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Open problems in light scattering by ice particles Chris Westbrook Department of www.met.reading.ac.uk/radar Meteorology Overview of ice cloud microphysics: Cirrus Particles nucleated l=8.6mm at cloud top vertically 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) ICE Aggregation RAIN 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 way • 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 scattering 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 2 E BCs: Far away from particle E = applied field n = normal vector to ice surface ice permittivity 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) so, (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 4 Scaling regime 1- 3 (kR) 2 (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 •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? 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 t • 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 only one study so far! BEM 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 doesn’t seem to affect the phase function much so probably T-matrix • 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 dipole) • Every dipole sees every other dipole, ie total field at the lth dipole is: So need a self-consistent solution for Ej applied 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 estimates Rayleigh-Gans Want to parameterise a multiple scattering correction so we can map R-G curve to the real data based on: 3 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 33 577 (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() - 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: (x=r/R) Rayleigh-Gans 2 corrected by d ie. 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 Dl 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 (NIST) Transition probability Efficiency may be poor for small ? 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)

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