Assignment 6 Due Mar 1

1. Read and summarize the paper by Khain et al. (2007)
2. Growth of Drops via condensation and coalescence
   1. Consider a drop of radius 10 $\mu$m growing by condensation at $T=5$ $\textdegree$C, S = 0.1%. Find $\tau_{1}$, the time it takes for the radius to double. You may, though you don't have to, make the useful simplifying approximation (suitable for the lower atmosphere) that
      
      $$a\frac{da}{dt}\simeq10^{-10}S_{\infty}\,\left(\frac{m^{2}}{s}\right)$$
      
      
      The terminal velocity $U_{t}$ of a spherical drop of radius $a$, $a<35$ $\mu$m is given by the Stokes flow approximation. That is, we assume the drag force on the drop, $F_{drag}=6\pi a\mu U_{t}$, balances the force on it due to gravity. ( $\mu\,\left(\frac{kg}{m-s}\right)=$air molecular viscosity)
   2. Find $U_{t}\left(a\right)$

      Consider a drop of initial radius $a=15$ $\mu$m falling through a cloud of liquid water density $\rho_{l}$= 0.2 g m$^{-3}$, contained in droplets of radius $a'\ll a$. The collection efficiency $E\left(a,a'\right)=1$.

   3. Find $\tau_{2}$, the time for the drop radius to double via collection of the smaller drops.

      For $35\leq a\leq300$ $\mu$m, $U_{t}\left(a\right)\, m/s\sim8\times10^{3}a$ where $a$ is in meters.

   4. Find $\tau_{3}$, the time for the drop radius to double under the conditions of part (c) if the initial radius is $a=70\,\mu$m.
   Useful information: $D_{v}\simeq\nu=1.5\times10^{-5}$ m$^{2}$ s$^{-1}$, where $\mu=\rho_{air}$$\nu$. The density of liquid water is 1000 kg m$^{-3}$.

3. Riming Growth of Hailstones

   Consider a cylindrical hailstone of radius $R$ falling with velocity $U_{i}$ relative to the air. The air temperature is $T_{\infty}$ and the liquid water density in the cloud is $lwc$ $\left(g/m^{3}\right)$. Supercooled drops (originally at $T=T_{\infty}$) land on the hailstone and freeze in two stages.

   Stage 1: Ice is nucleated inside the drop. The drop temperature rises to $T=T_{0}$ ($T_{0}=273.15$ K) as a result of partial freezing.

   1. Find $F$, the fraction of the drop that freezes without heat loss to the hailstone or the air, in terms of $L_{f}\left(T\right)$ $\left(J/kg\right)$, the latent heat of freezing/melting, and the specific heats of ice and water $c_{i}$ and $c_{w}$. (Note: dLf/dT = cw-ci: Kirchoff's Law).
   Stage 2: The rest of the drop freezes, losing heat via conduction to the ice substrate and to the air, and losing latent heat via evaporation. The following equation (Maclin and Payne (1967), QJRMS) uses five terms to parameterize the heat balance at the hailstone surface.
   
   $$ClwcU_{i}\frac{\left\{ L_{f}+c_{w}\left(T_{\infty}-T_{0}\rig... ...L_{v}D_{v}\left(\rho_{v,s}-\rho_{v,\infty}\right)\right\} }{2R}$$
   
   
   where $C$ is a numerical constant, $T_{s}$ is the steady state hailstone surface temperature, $Re$ the Reynolds number, $Pr=\nu/\kappa$ the Prandtl number, where $\kappa$ is the thermal diffusivity, $Sc=\nu/D_{v}$ the Schmidt number which is approximately unity for air, $L_{v}$the latent heat of evaporation, and $\rho_{v,s}$ and $\rho_{v,\infty}$ the densities of water vapor at the hailstone surface and the environment.

   Assume $\rho_{v,\infty}=\frac{e_{sat}^{l}\left(T_{\infty}\right)m_{v}}{kT_{\infty}}$ (note liquid vapor pressure) and $\rho_{v,s}=\frac{e_{sat}^{l}\left(T_{s}\right)m_{v}}{kT_{s}}$ (note ice vapor pressure), where the mass of a water vapor molecule = $m_{v}$ and the Boltzmann constant = $k$

   1. Explain in words the physical significance of each of the five terms (three on the left and two on the right) in this equation.
   2. Find the relationship between $lwc$ and $T_{\infty}$ that produces the transition from dry to wet growth
   3. Find the relationship between $lwc$ and $T_{\infty}$that give the transition between evaporation and vapor growth.