Scaling in ice crystal aggregation
Paul R. Field and Andrew J. Heymsfield
Ice particle size distributions (PSDs) can be scaled onto a single distribution for a wide range of observed conditions. This is demonstrated using data from ARM (Atmospheric Research Measurement, cirrus uncinus), TRMM (Tropical Rainfall Measurement Mission, tropical anvils) and FIRE1 (First ISCCP Regional Experiment, midlatitude cirrus) field programs. The form of the scaling is given by
where N(D,t) is the PSD as a function of particle diameter ,D ,and time, t, D1(t) is the number weighted average diameter for the PSD at time t,q is a scaling exponent and g is the scaled function.
Similar to other aggregating systems, it is seen that PSDs evolving through ice crystal aggregation are scalable when it is assumed that ice mass flux density (precipitation rate) is conserved during aggregation. This scalability occurs in spite of the fact that the numbers of component crystals in the aggregates are only of order 10 compared to other colloidal systems where aggregates can be composed of many orders of magnitude more monomers. The form of the scaled PSD (g) is exponential out to at least 9 times the number weighted mean diameter.
If power law expressions are used to relate aggregate mass and fallspeed to diameter in the usual manner (mass=a Db , fallspeed=aDb) then it can be shown that the scaling exponent satisfies the following relation q =b +b+1. Consideration of theoretical arguments allows the quartet of parameters (a , a, b, b) necessary to state the mass-diameter and fallspeed-diameter power law relations to be determined. It was also shown empirically that q is well correlated with precipitation rate. For large precipitation rates (>0.5 mm hr-1) q tends to ~3. These precipitation rates are dominated by large aggregates whose fallspeed relations are insensitive to particle size variations (i.e. b~0) and hence a minimum value of b~2 is suggested for the mass-diameter power law exponent.
Because precipitation rate is a good predictor of q it is clear that two variables, precipitation rate (or IWC given the power law relations) and D1(t), are required to provide an accurate estimate of ice particle size distributions, N(D,t). Consequently these two variables are required to make a good estimate of the radar reflectivity. Similarly the inverse problem of obtaining precipitation rate from radar measurements requires an estimate of D1(t) from dual wavelength retrievals, for example, as well as reflectivity.