Open problems in light scattering by ice particles

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					 Open problems in light
scattering by ice particles

                   Chris Westbrook

Department of
Overview of ice cloud microphysics:
                                   Particles nucleated                           l=8.6mm
                                       at cloud top
   Growth by diffusion of vapour                                                pointing
 onto ice surface – pristine crystals                                            radar

            Aggregation of crystals

                                        Evaporation of ice particles (ie. no rain/snow at ground)

• Redistributes water vapour in troposphere
• Covers ~ 30% of earth, warming depends on mi
• Badly understood & modelled
      Overview of ice cloud microphysics:
             thick stratiform cloud

                                    Nucleation & growth of pristine crystals
                                 (different temp & humidity -> different habits)



If supercooled water droplets   Melting layer
are present may get ‘riming’:   - snow melts into rain (most rain in the UK starts as snow)
                                - if T<0°C at ground then will precipitate as snow
                                - if the air near the ground is dry then may evaporate on the
                                • Important for precipitation
Need for good scattering models

• Need models to predict scattering from
  non-spherical ice crystals if we want to
  interpret radar/lidar data, particularly:

  – Dual wavelength ratios  size  ice content
  – Depolarisation ratio LDR
  – Differential reflectivity ZDR}Particle shape
                                   & orientation

         + analogous quantities for lidar
                     Radar wavelengths
                                             3 GHz     35 GHz     94 GHz
                                             l=10 cm   l=8.6 mm   l=3.2 mm

~ uniform E field across particle at 3, 35 GHz
                     -> Rayleigh Scattering

                                                                       Applied E-field varies over
               1 mm ice particle                                          particle at 94 Ghz
                                                                           ie. Non-Rayleigh

                Applied wave
                (radar pulse)

                                                 kR from 0 to 5 for realistic sizes
          Lidar Wavelengths
• Small ice particles from 5mm (contrails)
  to 10mm ish (thick ice cloud)

• Lidar wavelengths 905nm and 1.5mm

• Wavenumbers k=20 to k=70,000

• Big span of kR  need a range
  of methods
           Current methodology:

Radar: approximate to idealised shapes

                                Sphere     Mie theory for both Rayleigh
                                            and Non-Rayleigh regimes

                                Prolate spheroid (cigar)

                                               Exact Rayleigh solution
                                               T-Matrix for Non-Rayleigh

                                Oblate spheroid (pancake)
                   Rayleigh scattering
Applied field is uniform across the particle so have an electrostatics problem

     0
                    E                                           BCs: Far away from
                                                                         E = applied field
                                             n = normal vector to ice surface
                                     3.15
   E (applied)

                                                          ICE AIR
 BCs: On the particle surface:   ICE   AIR                 
                                                           n    n
                 Analytic solution for spheres, ellipsoids. In general?
      Non-Rayleigh scattering

• Exact Mie expansion for spheres
• So approximate ice particle by a sphere
• Prescribe an ‘effective’ permittivity
  – Mixture theories: Maxwell-Garnett etc.
• Pick the appropriate ‘equivalent diameter’

• How do you pick equiv. D? Maximum
  dimension? Equal volume? Equal area?
                 Non-spherical shapes
                                Rayleigh-Gans (Born) approximation:

                                Assume monomers much smaller than wavelength
                                (even if aggregate is comparable to l)

                                For low-densities, Rayleigh formula is reduced by
                                a factor 0 < f < 1 because the contributions from
                                the different crystals are out of phase

Crystal at point r sees the applied field at origin shifted by k . r radians
So for backscatter, each crystal contributes ~ K dv exp(i2k.r)

                                             (ie. essentially the Fourier transform of the
                                             density-density correlation function)

                                      this is great because it's just a volume
Rayleigh – Gans resultsWestbrook, Ball, Field Q. J. Roy. Met. Soc. 132 8

Guinier regime
                                                        Scaling regime
 1-   3 (kR) 2

fit a curve with the correct
asymptotics in both limits:
                                        Nice, but we've neglected coupling
                                        between crystals (each crystal see
                                        only the applied field).
     Current approach for lidar:
Geometric optics:
Ray tracing of model particle shapes

   •Hexagonal prisms
                              measured phase functions usually find no halos.
                              surface ‘rougness’ ?
                       Is this real? And if so, at what k does it become important?

