1. Cumulus, stratus, and stratocumulus are types of low-level, or boundary layer, clouds.
a) Describe the visual appearance of each of these three cloud types
for a ground-based observer. Include characteristics that distinguish the
three cloud types (you may want to consult another text or the Web).
b) Explain the physical basis behind the appearance of these cloud types.
Consider the roles of surface heating, vertical velocity, stability, and
the widespread humidity profile.
2. Suppose a sample of marine air has an aerosol size distribution described by
dN/d(lnr) = rdN/dr = c r-b
from radii r1 = 0.01 m m to r2 = 10 m m (with no aerosol outside this range), with constants b = 2.5 and c = 0.025 cm-3m m2.5.
(a) Calculate the total number concentration of aerosols (in cm-3).
(b) Calculate the mass of aerosols per cubic meter of air (g m-3). Use a density of 2.0 g cm-3 for this aerosol material. Hint: find the mass of each aerosol radius and then integrate to find the total mass of aerosol per volume of air.
(c) Suppose all aerosol with dry radii above rc=0.03 m
m are cloud condensation nuclei (CCN) for typical cloud supersaturations.
Calculate the number concentration of CCN for this marine air.
3. (a) What is meant by the activation of a cloud condensation nuclei?
(b) Using the shape of the Kohler curves as given by
e'r/es = 1 + A(rT)-1 - B ms/r3
derive the radius at which a salt aerosol particle becomes activated. Also derive a simple expression for the supersaturation at which this occurs. Hint: Think about the point at which activation occurs on a Kohler curve and then use calculus to find a mathematical expression for this point. For a droplet in equilibrium the relative humidity is RH/100 = e'r/es.
(c) Calculate the activation radius (in (m) and supersaturation (in
percent) of a sodium chloride particle of mass ms=10-15
g. Use A = 3.25 x 10-7 m K for water at 0 oC and
B = 1.47x10-7 m3 g-1 for NaCl. You should
check your results with the Kohler curve figure from the book.
4. A concentration of N=100 cm-3 haze droplets all activate at a radius of 0.3 m m and grow in an updraft having an average supersaturation of 0.1% (S=0.001). The temperature is -15 oC and pressure is p = 800 mb, so the diffusion constant is D= 2.54x10-5 m2/s.
(a) What is the radius of the cloud droplets after 10 minutes?
(b) What is the liquid water content (g m-3) of the cloudy parcel at this time?
(c) How would the growth rate and liquid water content compare for a warmer cloud with the same average supersaturation? Explain.
5. (5710 question) In a real updraft the supersaturation is not constant because there is a competition between the supersaturation increasing as the parcel rises (and cools) and the supersaturation decreasing as the water vapor condenses onto droplets.
(a) Derive the following expression for the droplet growth rate assuming all of the cloud droplets are the same size:
dr/dt = D(r a/rl) a (t/r) - 4p /3 (DNr2)
where N is the number concentration of droplets, r is the droplet radius, t is time, r a is the density of air, r l is the density of liquid water, D is the diffusion coefficient, and a is the rate of change of the saturation mixing ratio, ws=ws(t=0)-at. Hint: start with the condensational growth equation in terms of vapor density, rather than supersaturation.
(b) Numerically integrate the above growth equation. Use a cloud base pressure of p=800 mb and temperature of T=10 oC. Obtain the constant a from assuming a linear change in saturation mixing ratio (ws) for a small change in height ((z)
a = [ws(zb) - ws(zb + D z)](vu/D z)
where zb is the cloud base height and vu is the updraft
speed, and (z=500 m. The saturated adiabatic lapse rate for these conditions
is about dT/dz= -5.0 oC/km. Assume the diffusion coefficient
is constant at D=3.0x10-5 m2/s and the air density
is constant at r a= 0.98 kg m-3.
Assume the CCN are relatively large, so the droplets activate at rather
low supersaturations, and the starting droplet radius is r0=1.0
m
m.
(c) Plot the droplet radius and supersaturation as a function of time for three cases: 1) N=100cm-3 and vu=1.0 m/s, 2) N=100 cm-3 and vu=0.4m/s, and 3) N=200 cm-3 and vu=1.0m/s. The maximum time should be tmax = D Z/vu.