Name: SN:

ATSC 301 Final – December 8, 2006

Answer all four questions and show all your work (note weights in parenthesis).Sketches are particularly helpful in assessing partial credit. If you are stumped overone part of a multipart question and need that intermediate result, make up a likelynumber and continue, explaining what you’re doing.

1. (12 points) A 100 m thick nocturnal cloud layer with a temperature of 270 K floats over a300 K surface. The volume absorption coefficient of the cloud is βa = 0.01 m−1, and thedownward longwave flux density from the air above the cloud is 100 W m−2. Assuming thatthere is no absorption between the surface and the cloud base:

(a) (1) Find the total optical depth of the cloud

(b) (6) Sketch the upward and downward fluxes at the top (ztop) and the bottom (zbase) ofthe cloud

(c) (3) Calculate the heating/cooling rate in K/hour between (z)top and zbase, assumingρ = 1 kg m−3

(d) (2) What is the total greenhouse effect of this cloud?

2. (14 points) For the situation described in question 1:

(a) (6) Find the upward (µ = 1) and downward (µ = −1) monochromatic radiance I (W m−2 µm−1 sr−1) at a height z = 75 m above cloud base and a wavelength of 10 µm,assuming that the downward radiance at cloud top is I10 µm = 4 W m−2 µm−1 sr−1 forµ = −1.

(b) (4) Suppose you are in an aircraft with an instrument that measures radiation between9.75 - 10.25 µm with a field of view of 0.01 sr, flying above the top of this cloud. Ne-glecting any absorption in the overlying atmosphere, what does the instrument measurefor:

i. The flux density ( W m−2)

ii. Brightness temperature (K)

(c) (2) For this isothermal cloud layer, write down two equations (one for µ > 0 and onefor µ < 0) that take as parameters: Iλ(zbase, µ) (where µ > 0), Iλ(ztop) (where µ < 0),and Bλ cloud and βa (both constant with height), and calculate the resulting Iλ(z, µ) asa function of height for zbase < z < ztop and any angle µ.

(d) (2) What have you assumed about the cloud structure in your answers above? Explain.

3. (12 points) A cloud radar operates at λ = 10 cm with a PRF of 400 s−1. It records areturned pulse with I = 0.8 and Q = −1, followed immediately by a pulse with I = −0.2and Q = 1.

(a) (6) Draw a phasor diagram showing the phase angles for these pulses.

(b) (4) Calculate a “first guess” radial velocity with direction, plus two other velocities thatare also consistent with this pulse pair, explaining your reasoning.

(c) (2) Explain the connection between the existence of three possible velocities from thispulse pair measurement, and the fact that the spokes on a wagon wheel look like they’regoing backwards in old movies.

Name: SN: page 2/5

4. (12 points) Sounding

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

10−1

100

101

102

103

Weighting function

pres

sure

(hP

a)Weight functions, HIRS sounder

15.14.714.514.214.13.713.4

The figure above shows the weighting functions for 6 sounder channels between 13.4−15 µm.Discuss in a couple of paragraphs:

(a) How these curves were calculated from atmospheric measurements of mixing ratio, tem-perature, and pressure.

(b) Why the 13.4 − 15 µm wavelength range was chosen for the sounder channels.

(c) Why they peak at different pressure levels.

(d) How these weighting functions, together with the sounder radiance measurements ineach channel, can be used to retrieve a vertical profile of temperature (you can chooseany of approaches we discussed in class).

Useful Equations:

Name: SN: page 4/5

Beer’s law for extinction:

dIλ

Iλ

= −κλ sρgds − κλ aρgds

= −κλeρgds (15)

(assuming ρa=ρs=ρg).

Hydrostatic equation:

dp = −ρd g dz (16)

Equation of state

p = RdρdT (17)

optical thickness:

τ(s1, s2) =∫ s2

s1

βa ds′ (18)

vertical optical thickness:

τ(z1, z2) =∫ z2

z1

βa dz′ (19)

Vertical optical thickness for downward radia-tion at the surface:

τ ↓(0, z) =∫ z

0

βa dz′ (20)

Vertical optical depth for upward radiation atthe top of the atmosphere:

τ ↑(z, zT ) =∫ zT

zβa dz′ (21)

Liebniz’ rule:

dτ ↑(z, zT )

dz= −βa dz

dτ ↓(0, z)

dz= βa dz (22)

Transmission functions (plane parallel atmo-sphere):

t↑(z, zT ) = exp(−τ ↑(z, zT )/µ)

t↓(0, z) = exp(−τ ↓(0, z)/µ) (23)

Weighting functions:

W ↑(z) =dt↑(z, zT )

dz=

βa(z)

µt↑(z, zT )

W ↓(z) = −dt↓(0, z)

dz=

βa(z)

µt↓(0, z)

(24)

Schwarzchild’s equation

dI = −I βa ds + Bλ(s) βa ds (25)

Some integrated forms of (25):

Downwelling intensity at the surface:

I↓(0, µ) = I↓(zT , µ) tT +∫ zT

0

B(z) W ↓(z, µ) dz

(26)

Upwelling intensity at the top of the atmosphere:

I↑(zT , µ) = I↑(0, µ) tT +∫ zT

0

B(z) W ↑(z, µ) dz

(27)

In terms of transmission

I↑(zT , µ) = I↑(0, µ) tT +∫ zT

0

B(z) dt (28)

In terms of the temperature profile (assuming ablack surface and no temperature jump betweensurface and bottom of the atmosphere):

I↑(zT , µ) = I↑(0, µ) +∫ zT

0

a(z)dB

dzdz (29)

Radar

Name: SN: page 5/5

Rayleigh scattering

I

I0

∝D6

λ4(30)

Returned power ∝ |k2|Z

r2(31)

where |k2| ≈ 0.2 for ice and 0.9 for liquid water.

Z =∫ ∞

0

n(D)D6dD (32)

Z-R relationship

Z = 300RR1.4 (33)

(rain, RR in mm/hr, Z in mm6/m3)

MUR =c

2 · PRF(34)

Doppler radar

∆ν =2 Mr

λ(35)

Mr max = λ · PRF/4 (36)

Useful constants:

cpd = 1004 J kg−1 K−1,σ = 5.67 × 10−8 W m−2 K−4

kb = 1.381 × 10−23 J K−1

c = 3 × 108 m s−1

h = 6.626 × 10−34 J sπ ≈ 3.14159Rd = 287 J kg−1 K−1