Science A-30
Solution Set 3
Due: Thursday, February 28, 2008, at the beginning of lecture
3 questions: 28 pts
Formulas and constants:
Adiabatic Lapse Rate: ΔT/ΔZ = -9.8 °C /km
Ideal gas law: PV = NkT, P=pressure, V=volume, N=number of molecules, T=temperature,
k=Boltzmann’s constant=1.38×1023 J/ K
Number density: n=N/V
ppm= parts per million
Absolute temperature T [K]=273 + t [°C]
1. Basic Elements of Atmospheric Stability [9 pts]
Below is a typical vertical atmospheric profile over Los Angeles (LA).
D (19.0°C, 2000m)
2
Altitude [km]
1.5
C (22.0°C, 1500m)
B (20.3°C, 1000m)
1
0.5
A (30.0°C, 0m)
18
21
24
27
Temperature [°C]
30
(a) [2 pts] What is the environmental lapse rate ([°C /km]) between point A and point B? Is this
stable, unstable, or neutral with respect to the dry adiabatic lapse rate.
The dry adiabatic lapse rate is -9.8 °C /km. Between A and B, ΔT/ΔZ = (20.3°C – 30.0°C)/(1000m
– 0m) = -9.7°C /1 km. Since the environmental lapse rate between A and B is less than the dry
adiabatic lapse rate the atmosphere is stable (and is very close to neutral stability).
(b) [3 pts] What is the environmental lapse rate ([°C /km]) between point B and point C? Such a
situation in which the lapse rate is positive is called an inversion (between B and C, the temperature
increases with height). Does an inversion in the environmental lapse rate denote extremely stable
or extremely unstable conditions?
Between B and C, ΔT/ΔZ = (22.0°C – 20.3°C)/(1500m – 1000m) = 1.7°C /0.5 km = 3.4°C/km. An
inversion would imply extremely stable conditions
(c) [2 pts] What is the environmental lapse rate ([°C /km]) between point C and point D?
Between C and D, ΔT/ΔZ = (19.0°C – 22.0°C)/(2000m – 1500m) = -3.00°C /0.500 km =
-6.00°C/km.
(d) [2 pts] What would happen to a parcel of smog-filled air if it were lifted into the
inversion? Would it be warmer, cooler, or the same temperature as the surrounding atmosphere?
What direction will the net vertical force be?
As the parcel is lifted through the inversion it will be much colder then the environment and will
therefore be much denser then the environment. This will produce a strong downward net force
trapping the air below the inversion.
2. Air Conditions Determined by Geography and Local Weather [11 pts]
British Columbia (BC), which is Canada’s
westernmost province, directly north of Washington
State, has attracted many people. Over 20,000 people
have moved there from other parts of Canada every year,
and 13% of the country’s total population lives there.
However, most of them live in Victoria and Vancouver,
which are right by the sea, instead of the regions farther
east. Let's quantitatively examine why they move to these
coastal regions.
Things to know:
- The prevailing wind in the region is “westerly,” meaning that it blows from the west, toward the
east – that is, from the Pacific Ocean, onto the land.
- Air passing over the Pacific Ocean picks up a lot of moisture.
- There’s a big 3,000 m high mountain range to the east of Vancouver. To the west of it, the land
is green and lush. To the east, it’s arid.
40
Saturation Vapor Pressure(mb)
20
30
10
0
5
10
15
20
25
30
Temperature(°C)
Temperature (C)
(a) [2 pts] We’ll start with a moist air parcel, happily enjoying life in Vancouver harbor. The
temperature of the parcel is 15 °C, and its dew point is 7 °C. Using the Clausius-Clapeyron graph,
calculate the parcel’s humidity in the unit of [%].
From the graph, we see that the saturation vapor pressure at 15˚C is ~ 17 mb, while a 7˚C
dewpoint corresponds to a partial pressure of ~10 mb. R.H. is then 10mb/17mb = 59%.
(b) [2 pts] The westerly wind begins to blow, and the air parcel is carried up the
mountain. Assuming that dew point within the parcel remains constant, at what altitude will
clouds begin to form? Give your answer in the unit of [m].
Condensation will begin when the temperature of the air parcel equals the dew point (7 ˚C). Before
that point, the air parcel will rise and its temperature will decrease according to the dry adiabatic
lapse rate (-9.8 ˚C /km), given that the relative humidity of the parcel < 100%:
15 ˚C – (9.8 ˚C/km)(altitude) = 7˚C. Thus, altitude = 0.82 km = 820 m
(c) [1 pt] Assume the mountain tops are cloud covered (as they often are!). What’s the relative
humidity at the mountain top?
Since condensation continues to occur and the air parcel is not depleted of water vapor, R.H.
remains at 100% when the moist air parcel is rising.
(d) [4 pts] What is the air temperature and dew point of the parcel at an altitude of 3,000 m in the
unit of [°C]?
Hint: Use a moist adiabatic lapse rate of -5°C/km; assume a cloud base altitude of 0.80km. Note:
the 0.80km cloud base is not what you should have obtained in part b, but it’s reasonably close.
The air parcel will rise from 0.8 km to 3 km (a total of 2.2 km), and its temperature will decrease
(from 7 ˚C) according to the moist adiabatic lapse rate given (-5.0 ˚C /km). Since R.H. remains at
100% , the actual temperature remains equal to the dew point temperature.
T (and dew point) = 7 ˚C + (-5.0 ˚C/km)(2.2km)
= -4 ˚C
(e) [2 pts] Explain in conceptual terms why it’s so much more arid east of the mountains in 2
sentences.
The dew point temperature has decreased because the water vapor pressure has decreased due to
condensation of some of the water vapor. Cooling of the oceanic air as it’s carried up the mountain
causes lots of rain; this rain keeps Vancouver well watered. By the time the air has come down the
backside of the mountains, it’s dried out. Hence, the regions just east of Vancouver are
surprisingly arid.
3. Conditional Convective Instability [8 pts]
3. Convective Instability [8 pts]
Assume that the environmental lapse rate is -6.5 [℃/km], a dew point lapse rate is -1 [℃/km],
and a moist pseudo-adiabatic lapse rate is -5 [℃/km]. Suppose the ground temperature is 10 [℃]
and ground dew point is 7 [℃].
(a) [2 pts] Draw a graph of temperature vs. height that a completely dry air parcel will have if it is
lifted adiabatically (The environmental lapse rate is -6.5 [℃/km]). On the same graph, draw the
dew point temperature vs. height.
(b) [1 pts] What happens at the intersection of this graph? Is cloud formed? Why or why not?
No condensation is occurring. Only if there is water, there is condensation at the intersection.
(c) [2 pts] On a new graph draw temperature vs. height for an air parcel being lifted from the
ground with dew point 7 [℃] and initial temperature of 10 [℃]. On the same graph draw the
environmental temperature as a function of height.
(d) [1 pts] What happens at the intersection of this graph? Is cloud formed? Why or why not?
The air parcel follows the dry lapse rate from the surface to the intersection between the dry
lapse rate and moist lapse rate, then moist lapse rate to the intersection with environmental lapse
rate, and environmental lapse rate. The intersection is the boundary between instability (at upper
height) and stability (at lower height).
(e) [2 pts] Say we were to lift the parcel a little higher then this intersection and let go. Is the parcel
above this intersection stable, unstable, neutrally buoyant with respect to its surrounding?
The air parcel will rise because it will be warmer then its surroundings since it is cooling at a
slower rate then its environment. If it is warmer then its surroundings, it less dense and therefore
will experience a buoyancy force sending it upward.