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3.4 Graupel and hailstone density
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Bulk density of large rimed ice particles varies greatly, depending on denseness of
packing of cloud drops frozen on the ice crystal, growth mode (dry vs. wet),
surface (dry vs. wet), and internal state (solid ice, air/ice mixture, ice/water
mixture)
Density of graupel particles range from 0.05 g cm-3 to as high as 0.89 g cm-3. See
Table 2.8 Pruppacher and Klett (1997)
Density of hailstones usually > 0.8 g cm-3 and approaches solid ice (0.917 g cm-3),
especially if in wet growth
– Growth mode and history (and melting/freezing) matters
– External wet surface during wet growth or melting can slightly increase bulk density of
particle
– Earlier dry growth can reduce overall bulk density
– But water can soak into ice/air matrix and dramatically increase bulk density of particle
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3.5 Graupel and hail dielectric (or refractive index)
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Use Debye mixing theory, Debye (1929), for ice and air mixtures (e.g., graupel
or dry growth hail) (e.g., Battan 1973)
ρ
M=
Ki
ρi
Mi +
Ka
ρa
Ma
m −1
[4]
K= 2
m +2
2
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0.0025
0.002
0.0015
n
0.001
k
0.0005
0
0
0.2
0.4
0.6
0.8
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Ice density (ρi, g cm-3)
Where M:mass, ρ: density, m: refractive index; subscript i=ice and a=air (no
subscript=mixture)
Can simplify [3] by noting that ma in [4] is ≈ 1 so Ka ≈ 0 and M ≈ Mi ∴→ K/ρ is
constant. Hence, K for mixture is
K 
K =  i  ρ
 ρi 
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[3]
0.003
Imaginary component of refractive
index (k)
K
Real component of refractive index (n)
Refractive index of bulk ice, m=n+ik
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
[5]
Combine [4] and [5] to solve for refractive index of mixture (m)
2χ + 1
m =
where
1− χ
2
 Ki 
χ =   ρ
 ρi 
[ 6]
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For melting hail, you can model it as 1) concentric
oblate spheroids with ice inside and liquid melt
water outside or 2) spongy ice
For 2) spongy hail (water+ice), Deybe mixing theory
does NOT apply. Cannot be strongly absorbing.
For 2) spongy hail, must use different theory like
Maxwell Garnett (1904) mixing theory to calculate
dielectric (e.g., Bohren and Battan 1980; Longtin et
al. 1987 JTECH)
– Dielectric of spongy ice εsi is a function of
dielectric constant of solid ice εi, liquid water εw
and volume of water fraction (f) in ice-water
mixture
C-band
[7]
– Where assumed ice inclusions in water matrix best
simulates spongy ice, especially where f is high
Longtin et al. (1987)
Bohren and Battan (1980)
“MG ice in water” = spongy ice
Where ƒ is high
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3.5 Impact of hailstone properties on polarimetric radar observables
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Balakrishnan and Zrnic (1990, JAS): Large hail discrimination at S-band (mostly ρhv)
• To understand size effect on ρHV,
first recall definition of correlation
coefficient from radar
ρ hv =
Shh Svv e
(S
2
hh
− j ( δ +φdp )
Svv
)
2 1/ 2
• Where δ is differential phase
shift on backscattering (or
backscatter differential phase)
• Variability in δ (or increasing
the effective width of
distribution of δ) reduces ρHV
• δ is non-zero during resonant
(Mie) scattering
• ∴ presence of Mie scatters
lowers ρHV
[8]
• Resonant scattering (non-zero δ) occurs for wet
or spongy hail as D approaches and exceeds
about 10 mm (D >≈ 10 mm)
• Not much δ for dry hail except when very large
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• Wet hail (solid ice surrounded by
water film)
• 10 cm wavelength (S-band)
• Minor axis in vertical (i.e., largest
dimension in horizontal)
• Shape according to Knight (1986)
for Oklahoma storms
• Vary size
• Specify fraction of hailstones that
are randomly tumbling
• Increasing fraction of randomly
tumbling hail lowers ρhv
• Tumbling increases variability
of effective shapes in RRV
• Extent of tumbling is enhanced by
resonance at particular size
ranges once D >≈ 10 mm
• Resonance causes further
reduction in ρHV
• See δ for extent of resonance
effects on ρHV
Effect of tumbling and size
(resonance) on S-band ρHV
Balakrishnan and Zrnic (1990)
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Balakrishnan and Zrnic (1990)
• Effect of size, shape, dielectric on
S-band Zdr and ρhv
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Dielectric: dry, wet, spongy
Shape: axis ratio = 0.6 or 0.8
Fall mode: minor axis in horizontal plane
Size: varied up to 60 mm
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Dry hail has little response in ρHV except
for the largest sizes in resonance
Wet hail and especially spongy hail show
much more lowering (troughs) in ρHV
associated with resonance peaks
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ρHV
