Wave phenomena in radar meteorology

Chris Westbrook

Reflectivity, Z

- Measure of intensity reflected back to radar:

4back scatter constantZ

( )fallZ air

Z vv v

Z

back scatter cross section(intensity)

radar wavelength

Doppler radar

Measure frequency shift (the police car effect) – gives reflectivity weighted average fall speed:

Includes (unknown) contributionfrom velocity of the air (updraft/downdraft)

- can remove this effect usingtwo wavelengths (soon!)

Rayleigh (small size compared to wavelength)

2

2

4

amplitude ~ volume /

volumeintensity ~

Almost no phase difference (uniform E - field)across particle: wave blind to details, scattersIsotropically according to how much material is there

eg. Rain Radar~10cmdrop~1mm

RAL Chilbolton

Doppler velocity

RED = TOWARDS RADARBLUE = AWAY FROM RADAR

Reflectivity

RED / PURPLE = HEAVY RAIN

Strong ascending motion can be seen in the regions of heaviest precipitation.

At the tropopause, the cloud spreads out horizontally to form cirrus anvil clouds.

At the tropopause, the cloud spreads out horizontally to form cirrus anvil clouds.

EXAMPLE: THUNDERSTORM 28th JULY 2000

phase shift

ZDR 0 dB (ZH = ZV) 1 mm

3 mm

4.5 mm

ZDR = 1.5 dB (ZH > ZV)

ZDR = 3 dB (ZH >> ZV)

Differential polarisability

Bigger drops aren’t spherical, but oblate (pancake)

drop is more easily polarised in horizontal direction than in the vertical, so ZH > ZV

Look at ratio ZDR provides estimate of drop size

Differential phase shift dp

Flattened drops -

Horizontally polarised component is slowed down more than the vertically polarised component

difference in phase between H and V

Polarisation measurements

Less influenced by largest drops, but data can be noisy.Helps distinguish big rain drops from hailstones (which are spherical and so dp=0)

ICE

RAIN

ICE

RAIN

Observations

Cold front 20th October 2000

(time scale ~ 1¼ hours)

melting layer

Clear air returns

Refractive index variations in clear air can produce radar returns if length scale is /2

Waves from each layer add up constructively and in phase (like Bragg scattering in crystals) Allows you to see the boundary layer,Edges of cloudsTurbulence. insects?

Boundary layer

Bulge, indicative of storm

Ice clouds

Want to interpret observations in terms of :

how much ice is in the cloud,

how big the ice particles are,

how fast they’re falling etc…

Cover about ¼ of earth’s surface typically – important for radiation budget / climate etc.

Need to model the scattering properties of ice particles in clouds

Pristine crystals –Columns, Plates, Bullet rosettes

First fewhundred metres

AGGREGATES -(complicated!)

LowerAltitudes

Diffusion ofwater vapouronto ice

Sedimentationat different speeds

Cloud radars: wavelength and ice particle size are comparable

PARTICLE SHAPE MATTERS!

Previous studies concentrate only on pristine crystals

We want to try and model the scattering from aggregates

Timely, since next month ‘CloudSat’ 3mm radar will be in space.

Aggregation model

2rate of close approach = i j i jr r v v

relative fall speedtotal possible collision area

Mean field approach – big box of snowflakes, pick pairs to collide with probability proportional to:

Then, to get the statistics right, pick a random trajectory from possible ones encompassed byand track particles to see if they actually do collide – if so, stick them together.

UNIVERSALITY:

Statistically self-similar structure – fractal dimension of 2

Also self-similar size distribution.

Real ice aggregatesfrom a cirrus cloudIn the USA

Simulated aggregates

(aggregates of bullet rosette crystals)

Rayleigh-Gans theory (‘Born approximation’)

r

dv

k

Assume: 1. each volume element sees only the applied wave 2. the elements scatter in the same way as an equivalent volume sphere (amplitude ~ dv / 2)NO INTERACTION BETWEEN VOLUME ELEMENTS.

So just add up the scattered amplitude of each element × a phase factor exp(I 2k•r) …

2

1scattered amplitude ~ exp( 2 )d

v

i v

k r

[ Small particle limit (kr0) reduces to Rayleigh sphere formula, ie. intensity ~ volume2 / 4 ]

For small particles, wave only seesparticle volume, not particle shape. (Rayleigh regime)

Phase shift between centre of mass and element at position r is k•r

So for back-scatter the total phase difference in the scattered wave is 2k•r (ie. there and back)

If particle size and wavelengthare comparable (cloud radar / ice particles) then we need a more sophisticated theory…

|k| =

APPROXIMATE PARTICLE BYASSEMBLY OF SMALL VOLUMEELEMENTS dv

So the scattered intensity (radar cross section) is:

2

2

~ ( )

1where ( ) exp( 2 )d

v

v f kr

f kr i vv

k r Dimensionless function f - tells you the

deviation from the Rayleigh formula withIncreasing size r

The ‘form factor’ f is easy to calculate, and allows you (for a given shape) to parameterise the scattering in terms of:1. the particle volume, 2. characteristic particle length r (relative to the wavelength).

f =

/

(

Ra

yle

igh

fo

rmu

la)

to appear in the ‘January’ edition of Q. J. Royal Met. Soc. (Westbrook CD, Ball RC & Field PR ‘Radar scattering by aggregate snowflakes’)

LIMITATION OF RAYLEIGH-GANS:

Assumes no interaction between volume elements(low density / weak dielectric / small kr limit)

Good for first approximation, but is this really all ok?

Universal form factor for aggregates irrespective ofthe pristine crystals that compose them(as long as crystals much smaller than wavelength)

The discrete (coupled) dipole approximation ‘DDA’

applied Each dipole is polarised in response to:1. The incident applied field2. The field from all the other dipoles

Approximate particle by an assembly of polarisable, INTERACTING dipoles:

etc..

Now instead of simple volume integral of Rayleigh-Gans, have 3N coupled linear equations to solve:

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

E r E r A E r

polarisability of dipole kElectric field at j

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

Electric field at k

Aggregates of: 100m hexagonal columns, aspect ratio = 1/2

discrete dipole approximation

(Increased backscatter relative to Rayleigh-Gans)

Rayleigh-Gans

f = Ratio of real backscatter to Rayleigh formula

w

lw/l =½

Results

Would like to parameterise the increased backscatter – must depend on:1. Volume fraction of ice ie. Volume / (4r3/3)2. Size of aggregate relative to wavelength kr

343

volume(Rayleigh-Gans) S(kr)

r

USE DDA CALCULATIONS TOWORK OUT FUNCTIONS S AND

-THEN ONLY NEED r AND v TOCACLULATE THE SCATTERING.

Acknowledgements / Ads etc.

John Nicol (sws04jcn@reading.ac.uk) for the clear air boundary layer image.

Robin Hogan (r.j.hogan@reading.ac.uk) for advice and an old presentation from which the animations were robbed.

Robin Ball (Physics, Warwick) and Paul Field (NCAR), collaboration on ice aggregation & Rayleigh-Gans work.

Photographs of pristine snow crystals were from www.snowcrystals.com (Caltech)

For more information on the radar group at Reading:

www.met.reading.ac.uk/radar

and on my work…

www.reading.ac.uk/~sws04cdw

Chris WestbrookRoom 2U04, Meteorologyc.d.westbrook@reading.ac.uk