Assignment 2
EOS340 / PHYS340, Spring 2021
J. Gemmrich
DUE: 23 February 2021, 6 pm
(hand in as single pdf-file, via Brightspace. Filename has to include course and your lastname.)
Throughout this assignment, when you need a value for the specific enthalpy of vaporization, you
can use L = 2.45× 106 J kg−1.
Question 1. More on Planet C
1. All assumptions regarding Planet C made in Assignment 1 are still valid.
(a) Compute relative humidity and plot it as a function of p. Assuming that RH = 1
corresponds to clouds, identify the pressure range of the cloud level in this profile.
(b) Compute the molar mass of the dry air md and plot it as a function of p.
(c) Compute the potential temperature (using the surface pressure as reference pressure
pr) and plot it as a function of p.
(5 points)
Question 2. Cloud droplet
1. Consider a cold cloud with an air parcel containing ice crystals and subcooled water droplets
of the same size. Briefly describe the evolution of the droplets, and justify your answer.
(3 points)
Question 3. Adiabatic change
1. A parcel of air moves adiabatically and without phase changes of water from 900 hPa (Point
A) to 700 hPa (Point B) and then to 800 hPa (Point C).
(a) At which of these three points is the relative humidity the largest, and at which point
is it the smallest? Justify your answer
(b) At which of these three points is the specific humidity the largest, and at which point
is it the smallest? Justify your answer
(4 points)
Question 4. Optical depth of a gas mixture
1. You are handed a sealed container with a sample of gas containing a mixture of CH4 and
CO2. The sides of the container are transparent to LW radiation, and are separated by a
distance of 0.2m. Light at wavelengths of 14 µm and 7.7 µm is shone through the gas sample.
CO2 absorbs at the first wavelength with an absorption cross-section of 10
−19 cm2, while
CH4 is approximately transparent. In contrast, CH4 absorbs well at the second wavelength,
with an absorption cross-section of 10−19 cm2, but CO2 is approximately transparent. It is
observed that, passing through the gas sample, 80% of the 14µm light is absorbed and 65%
of the 7.7µm light is absorbed.
(a) Assuming that both gases are uniformly distributed within the container, compute the
mass mixing ratios of the two gases and the effective molar mass and mass density
(in kg m−3) of the mixture. Hint: knowing the optical thickness of the gas, you can
compute its number density.
(b) Adding 1500 J of heat to a 1 kg sample of the mixture (with the volume held constant),
the temperature increases by 1.75K. Is this result consistent with your answer to (a)?
Justify your answer.
The specific heat capacities at constant pressure are 844 J kg−1 K−1 for CO2
and 2220 J kg−1 K−1 for methane .
(10 points)
Question 5. Altimeter on Planet C
1. Assume an altimeter is placed on Planet C. Altimeters estimate altitude from the mea-
surement of air pressure, based on the assumption that the atmosphere is in hydrostatic
equilibrium. For following calculations, the altitude estimated by the altimeter from the
pressure measurement p is denoted as za; e.g. za = 0 is reported when the air pressure
equals the mean sea level value, p = pmsl. Since the altimeter was made on Earth, let’s
assume pmsl = 1013 hPa.
(a) Show that ∫ za
0
m(z)
T (z)
dz = −R
g
ln
(
p
pmsl
)
(1)
where T (z) and m(z) are respectively the vertical profiles of temperature and effective
molar mass between z = 0 and z = za.
(b) Assuming that the composition and temperature of the air are constant between mean
ground level and altitude za, show that
za = −RT
mg
ln
(
p
pmsl
)
(2)
Furthermore, show that if p is close to pmsl:
za ' − RT
mg pmsl
(p− pmsl) (3)
Hint: you can rewrite p/pmsl to make use of the truncated Taylor series approximation
for |x| 1: ln(1 + x) ' x.
(c) Suppose the altimeter was placed on the ground when the profile data on PlanetC were
obtained (data file given on Brightspace). Assuming that Eqn. (3) is appropriate, what
altitude does the altimeter read?
(d) Suppose that at the same location a day later, the pressure has fallen to 982 hPa but
the temperature has remained the same. What altitude does the altimeter display?
(e) By definition, the precision of the altimeter δz and of the barometer δp are related by
δz
δp
=
dza
dp
(4)
Suppose you need the precision of your altitude measurements to be better than 2m.
Plot the resulting precision required in the measurement of p (in hPa) as a function of
T , from −20◦C to 40◦C, assuming that the air is perfectly dry.
(f) Repeat (e), using fixed relative humidities of 25%, 50%, 75%, and 95%. To do this,
you may calculate the molar mass of the moist air
m = md −RHes(T )
p
(md −mv) (5)
In your calculations of m, you can make the approximation p ' pmsl. Note, here md
is the molar mass of the dry air on Planet C at ground level, which you calculate in
Question1.
Based on your results, for the temperature and relative humidity range considered,
would the accuracy of the altimeter be improved more by building in a thermometer
or a hygrometer? Justify your answer.
(13 points)















































































































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