Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2017 / 2 / b / Solution

Using the adiabatic cloud droplet moments, define $N=\int n(r)dr$ and the mean volume radius by $r_v^3=N^{-1}\int r^{3}n(r)dr$. Summing the liquid mass of spherical droplets givesTherefore, with the shallow-cloud liquid-water content $w=c_{w}h$,The growth is $h^{1/3}$ at fixed number concentration, and the radius scales as $N^{-1/3}$ at fixed water content. The dimensions are consistent: $c_w$ has units $\mathrm{kg}{\mathrm{m}}^{-4}$, so $c_{w}h/({\rho }_{w}N)$ has units ${\mathrm{m}}^3$. The formula is a third-moment radius; a monodisperse distribution is not required. Fixed $N$ per volume is the shallow-cloud approximation, not exact conservation of number per parcel volume during an arbitrarily deep expansion.