Cloud optical depth at fixed liquid-water path ($\tau \simeq 3W/(2{\rho }_w{r}_{\mathrm{eff}})$)

For spherical liquid droplets in the geometric-optics regime, the extinction efficiency is approximately2. Comparing projected extinction area with liquid-water mass gives the displayed relation. At fixed liquid-water path, smaller effective radius means larger optical depth, the basis of the Twomey effect. Absorption spectra, phase and cloud adjustments can invalidate a simple inference from this optical-depth change to net climate forcing.