Perform Mie calculations at one wavelength (e.g. 550 nm) for gamma distributions with effective radii in the range from 1 to 20 µu. Use the option 'pmom' to output the Legendre polynomials of the phase function. Use the tool 'phase' to calculate the phase function. Finally plot the phase functions for the various particle sizes and describe the results!

Important: Please change in mie.c line 674 to

 radmax = 5.00 * input.r_eff_max; 

and line 684 to

 dx=0.03; 

and recompile in order to speed up the calculations!

Here is an input file for a Mie calculation including a size distribution which gives the Legendre polynomials as output (please see the libRadtran user manual for detailed description of the options):

mie.inp

mie_program MIEV0     # use Mie program MIEV0 by Wiscombe
refrac water          # use refractive index of water
r_eff 10              # Specify effective radius 
wavelength 550 550    # Specify wavelength
nmom 1000             # Number of Legendre polynomials to be calculated 
distribution GAMMA 7  # Specify gamma distribution with veff=1/(GAMMA+3)
 
output_user pmom      # Output Legendre polynomials

The mie tool is executed as follows:

 mie < mie.inp > mie.out

In order to calculate the phase function of the Legendre polynomials use

phase -c -d -s 0.1 mie.out > phase.dat

To get help try

phase -h

We want to investigate the mie scattering of water droplets with the libradtran tool mie.
For that the phase function is expanded into a series of Legendre polynomials which is for practical purposes truncated after a defined number of polynomials. The output of mie depends on wavelength (here 550nm) and the droplet size spectrum (gamma distribution, effective radius) as the scattering does. The input values are defined in the mie.inp file.

We use the output of mie (the Legendre polynomials) to calculate the scattering phase function with the libradtran tool “phase” for each effective radius value (ranging from 1 to 20 µm).

We replaced the effective radius in the input file by the variable “reff” and used a shell script to call the different libradtran tools as well as a python plotting routine.

mie.sh

# shell skript
 
for reff in 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0
 
  ##ersetze Reff
  do sed 's/REFF/'$reff'/g' < mie.inp > tmp/mie$reff.inp
 
##rufe mie auf
  ./mie < tmp/mie$reff.inp > tmp/mie$reff.out
 
## rufe phase auf
  ./phase -c -d -s 0.1 tmp/mie$reff.out > tmp/phase$reff.dat
 
### plotte
  python phase_all.py
 
 
done

phase_all.py

from pylab import *
from matplotlib import pyplot
 
 
# save or show? ### dont use show if run by shell-script!!!
save=1 # to show set 0, to save set 1
 
img_filename='phase_all.png'
 
for reff in range(1,20):
 
  filename='tmp/phase'+str(reff)+'.0.dat'
 
  print 'plot ...'
 
  data=loadtxt(filename)
  mu=data[:,0]
  phasefct=data[:,1]
 
  pyplot.semilogy(mu,phasefct,label=reff)
 
 
#pyplot.title('g='+str(g)+'    # moments='+str(nrofmoments))
pyplot.xlabel('theta')
pyplot.ylabel('phasefunction')
 
 
if save==0:
   pyplot.legend()
   pyplot.show()
 
if save==1:
   print 'save plot ...'
   pyplot.savefig(img_filename)
 
 
 
exit()

The results are shown in the following plot (y-axises are logarithmic):

reff=1µm

reff=10µm

reff=20µm

reff from 1 - 20µm

As you can see, the curves show more peaks when the effective radius gets bigger. You can see backward and forward scattering, but the forward scattering dominates. In general a low number of photons is scattered to the sides. This minimum can be seen more clearly with increasing effective radius. At approximately 140° the rainbow is clearly visible especially for larger water droplets. At 125° another - but smaller - peak occurs with increasing droplet size.