# 5.10. Notes on the Marshall-Palmer distribution and the Z-RR relation¶

A brief backgrounder in where Stull gets his equation 8.30 on page 247:

$Z = a_3 RR^{a_4}$

This equation works because the collision/coalescence process for rain formation produces a remarkably regular distribution of droplet sizes for different rainrates, essentially “fingerprinting” the rain rate with its drop-size distribution, which produces a unique value of the radar reflectivity.

In 1947 Marshall and Palmer published their measurements of rain drop size as a function of rain rate: These results are well fit by this equation:

$n(D) = N_0 \exp(-\Lambda D)$

where $$\Lambda=4.1 RR^{-0.21}$$ with D in mm, $$N_D$$ in $$m^{-3}\,mm^{-1}$$ and RR in $$mm/hr$$

I’ve put this equation into the following function in marshall_palmer.py:

def marshall_dist(Dvec,RR):
"""
Calcuate the Marshall Palmer drop size distribution

Input: Dvec: vector of diameters in mm
RR: rain rate in mm/hr
output: n(Dvec), length of Dvec, in m^{-3} mm^{-1}
"""
N0=8000  #m^{-3} mm^{-1}
the_lambda= 4.1*RR**(-0.21)
output=N0*np.exp(-the_lambda*Dvec)
return output


You can run it with:

python -m a301.scripts.marshall_palmer


and should see the following plot: Here’s a 2009 paper that presents the current leading contender for why the drop-size distribution behaves this way:

Will be assigned on Wednesday.

1. Integrate $$Z=\int D^6 n(D) dD$$ assuming a Marshall Palmer size distribution and show that it integrates to:

$Z \approx 300 RR^{1.5}$

with Z in $$mm^6\,m^{-3}$$ and RR in mm/hr. It’s helpful to know that:

$\int^\infty_0 x^n \exp( -a x) dx = n! / a^{n+1}$

1. Calculate $$P_r$$ for the Nexrad radar given $$|K|^2$$, Z, R, $$L_a$$.