A New Micro-physical Interpretation of the Variability of Z-R Relationships

It has been established by past studies that the variability of Z-R relationships (Z=aR^b) is related to different characteristics of raindrop size distribution (DSD). A new and simple micro-physical scheme based on the DSD formulation normalized by drop number concentration (Nt in m^-3) and drop mean diameter (Dm in mm) is proposed to explain the variation of the coefficient a and exponent b. Different from the previous studies focused on the shape parameter of the gamma function to explain b, this scheme doesn’t take any assumption of analytical function on DSD and b is determined by the correlation between log(Nt) and log(Dm), multiplied by the ratio of the standard deviation of log(Nt) to the standard deviation of log(Dm). The coefficient a depends on value of b, mean log(Nt) and mean log(Dm).

The scheme can well explain many ideal situations proposed in literature, such as 1) the number-controlled situation with b=1, 2) the size-controlled situation with b=1.63, 3) the scaled-number(Nt/Dc)-controlled situation with b=1.5 and 4) the inverse number-size situation with b=1.87. This scheme completes 5) the independent number-size situation with b=1.63, and 6) the number-controlled but inversely related to size situation with b<1. Different from the previous studies explaining b by the shape parameter (mu) of the gamma function which is a less physically meaningful parameter, this scheme suggests that b is independent to any absolute value of DSD moments and it should be involved with impacts of dominant micro-physical rain process through the correlation between Nt and Dm and the competition of their variability. For the coefficient a, the scheme predicts a small variation of a when b is close to 1 for the number-controlled situation, and a large variation of a when b is close to 1.63 for the independent number-size situation. These characteristics are consistent to the 69 power law Z–R relationships quoted by Battan (1973).

Two-year DSD data collected in the south of France is used to illustrate the impacts of Nt and Dm on the variation of Z-R relationships according to this scheme.