Cloud droplet

A liquid-water particle suspended in a cloud. Number, size distribution and liquid-water content are separate properties; increasing number at fixed total water shifts the characteristic radius downward.

For droplet number density $N$ and radius distribution $n(r)$, let $r_v^3=N^{-1}\int r^{3}n(r)dr$ and $r_s^2=N^{-1}\int r^{2}n(r)dr$. The effective radius is $r_e=r_v^3/r_s^2$. With liquid-water content $w=c_{w}h$, $r_v=[3c_{w}h/(4\pi {\rho }_{w}N){]}^{1/3}$. The shape factor $k=r_v^3/r_e^3$ gives $r_s^2=k^{1/3}{r}_v^2$; these are distribution moments, not a monodisperse assumption.

The number of cloud droplets per volume in each radius interval, often represented by $n(r)dr$. Its width controls differential terminal speeds and collection opportunities; condensational growth with a common $d(r^2)/dt$ narrows relative radius differences.

The ratio of the third to the second radius moment determines projected extinction area per liquid mass. For size-independent extinction efficiency $Q$, the local particle extinction coefficient is $3Qw/(4{\rho }_w{r}_e)$. It is not generally the mean radius or the mean-volume radius. In adiabatic cloud droplet moments, the shape factor gives $r_e=r_v/k^{1/3}$.

Growth of liquid cloud droplets by collisions followed by merging. Differences in droplet fall speeds and collection efficiency affect the process. Increasing droplet number at fixed liquid-water content can reduce typical sizes and delay drizzle, but cloud mixing and water-content adjustments prevent a universal precipitation response.

If a spherical collector falls with $u_R=k_{R}R$ through constant liquid-water content $L$ and efficiency $E$, neglecting condensation and small-droplet fall speed, then $\dot{R}={\alpha }_{R}R$ with ${\alpha }_R=k_{R}LE/(4{\rho }_l)$. A uniform updraft gives $h(t)=Ut-(k_R{R}_0/{\alpha }_R)(e^{{\alpha }_{R}t}-1)$. The formula is valid only while the assumed cloud conditions hold.

A gravitational collision kernel multiplies geometric encounter area, relative fall speed, collection efficiency and coalescence efficiency. Its units are volume per time. The number of encounters also depends on the droplet size distribution. Hydrodynamic deflection, breakup and turbulence change the simple settling picture. Condensational growth alone tends to reduce relative size differences, so a precipitation-growth calculation must explain how efficient collectors arise.