Climate and water
Climate and water - Homework #3
1. Radiation
a) (5 points)
Use the physical laws governing radiative heat transfer to explain why
the colors of the stars (such as the Sun) are related to their temperatures
but the colors of the planets are not. In other words why can we see the
sun (star) and Mars (planet)?
b) (5 points) Show how you would calculate the surface temperature
of a star if you knew the wavelength where its radiation is largest. What
is the Sun's surface temperature if that wavelength is 0.5012 mm?
c) (5 points) The Earth receives solar radiation with a flux
of 1370 W m-2. What is the solar flux on Mars, which is 1.542
times further away from the Sun.
d) (5 points) The Earth is about 3.5% closer to the Sun in winter
than is summer. Assuming all else stays the same, what is the implied difference
between the Earth's winter and summer effective temperatures? Which effective
temperature is higher winter or summer?
e) (5 points) Why is it that on a cloudy day, the temperature
tends to be lower than that during a clear day and why do clouds have an
opposite effect at night?
2. Atmospheric stability and moisture
a) (5 points)
Assume you have 1 kg of dry air, enclosed in a rigid container with a volume
of 1 m3. What is the pressure in the container when the temperature is
0 °C? 25°C? (Remember to convert T to °K.)
b) (5 points) Make a graph showing how the pressure of the air
inside the container (on the vertical axis) increases if the temperature
(horizontal axis) rises gradually from -20 °C to 100°C.
3. Hydrostatic balance
a) (5 points)
Write down the units of all the variables in the hydrostatic balance equation
and show that when similar terms are canceled on each side of the equation
the units on both sides are identical.
b) (5 points)
A mountaineer carries a barometer (a pressure measuring instrument) on
his climb. He notes the pressure at the beginning of his ascent [In meteorology
the pressure units used are usually Pascal (Pa) or millibar (mb) where
1 N/m2 = 1 Pa =10-2 Mb. The pressure at sea level is about 105
N/m2]. At his first rest stop, the climber takes another reading
of his barometer and finds that the pressure fell by 6000 Pa. Assuming
that the density of air is 1.225 kg/m3 (the nominal value at
sea level), how high is the climber above his starting point?
c) (5 points)
Can the climber assume that the pressure will continue to drop at the same
rate as he ascends up the mountain?
4. Temperature profiles
a) (5 points)The
table below has a set of temperature values as a function of height as
measured by a balloon carrying a temperature gauge. Make a graph of this
profile, with temperature on the horizontal axis and height on the vertical.
|
Temp
(°C)
|
Height
(km)
|
|
16
|
0
|
|
19
|
0.5
|
|
20
|
1
|
|
19
|
1.5
|
|
14
|
2
|
|
10
|
2.5
|
|
5
|
3
|
|
-15
|
4
|
|
-25
|
5
|
|
-35
|
6
|
|
-43
|
7
|
|
-50
|
8
|
|
-55
|
9
|
|
-57
|
10
|
b)
(5 points) Draw a line that shows how the temperature varies in a parcel
of air that ascends from the surface to a height of 5 km.
c)
(5 points) In the lower section of the atmosphere shown in the graph,
temperature is increasing with height (in what is called an inversion).
Does this make the lower atmosphere in this example stable or unstable?
d)
(5 points) In the middle section of the graph temperature falls 10°C
with each kilometer. Does this make this part of the atmosphere stable
or unstable?
5. Homework problem
a) (5 points)
Air with properties described in problem #D part 2 (temperature of 17°C
and 50% relative humidity) is rising up from the surface. At what elevation
will the vapor in the air begin to condense (use a value rounded up to
the nearest km)?
b) (5 points)
Assuming
a moist adiabatic lapse rate of 6.5°C/km, what will be the temperature
in the rising air at an elevation of 2km?
c) (5 points)
What will the relative humidity be at 2km?