(week9:marshall)=
# Notes on the Marshall-Palmer distribution and the Z-RR relation

Reading:  Stull chapter 8, pages 245 - 248 on radar reflectivity
rain rate and bright band.

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:

```{image} figures/marshall_size.png
```

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: 



```python
import numpy as np
from matplotlib import pyplot as plt

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
```

```python
def plot_marshall():
    Dvec = np.arange(0, 5, 0.1)  # mm
    rr_1 = marshall_dist(Dvec, 1.0)
    rr_5 = marshall_dist(Dvec, 5.0)
    rr_25 = marshall_dist(Dvec, 25.0)

    fig = plt.figure(1)
    fig.clf()
    ax1 = fig.add_subplot(111)
    ax1.semilogy(Dvec, rr_1, label="1 mm/hr")
    ax1.semilogy(Dvec, rr_5, label="5 mm/hr")
    ax1.semilogy(Dvec, rr_25, label="25 mm/hr")
    ax1.set_xlabel("Drop diameter (mm)")
    ax1.set_ylabel("$n(D)\ m^{-3}\,mm^{-1}$")
    ax1.set_title("Marshall Palmer distribution for three rain rates")
    ax1.set_ylim([0.1, 1.0e4])
    ax1.legend()
    
plot_marshall()

```

Here's an introduction to 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](https://www.nature.com/articles/nphys1385)

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