image for a visual definition, taken from this site that also discusses it).

When the sun is directly overhead the solar zenith angle is zero degrees.

You can use this convenient website to calculate the solar zenith angle for your location. It has UNR as one option in its coordinates.

To determine the solar zenith angle at any latitude, longitude, and time of day, first start with 1 finding your latitude and longitude. (You can use UNR's zip, 89557, or your own if you do measurements from elsewhere.) Then enter your coordinates and the time at this website, and get the cosine of the solar zenith angle.

Rayleigh Scattering Optical Depth: You will need this for the homework also.
Molecules in the atmosphere scatter light as dipole scatterers
. The scattering amount is much stronger at shorter wavelengths than at longer wavelengths.
Table of values of Rayleigh Scattering Optical Depths for the atmosphere. Top of the atmosphere solar radiation.

Specific Deliverables:
1. On as many days as possible, days with minimal clouds, use the sun photometer every 10 minutes to measure the amount of sunlight using your sun photometer, especially early in the morning and later in the day when the solar zenith angle changes rapidly with time. A data table for your sun photometer observations is available as a PDF, or as a Word doc. Be sure to line up the inlet to the LED so that the sun comes straight in, using the bulls eye target. OF COURSE, DON'T LOOK DIRECTLY AT THE SUN! Find the maximum signal at each measurement time, and record this value. Write up your procedure for doing these measurements (as if it were a manual for another person to use in learning how to run the sun photometer). You may find a number of sites on the internet that also describe sun photometer measurements, for example this one.

2. Make a graph of the natural logarithm of the voltage measurements in part 2 on the y-axis, and the air mass on the horizontal axis (air mass is 1/cos(solar zenith angle).) Do this on several days and see if you can extrapolate backwards to get the solar constant on days when the atmospheric optical depth is constant all day. You can look up the amount of sunlight at the top of the atmosphere for the particular wavelength of your instrument and use this for an absolute calibration of your instrument. This graph is called a Langley plot (see site 1, site 2, site 3).

3. Derive a slope for the Langley plot in part 2, and compare the value of the slope you get with this table of values of the clear sky optical depth from Rayleigh Scattering by molecules in the atmosphere (no aerosols, clouds, or gaseous absorption). Do this for several days, and compare your results. Is it reasonable to conclude that one day is 'cleaner' than another based on sun photometer measurements?

ASSIGNMENT 2 due September 9th before class.

Problems 2.7, 2.8, 2.9, 2.10, 2.14, 2.20. Also, define and discuss actinic flux; why is it important?

ASSIGNMENT 1 due September 2nd before class.

Read chapters 1 and 2. Problems 1.1, 1.2, 2.1, 2.2 and 2.3. On problem 2.1, add a part d) as follows. 2.1 d) What is the wavelength of a microwave oven, and why is it that wavelength? 