The End of N0

The Marshall-Palmer drop-size distribution (DSD) parameters N₀ and Λ have outlived their usefulness. These are graphical parameters that arose when those investigators analyzed empirical DSD data by plotting them on semi-logarithmic scales (the use of semi-log scales now appears to be cast in a stone tablet). Those parameters lack any useful physical connotation, and even lose their graphical meaning when the same notation is used in describing gamma DSDs. The “intercept parameter” N₀ has nothing to do with the intercept of a plotted gamma DSD function (whether on linear or semi-log scales), and the slope of the gamma DSD plot is not determined by the “slope parameter” Λalone.

The science is better served by reformulating the DSD functions in terms of physically meaningful parameters. For an exponential DSD function the parameters N_T, denoting the total drop number concentration, and D_m, the mass-weighted mean drop diameter, serve nicely:

n(D) = N_T Λ exp(-ΛD) = (4N_T /D_m) exp(-4D/D_m)

Statisticians recognize as the exponential probability density function (PDF), and would rarely if ever think of plotting it on semi-log scales. Use of the parameters N_T and D_m allows for more meaningful analysis and discussion, as will be illustrated by a reinterpretation of Albert Waldvogel’s famous “N₀jump” story and reformulation of an expression used in numerical cloud modeling.

For the case of a gamma DSD, a corresponding reformulation in terms of physically-meaningful parameters is

 n(D) = N_T f(µ) D^µ (D_m)^µ+1 exp(-(µ+4)/D_m) 

This requires the gamma shape parameter µ, which in one sense is a graphical parameter but in physical terms is an index of the width of the distribution; the coefficient of variation is 1/(SQRT(1 + µ). The part of n(D) following N_T is the gamma PDF. The utility of this formulation will be illustrated by application to the question of finding relationships between µ and Λ, and to other expressions used in numerical cloud modeling.

To be sure, the expressions for n(D) become a bit more cumbersome with these parameterizations. However, the advantage gained in more meaningful analyses and discussions is well worth the small effort involved.