EAS 6140-Thermodynamics of Atmosphere and Ocean
Fall 2007 - EXAM II
A parcel of air consisting of dry air, water vapor, and condensed liquid water is sinking,
undergoing to reversible saturated adiabatic compression. The air parcel is a closed
thermodynamic system (i.e. no mass enters or leaves the system). As the parcel sinks from p2 to
p1 (p2 < p1), compare the thermodynamic state of the parcel at p1 with its state at p2 (i.e., state
whether the following thermodynamic variables increase, decrease, or remain the same as a result
of the sinking):
1. T (temperature)__increase______________
2. wv (water vapor mixing ratio)___increase_____________
3. wl (liquid water mixing ratio)_____decrease_________
4. H (relative humidity)_____same___________
5. e (vapor pressure) _______increase________
6. TD (dewpoint temperature)______increase___________
7.  (potential temperature)_______decrease______________
8. e (equivalent potential temperature)_____same_______________
A parcel of air consists of dry air, water vapor, condensed liquid water. At initial conditions
T=-10oC and p=500 mb, ice freezing nuclei are introduced into the parcel of air, and the liquid
water in the parcel glaciates (freezes) adiabatically and isobarically (as the result of the activation
of ice freezing nuclei). Apart from the introduction of ice freezing nuclei, you may consider the
air parcel to be closed, isolated thermodynamic system (no heat or mass is added or removed
from the system). Compare the thermodynamic state of the parcel of air at 500 mb before
glaciation with its state once the parcel of air has achieved thermodynamic equilibrium after the
glaciation (i.e., state whether the following thermodynamic variables increase, decrease, or
remain the same):
9. T (temperature) ____increase______________
10. wv (water vapor mixing ratio)___decrease______________
11. wl (liquid water mixing ratio)_____decrease_______________
12. wi (ice water mixing ratio)______increase____________
13. RH (relative humidity)______decrease____________
14. (entropy) _________decrease_________________
15. If the tropical sea surface temperature increases from 299 to 301K and the atmospheric lapse
rate follows the saturated adiabat, the atmospheric temperature at 300 mb for conditions with
SST=301 K relative to SST= 299K will be:
a) > 2oC cooler
b) = 2oC cooler
c) < 2oC cooler
d) same temperature
e) < 2oC warmer
f) = 2oC warmer
g) > 2oC warmer
A parcel of air is ascending in a conditionally unstable cumulus cloud. Using the
elementary parcel theory, the vertical velocity at a height of 600 mb is determined to be
50 m/s. Describe whether the following processes would result in an increase, decrease,
or no change to the vertical velocity. Briefly explain your answers.
16. lateral entrainment of environmental air. - decrease
Lateral entrainment mixes in cooler drier air through the cloud’s lateral boundaries. This
causes the density differences between the cumulus cloud and the environment to
decrease and as a result the buoyancy force is also decreased.
17. radiative cooling at cloud top. – decrease
If we consider the cloud and the environment the cooling of the cloud would work
towards the equilibrium between the temperature of the cloud and the temperature of the
environment and that would decrease the vertical velocity of the cloud with respect to the
environment.
18. Compensating downdrafts - decrease
Downdrafts act in the opposite direction of the updraft; it will decrease the vertical
velocity.
19. Water drops remain in cloud - decrease
Water drops will decrease the vertical velocity because condensed water affects cloud
buoyancy and thus vertical accelerations by generating a downward directed drag force
equivalent to the weight of the suspended water.
If the number of aerosol particles in the Earth's atmosphere doubled (composition
remains the same), what would the impact be (increase, decrease, remain the same) on
the following. Briefly explain your answers. Note, you will get credit for correct
reasoning even if the answer is incorrect.
20-21. Concentration of cloud particles - increase
If the number of aerosols double the number of CCN is likely to double as well. This
leads to more cloud particles.
22-23. Size of cloud particles - decrease
When the number of CCN increases, the moisture available to nucleate CCN must be
distributed over a larger number of particles. Without increasing saturation the particles
will not be able to grow via diffusion. Hence the average size of the particles will
decrease in comparison to a scenario with less CCN.
24-25. Cloud cover (fraction) - increase
Cloud cover will increase because the particles will tend not to grow large enough to
precipitate out of the cloud. Hence the cloud will have a longer lifetime and cloud cover
will increase.
26-27. Cloud liquid water content - stays the same
Increasing the amount of CCN will not change the amount of cloud liquid water content.
28-29. Rainfall amount received at the Earth's surface - decrease
Increasing the aerosol loading will cause smaller sized particles that will not grow large
enough to precipitate out of the cloud. Hence precipitation will be suppressed.
30-34. Consider the following version of the first and second laws combined:
Td = cpd dT + Llvdwv - vdp
For reversible saturated adiabatic processes, is the following equation correct? Show your work.
s = - dT = - g + Llv dws
dz cpd cpd dz
Td  dq  0
c pd dT  Llvdws  vdp  0
c pd dT  Llvdws  dzq  0
dT Llvdws
g


dz
c pd dz c pd
35-39. Derive an expression for precipitation rate P (in meters of liquid water per
second) due to snow crystals of mass m (in kilograms) which are present in concentration
N (per cubic meter) and falling with a speed uT (in meters per second).
dR  aE

wl uT (R)
dt
4 l
Consider a cloud of depth h that consists of drops of constant radius r with number
concentration N.
wl 
l  4
n(r )r 3dr
 a 0 3
wl 
l 4 3
mN
r N 
a 3
a
uT wl  uT
mN
a
40. Write an expression for the mass of a single liquid cloud drop of radius r.
4
m  r 3  l
3
41. Write an expression for the liquid water mixing ratio of N drops of radius r.
wl 
l 4 3
r N
a 3
42-45. In the geometrics optics regime (Qext =2), determine an expression for the optical
depth of this cloud in terms of h, N, and r.
 ext  2 Nr 2
h
*
 ext
   ext dz  2 Nr 2h
0
46-50. The growth of a water droplet due to diffusion may be written as
r(dr/dt) = A (S-1)
where A can be considered constant. The terminal velocity of a droplet, vt, is given by
vt = kr2
where k is a constant. Derive an expression for the height above cloud base, h, of a
droplet at time t, which is growing only by condensation in a cloud with a steady updraft
velocity, w, and constant saturation ratio, S.
r
dr
 A( S  1)
dt
t
 rdr   A(S  1)dt
0
r  2 A( S  1)t
2
 dh 
cldtop
 (w vt )dt 
0
cldtop

0
( w  kr2 )dt 
cldtop
 (w  k 2 At(S  1))dt wt  Ak (S  1)t
0
2