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:

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:

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:

• Single-drop fragmentation determines size distribution of raindrops

  Basically: large drops, formed by collision coallescence are unstable bags of water that split into smaller drops with this size distribution.