[Alferd]

Here is an interesting abstract of a research paper on contrail formation

This study examines how jet aircraft contrails develop precipitation trails, using data collected on 12 May, 1996 during SUCCESS. The DC-8 sampled the precontrail conditions, produced a contrail largely in clear air at -52°C, and sampled the contrail and developing trails for over an hour. The environment was highly ice-supersaturated, reaching nearly water saturation in some locations. Inside the contrail core, almost all ice particles remained small (∼ 1 to 10 μm) due to high crystal concentrations (∼ 101 to 102 cm-3) which reduced the vapor density to saturation. Mixing of moist environmental air and vapor-depleted contrail air produced localized regions of supersaturation along the contrail periphery, where crystals grew to several hundred microns at about 0.1 μm s-1. These particles could then fall from the contrail into the vapor-rich, undepleted, supersaturated environment below. As heavier crystals left the contrail, others moved into the regions of ice supersaturation. Precipitation trails developed as this process continued over time.

ETA left the link out:

Growth of ice crystals in a precipitating contrail : SUCCESS, 2