1. Open problems in light scattering by ice particles Chris Westbrook www.met.reading.ac.uk/radar Department of Meteorology

2. Overview of ice cloud microphysics: Cirrus Particles 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 vertically pointing radar =8.6mm

3. 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

4. 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

5. Radar wavelengths 3 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

6. Lidar Wavelengths Small ice particles from 5 m (contrails) to 10mm ish (thick ice cloud) Lidar wavelengths 905nm and 1.5 m Wavenumbers k=20 to k=70,000 Big span of kR need a range of methods

7. 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 solution T-Matrix for Non-Rayleigh

8. Rayleigh scattering Applied field is uniform across the particle so have an electrostatics problem: 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?

9. 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?

10. Non-spherical shapes Rayleigh-Gans (Born) approximation: Assume monomers much smaller than wavelength (even if aggregate is comparable to ) 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, this is great because it's just a volume integral :-) (ie. essentially the Fourier transform of the density-density correlation function)

11. Guinier regime Scaling regime (kR) -2 1- (kR) 2 3 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 correct asymptotics in both limits:

12. 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?

13. 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

14. 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 2 nd derivs in E and H This doesnt seem to affect the phase function much so probably ok. only one study so far! Mano (2000) Appl. Opt.

15. 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 isnt too extreme

16. 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

17. DDA continued… 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 Model complex particle with many point dipoles Each has a dipole moment of (E j is field at j th dipole) Every dipole sees every other dipole, ie total field at the l th dipole is: applied etc.. So need a self-consistent solution for E j at every dipole - Amounts to inverting a 3N x 3N matrix A

18. DDA for ice crystal aggregates Discrete dipole calculations allow us to estimate the true non-Rayleigh factor: Rayleigh-Gans discrete dipole estimates 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

19. 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

20. 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

21. 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 ~ N 3 (the real killer) Up to kR~40 for simple shapes

22. 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) Efficiency may be poor for small - + Transition probability at boundary

23. 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)