Open problems in light scattering by ice particles

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Open problems in light scattering by ice particles

• Chris Westbrook
• www.met.reading.ac.uk/radar

Department of Meteorology
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Overview of ice cloud microphysicsCirrus
Particles nucleated at cloud top
l8.6mm
vertically pointing radar
Growth by diffusion of vapour onto ice surface
pristine crystals
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
  microphysics
• Badly understood modelled

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Overview of ice cloud microphysicsthick
stratiform cloud
Nucleation growth of pristine
crystals (different temp humidity -gt different
habits)
ICE
Aggregation
RAIN

• Melting layer
• snow melts into rain (most rain in the UK starts
  as snow)
• if Tlt0C at ground then will precipitate as snow
• if the air near the ground is dry then may
  evaporate on the way

If supercooled water droplets are present may
get riming

• Important for precipitation forecasts

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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
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Radar wavelengths
3 GHz l10 cm
35 GHz l8.6 mm
94 GHz l3.2 mm
uniform E field across particle at 3, 35
GHz -gt Rayleigh Scattering
Applied E-field varies over particle at 94
Ghz ie. Non-Rayleigh scattering
1 mm ice particle
Applied wave (radar pulse)
kR from 0 to 5 for realistic sizes
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Lidar Wavelengths

• Small ice particles from 5mm (contrails)
• to 10mm ish (thick ice cloud)
• Lidar wavelengths 905nm and 1.5mm
• Wavenumbers k20 to k70,000
• Big span of kR ? need a range of methods

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Current methodology
Radar approximate to idealised shapes
?
Mie theory for both Rayleigh and Non-Rayleigh
regimes
Sphere
?
Prolate spheroid (cigar)
Exact Rayleigh solution T-Matrix for Non-Rayleigh
?
Oblate spheroid (pancake)
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Rayleigh scattering
Applied field is uniform across the particle so
have an electrostatics problem
BCs Far away from particle E applied field
n normal vector to ice surface
ice permittivity e 3.15
E (applied)
BCs On the particle surface
Analytic solution for spheres, ellipsoids. In
general?
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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?

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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 lt f lt 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)
so,
(ie. essentially the Fourier transform of
the density-density correlation function)
this is great because it's just a volume integral
-)
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Rayleigh Gans results
Westbrook, Ball, Field Q. J. Roy. Met. Soc. 132
897
Guinier regime
Scaling regime
4
2
1- (kR)
3
-2
(kR)
fit a curve with the correct asymptotics in both
limits
Nice, but we've neglected coupling between
crystals (each crystal sees only the applied
field).
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Current approach for lidar
Geometric optics

• Ray tracing of model particle shapes
• Hexagonal prisms
• Bullet-rosettes
• Aggregates
• etc.

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?
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Better methods FDTD

• Solves Maxwell curl equations
• Discretise to central-difference equations
• Solve using leap-frog method
• (ie solve E then H then E then H)
• Nice intuitive approach
• Very general
• But
• Need to grid whole domain and solve for E and H
  everywhere
• Some numerical dispersion
• Fixed cubic grid, so complex shapes need lots of
  points
• Stability issues
• Very computationally expensive, kR20 maximum

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BEM
only one study so far! Mano (2000) Appl. Opt.

• Boundary element methods
• Has been done for hexagonal prism crystal
• 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 doesnt seem to affect the phase function
  much so probably ok.

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T-matrix

• Expand incident, transmitted and scattered fields
  into a series of spherical vector wave functions,
  then find the relation between incident (a,b) and
  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 kR50 if the shape isnt too extreme

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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

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DDA continued

• Model complex particle with many point dipoles
• Each has a dipole moment of
  (Ej is field at jth dipole)
• 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

applied
polarisability of dipole k
etc..
Applied field at j
Tensor characterising fall off of the E field
from dipole k, as measured at j
Electric field at j
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DDA for ice crystal aggregates
Discrete dipole calculations allow us to estimate
the true non-Rayleigh factor
discrete dipole estimates
Rayleigh-Gans
Want to parameterise a multiple scattering
correction so we can map R-G curve to the real
data
based on
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volume fraction of ice (v/R ) size relative to
wavelength (kR)
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Mean field approach to multiple scattering
following the approach of Berry Percival Optica
Acta 33 577
(essentially d.d.a. with 1 dipole per crystal)
Mean-field approximation every crystal sees
same scalar multiple of applied field
ie. multiple scattering increases with -
Polarisability of monomers via K(e) - Volume
fraction F - Electrical size via G(kR)
v
so what's G(kR) ?
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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
(xr/R)
Rayleigh-Gans corrected by d
2
ie.

• This crude approximation
• seems to work pretty well
• strong clustering and fact that kR is
• fairly moderate have worked in our favour

Rayleigh-Gans
Fit breaks down as D?l
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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 ltlt l
• Takes a lot of processor time, hard to //ise
• Takes a lot of memory N3 (the real killer)
• Up to kR40 for simple shapes

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Rayleigh Random Walks

• Well known that can use random walks to sample
  electrostatic potential
• at a point.
• For conducting particles (e??) 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 (NIST)
• Efficiency may be poor for small e ?

Transition probability at boundary
-

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Conclusions

• Lots of different methods which are best?
• 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)