Adiabatic cloud droplet number retrieval ($N\propto c_w^{1/2}{\tau }_c^3{W}^{-5/2}$)

A shallow liquid cloud with $w=c_{w}h$ has liquid-water path $W=c_w{H}^2/2$. If $N$, extinction efficiency $Q$ and shape factor $k$ are constant with height, integrating droplet extinction gives ${\tau }_c=(3/5)\pi Q{k}^{1/3}[3c_w/(4\pi {\rho }_w){]}^{2/3}{N}^{1/3}{H}^{5/3}$. Eliminating $H$ yields $N=[250\sqrt{2}/(243\pi Q^{3}k)]{\rho }_w^2{c}_w^{1/2}{\tau }_c^3/W^{5/2}$. Nonadiabatic water profiles and biased water paths strongly affect the retrieval.