1. (a) Derive the expression for the time it takes a drop to grow from an initial radius ri to a final radius rf by collection of cloud droplets in a cloud of liquid water content W (g m-3). Assume that for the drop sizes in this problem the terminal velocity can be obtained from v=ar where a=8000 s-1. Ignore the fallspeed of the cloud droplets, but include the collection efficiency Ec (assume it is constant).
Equation 4.14 in the text allows us to derive the following:
dr/dt = (ar) W E / (4r) where v is the terminal velocity, we have ignored v2 (it is too small to matter...i.e. v1-v2 ~ v1), and W is the water content, E is the collection efficiency, and r is the density of liquid water. This can be rearranged to
1/r dr = a W E dt/(4r), which we can integrate to
ln (rf/ri) = a W E t / (4r)
or, t = 4r ln (rf/ri) / (a W E)
(b) Find the time it takes for a drop with an initial radius of 100 mm to grow to 1 mm in a cloud with a liquid water content of 2.0 g m-3. Use a collection efficiency of 0.8. Check your units.
Plug in the following numbers: W=0.0020 kg m-3, E=0.8, a=8000 s-1, r=1000kg m-3 (the density of liquid water), rf=1000 microns, and ri=100 microns, the result is 720 seconds.
2. A concentration of Nw=100 cm-3 cloud droplets all of radius r=8.0 m m are evaporating in subsaturated environment with a relative humidity of 99.9%. Meanwhile sector plate ice crystals are growing by vapor deposition. The temperature is T = -16 oC and the pressure is p=700 mb.
(a) What is this process (droplets evaporating and ice crystals growing by deposition) called?
Bergeron-Findeisen process (sublimation/deposition)
(b) Find the supersaturation with respect to an ice surface.
From the Clausius-Clapeyron relation, the vapor pressure over water at -16 C is 1.51 mbar. The difference between saturation vapor pressure over ice versus over water is .27 mbar, so the supersaturation over ice is
0.27 mbar/(1.51)mbar = 17.9%
(c) Calculate the mass growth rate of a plate-like ice crystal in this cloud. Approximate the crystal as a disk of radius r=50 m m and thickness h=10 m m, having capacitance C=2r/p (C/4p e0 in the text). Use a diffusion coefficient of D=2.8x10-5 m2 s-1.
dM/dt = C/eo D (r at infinity - r at the interface) x supersaturation ratio
D=2.8x10-5 m2 s-1
C/eo =
8r = 4x10-4 m
r at infinity = P/RT
= 1.48x10-3 kg m-3 (have to use R(water))
supersaturation = 0.18
Plug these in to get 3.0x10-12 kg sec-1
(d) Using the volume of a disk and the density of ice ri=916 kg m-3, calculate the radius growth rate (dr/dt in mm/min) of this ice crystal. Assume the disk thickness stays constant. Show that the radius growth rate is constant.
dM/dt = 1x10-5 m (2pr)
(dr/dt) (916 kg m-3)
= 3x10-12 kg s-1
dr/dt = 1.04x10-6 m s-1 = 60 mm min-1
(e) Compare this ice crystal growth rate (dr/dt) with the condensational
growth rate of a 50 micrometer radius liquid droplet. Why is there such
a large difference for what is essentially the same process?
3. A graupel particle of 1 mm diameter is growing by riming in an updraft.
(a) Calculate the mass growth rate of the approximately spherical graupel particle in a cloud with a droplet liquid water content of W=1.0 g m-3. The graupel fallspeed is 90 cm/s. Assume a collection efficiency of 1.
From equation 4.13 (which applies for any sphere)
dM/dt = p r2 (delta v) x W x E
pr2 = 7.85x10-7
m2
v = 90 cm s-1
E = 1
dM/dt = 7.07x10-7 g s-1
(b) Calculate the latent heat release rate (J s-1) from the droplets freezing. If the graupel was not being cooled by the air flow, how fast would it warm (oC s-1)? Assume the graupel density is rg=500 kg m-3 (it is a mixture of air and ice). The heat capacity of ice is 2106 J kg-1 K-1.
latent heat = 3.3x105 J kg-1, and the mass accumulation rate is 7.1x10-7 g s-1, so the energy release is the product of these two, with appropriate conversion from grams to kg....
energy released = 3.3x10-5 x 7.1 x 10-7 x 0.001 kg g-1 = 2.4x10-4 J s-1
Using energy = m x Cp x delta T, solve for delta T,
delta T per time = 2.4x 10-4 J s-1
/ (2.62x 10-7 kg x 2106 J kg-1 K-1) =
0.44 degress per second (or 26 degrees per minute)
4. (5710 question) If the relative humidity is not changing in Problem 2, what is the number concentration of disk-shaped ice crystals with r=50 mm?