                          Q. is how good is G.O. at lidar wavelengths
                          where size parameter is finite?
            Better methods: FDTD
• Solves Maxwell curl equations                              D
                                                       H 
• Discretise to central-difference equations                 t
• Solve using leap-frog method                                    H
  (ie solve E then H then E then H…)                     E  m
• Nice intuitive approach
• Very general

• But…
    – Need to grid whole domain and solve for E and H
    – Some numerical dispersion
    – Fixed cubic grid, so complex shapes need lots of points
    – Stability issues
    – Very computationally expensive, kR~20 maximum
                               only one study so far!
                 BEM           Mano (2000) Appl. Opt.

• Boundary element methods
• Has been done for hexagonal prism
• E and H satisfy the Helmholtz equation
• Problem with sharp edges/corners of
  prism (discontinuities on boundary)
• Have to round off these edges & corners
  to get continuous 2nd derivs in E and H
• This doesn’t seem to affect the
  phase function much so probably
• Expand incident, transmitted and scattered fields   p   T11   T12   a 
  into a series of spherical vector wave functions,   q    T   T22  b 
  then find the relation between incident (a,b) and      21          
  scattered (p,q) coefficients
• Once know transition matrix T then can compute
  the complete scattered field
• Elements of T essentially 2D integrals over the
  particle surface
• Easy for rotationally symmetric particles
  (spheroids, cylinders, etc)
• But…
    – Less straightforward for arbitrary shapes
    – Numerically unstable as kR gets big

• OK up to kR~50 if the shape isn’t too extreme
  Discrete dipole approximation
• Recognise that a ‘point
  scatterer’ acts like a dipole

• Replace with an array of
  dipoles on cubic lattice

• Solve for E field at every
  point dipole  know
  scattered field
                   DDA continued…
• Model complex particle with many point dipoles
• Each has a dipole moment ofp j  3dvK ( )E j              (Ej is field at jth
• Every dipole sees every other dipole, ie total field at the lth dipole is:

                                                                               So need a self-consistent solution for Ej
                                                                               at every dipole

                                                                               - Amounts to inverting a 3N x 3N matrix A
                                                                                 E(r j )  Eapplied (r j )   A jk  k E(rk ) 
                                                                                                                          k j

                                                                             etc..                                                          polarisability
                                                                                                Applied field at j                          of dipole k
                          E(r j )  Eapplied (r j )   A jk  k E(rk ) 
                                                    k j

                                                                                 Electric field at j            Tensor characterising fall off of the E field
                                                                                                                from dipole k, as measured at j
DDA for ice crystal aggregates
 Discrete dipole calculations allow us to estimate the ‘true’ non-Rayleigh factor:

                                          discrete dipole

Want to parameterise a multiple scattering correction
so we can map R-G curve to the real data            based on:
                                                      volume fraction of ice (v/R )
                                                      size relative to wavelength (kR
Mean field approach to multiple scattering
                       following the approach of Berry & Percival Optica Acta 33

                                                (essentially d.d.a. with 1 dipole per crystal)

Mean-field approximation – every crystal sees same scalar multiple of applied

                          ie. multiple scattering increases with:
                                 - Polarisability of monomers via K()
                                 - Volume fraction F  v
                                 - Electrical size via G(kR)
                                                                 so what's G(kR)
Leading order form for G(kR)
 Fractal scaling leads to strong clustering and a probability density
 of finding to crystals a distance r apart:

     this means that, to first order:

                                                                  corrected by d

 This crude approximation
 seems to work pretty well

 -strong clustering and fact that kR is           Rayleigh-Gans
 -fairly moderate have worked in our

 favour                                             Fit breaks down as Dl
              DDA pros & cons
• Physical approach, conceptually simple
• Avoids discretising outer domain
• Can do any shape in principle

• Needs enough dipoles to
   – 1. represent the target shape properly
   – 2. make sure dipole separation << l
• Takes a lot of processor time, hard to //ise
• Takes a lot of memory ~ N3 (the real killer)

• Up to kR~40 for simple shapes
              Rayleigh Random Walks
    Well known that can use random walks to sample electrostatic potential
    at a point.

    For conducting particles () Mansfield et al [Phys. Rev. E 2001] have
    calculated the polarisability tensor using random walker sampling.

    Advantages are that require ~ no memory and easy to parallelise (each
    walker trajectory is an independent sample, so can just task farm it)

    Problems: how to extend to weak dielectrics (like ice)? Jack Douglas
                                                             Transition probability
    Efficiency may be poor for small  ?
                                                             at boundary 
                                                                            1

-                                                                +
• Lots of different methods – which are
• Computer time & memory a big problem
• Uncertain errors
• Better methods? FEM, BEM…?
• Ultimately want parameterisations for
  scattering in terms of aircraft observables
  eg. size, density etc.
• Would like physically-motivated scheme to
  do this (eg. mean-field m.s. approx etc)

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