Open problems in light scattering by ice

particles

Chris Westbrook

www.met.reading.ac.uk/radarDepartment ofMeteorology

Overview of ice cloud microphysics:

CirrusParticles nucleated

at cloud top

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

verticallypointing

radar

=8.6mm

Overview of ice cloud microphysics:

thick stratiform cloud

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

Aggregation

Melting layer- 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 way

ICE

RAIN

If supercooled water droplets are present may get ‘riming’:

• Important for precipitation forecasts

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 wavelengths3 GHz=10 cm

35 GHz=8.6 mm

94 GHz=3.2 mm

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

Lidar Wavelengths

• Small ice particles from 5m (contrails) to 10mm ish (thick ice cloud)

• Lidar wavelengths 905nm and 1.5m

• Wavenumbers k=20 to k=70,000

• Big span of kR need a range of methods

Current methodology:

Radar: approximate to idealised shapes

Prolate spheroid (cigar)

Oblate spheroid (pancake)

Sphere Mie theory for both Rayleigh and Non-Rayleigh regimes

Exact Rayleigh solutionT-Matrix for Non-Rayleigh

Rayleigh scattering

2 0

ICE AIR

n n

E

Applied field is uniform across the particle so have an electrostatics problem:

ICE AIR

ice permittivity

BCs: On the particle surface:

BCs: Far away from particle

E = applied field

E (applied)

n = normal vector to ice surface

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 shapesRayleigh-Gans (Born) approximation:

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

For low-densities, Rayleigh formula is reduced bya 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 radiansSo for backscatter, each crystal contributes ~ K dv exp(i2k.r)

so,

this is great because it's just a volume integral :-)

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

Guinier regime Scaling regime

(kR)-21- (kR)23

4

Rayleigh – Gans results

Nice, but we've neglected coupling between crystals (each crystal sees only the applied field).

Westbrook, Ball, Field Q. J. Roy. Met. Soc. 132 897

fit a curve with the correctasymptotics in both limits:

Current approach for lidar:

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?

Geometric optics:

Q. is how good is G.O. at lidar wavelengths,where size parameter is finite?

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, kR~20 maximum

t

t

DH

HE

BEM

• 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 doesn’t seem to affect the phase

function much so probably ok.

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

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 kR~50 if the shape isn’t too extreme

2

2 22

11 1

1

T Tp a

T Tq b

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…

( ) ( ) ( )j applied j jk k kk j

E r E r A E r

polarisability of dipole k

Electric field at j

Applied field at j

Tensor characterising fall off of the E field from dipole k, as measured at j

( ) ( ) ( )j applied j jk k kk j

E r E r A E r

• 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:

applied

etc..

3d ( )j jp vK E

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

- Amounts to inverting a 3N x 3N matrix A

DDA for ice crystal aggregates

Discrete dipole calculations allow us to estimate the ‘true’ non-Rayleigh factor:

Rayleigh-Gans

discrete dipoleestimates

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)

3

Mean field approach to multiple scattering following the approach of Berry & Percival Optica

Acta 33 577

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

ie. multiple scattering increases with: - Polarisability of monomers via K()- Volume fraction F- Electrical size via G(kR)

so what's G(kR) ?

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

v

Leading order form for G(kR)

Rayleigh-Gans corrected by d

2

Rayleigh-Gans

Fractal scaling leads to strong clustering and a probability density of finding to crystals a distance r apart:

this means that, to first order:

ie.

This crude approximation seems to work pretty well

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

(x=r/R)

Fit breaks down as D

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 <<

• 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 WalksWell known that can use random walks to sample electrostatic potentialat a point.

For conducting particles () Mansfield et al [Phys. Rev. E 2001] havecalculated 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

- +

Transition probability at boundary

1

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)