A scaling relation for warm-phase orographic precipitation: a Lagrangian analysis for 2D mountains

The precipitation produced by orographic clouds is the result of the interaction between nonlinear dynamical and microphysical processes. Focusing on warm-phase stable clouds, we propose that the precipitation efficiency is essentially controlled by the bulk Damköhler number of the system, i.e. the ratio of the characteristic advective to the microphysical time-scale of the cloud. These time-scales are investigated thoroughly for single air parcels along temporally highly resolved trajectories from quasi-two-dimensional numerical simulations with vertical Froude numbers between 0.38 and 2.31. Based on these results, analytical formulations for the Damköhler number of single parcels and for a bulk Damköhler number for the entire cloud are developed, which depend only on the far upstream flow properties and the cloud droplet number density. Relating the bulk Damköhler numbers to the precipitation efficiencies from the numerical simulations results in a unique scaling relation between the two non-dimensional numbers: no precipitation is observed for very small Damköhler numbers, while for large Damköhler numbers the precipitation efficiency asymptotically approaches a maximum value of about 0.8. For intermediate Damköhler numbers, the precipitation efficiency depends strongly on the microphysical properties of the clouds.