1. (a) Derive the expression for the time it takes a drop to grow from an initial radius r_i to a final radius r_f 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 E_c (assume it is constant).
 

(b) Find the time it takes for a drop with an initial radius of 100 m m 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.

2. A concentration of N_w=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?
 

(b) Find the supersaturation with respect to an ice surface.
 

(c) Calculate the mass growth rate of a plate-like ice crystal in this cloud (e.g. use equation 4.22). Approximate the crystal as a disk of radius r=50 mm and thickness h=10 mm, having capacitance C = 8re_o (the capacitance for a cylindrical disk - see problem 4.23 in the text). Use a diffusion coefficient of D=2.8x10^-5 m² s^-1.

(d) Using the volume of a disk and the density of ice r_i=916 kg m^-3, calculate the radius growth rate (dr/dt in m m/min) of this ice crystal. Assume the disk thickness stays constant. Show that the radius growth rate is constant.

(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.

(b) Calculate the latent heat release rate (J/s) 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 r_g=500 kg m^-3 (it is a mixture of air and ice). The heat capacity of ice is 2106 J kg^-1 K^-1.
 

(5710 question)

4. If the relative humidity is not changing in Problem 2, what is the number concentration of disk-shaped ice crystals with r=50 m m?