Use of Mass- and Area-Dimensional Power Laws for Determining Precipitation Particle Terminal Velocities

Based on boundary layer theory and a comparison of empirical power laws relating the Reynolds and Best numbers, it was apparent that the primary variables governing a hydrometeor's terminal velocity were its mass, its area projected to the flow, and its maximum dimension. The dependence of terminal velocities on surface roughness appeared secondary, with surface roughness apparently changing significantly only during phase changes (i.e., ice to liquid). In the theoretical analysis, a new, comprehensive expression for the drag force, which is valid for both inertial and viscous-dominated flow, was derived. A hydrometeor's mass and projected area were simply and accurately represented in terms of its maximum dimension by using dimensional power laws. Hydrometeor terminal velocities were calculated by using mass- and area-dimensional power laws to parameterize the Best number, X. Using a theoretical relationship general for all particle types, the Reynolds number, Re, was then calculated from the Best number. Terminal velocities were calculated from Re. Alternatively, four Re–X power-law expressions were extracted from the theoretical Re–X relationship. These expressions collectively describe the terminal velocities of all ice particle types. These were parameterized using mass- and area-dimensional power laws, yielding four theoretically based power-law expressions predicting fall speeds in terms of ice particle maximum dimension. When parameterized for a given ice particle type, the theoretical fall speed power law can be compared directly with empirical fall speed-dimensional power laws in the literature for the appropriate Re range. This provides a means of comparing theory with observations. Terminal velocities predicted by this method were compared with fall speeds given by empirical fall speed expressions for the same ice particle type, which were curve fits to measured fall speeds. Such comparisons were done for nine types of ice particles. Fall speeds predicted by this method differed from those based on measurements by no more than 20%. The features that distinguish this method of determining fall speeds from others are that it does not represent particles as spheroids, it is general for any ice particle shape and size, it is conceptually and mathematically simple, it appears accurate, and it provides for physical insight. This method also allows fall speeds to be determined from aircraft measurements of ice particle mass and projected area, rather than directly measuring fall speeds. This approach may be useful for ice crystals characterizing cirrus clouds, for which direct fall speed measurements are difficult.