More oblate shapes enhance effect
For non-Mie, Zdr is negative due to
Zdr
orientation (major axis in vertical)
Resonance complicates Zdr. Some
resonance peaks cause enhanced
negative Zdr while at larger sizes
resonance can also cause positive Zdr for
hail as modeled here
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• Effect of hailstone lobes or
protuberances on ρhv
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Balakrishnan and Zrnic (1990)
Modeled protuberance size σD
relative to hailstone diameter (D)
Increasing lumpiness of hail
lowers ρhv
(σD /D)=0.1 can lower ρhv to
0.92
Very lumpy hail can lower ρhv
significantly
When considering ρhv of hail,
you must integrate all the
effects shown so far (size or
resonance, shape, dielectric
and lumpiness)
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Possible for some hailstones
to cause S-band ρhv to lower
enough to be confused with
debris (even more so at Cband)
Increasing Lumpiness
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Enhanced S-band LDR aloft
before surface hail fall
Red arrow: LDR=-25 to -22dB
Zdr > 1 dB color shaded; every 1 dB
• Wet Growth Hail (Aloft) Canting Angle and Water
fraction (dielectric effects) on S-band LDR and Zdr
(Kennedy et al. 2001)
• D = 1 cm (mono), axis ratio=0.8, Canting angle
Std Dev=35°-55° , minor axis vertical in mean
• T=0°C, S-band, radar elevation angle=5°
• Spongy hail (left) and wet hail (right)
• LDR (contoured), Zdr (#’s)
Spongy
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Wet hail
Increasing spongy water
fraction (lesser extent
water thickness) increases
LDR
Increased canting decrease
Zdr. Not big effect on LDR
since all values near 45°.
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Monodisperse PSD with
hailstone diameter
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0.45 g cm-3
0.91 g cm-3 (solid ice)
0.91 g cm-3 (solid ice) with water
coating
Spongy ice (40% water)
Shape = Oblate spheroid
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Effect of dielectric/density, size,
shape of hail on S-band Zdr and LDR
Varying ice density
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D=15 mm (left)
D=35 mm (right)
0.75 axis ratio
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Axis ratio =0.75 except
Axis ratio = 0.6 (last row)
Gaussian canting with 0° mean
and 75° standard deviation
In mean, minor axis in the
vertical
Zdr and LDR increase with
increasing dielectric strength
(from low to high density ice
to wet to spongy)
LDR increase with size except
when spongy
Zdr and LDR increase with
decreasing axis ratio
Depue et al. (2007)
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Zdr
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LDR
S-band Zdr (left) and LDR (right) as a function of hail diameter (monodisperse,
abscissa) and various dielectric and canting assumptions
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Resonance peaks are apparent, depending on dielectric and size
LDR increases with diameter generally except resonance peaks/troughs when wet or spongy
Increasing dielectric increases LDR and Zdr
Canting tends to decrease Zdr. Above a certain value, increasing canting doesn’t change LDR
too much (saw that in Kennedy et al. 2001)
Depue et al. (2007)
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Exponential size distribution. Recall:
N ( D ) = N 0 exp( − ΛD ) [1]
N 0 = AΛb [2]
– Cheng and English (1983)
– set Λ = 14/Dmax where Dmax varied from 8 to 53
mm (Ulbrich and Atlas 1982)
– One parameter size distribution terms of Dmax
• S-band Zdr and HDR increase
systematically with increasing
Dmax
• Zdr and HDR larger when wet
• Exponential PSD integrates out
resonance peaks seen earlier
Depue et al. (2007)
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C-band polarimetric radar
observations of melting hail
Zh (dBZ)
Anderson et al. (2011)
Zdr (dB)
CFAD Zdr (dB)
0.7°
0°C
melting
4.2°
11.0°
Zdr= 3-8 dB in melting
hail at C-band. Why?
Water torus around melting
hail whose size is resonant
at C-band
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Sensitivity study of varying wet hail properties on C-band Zdr
(Anderson 2010)
Answer Question: Why Zdr= 3-8 dB in melting hail at C-band?
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Shape: Oblate spheroid with axis ratio of 0.6 or 0.8
Fall Mode: Major axis either in the horizontal or vertical. Gaussian canting
angle distribution with 0° mean and varying standard deviation (STD=5° to
45°)
Density/dielectric: Assume 0.5 mm outer water torus with inner oblate
spheroid solid ice at T=20°C.
– Could be wet due to wet growth (aloft) or melting (at lower heights). More on these topics
later.
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Particle Size distribution: mono-disperse, exponential, or gamma.
– Fixed diameters ranged from 0.5 to 2.2 cm
– For exponential and gamma, median volume diameter (D0) varied from 0.5 cm to
2.2 cm. Maximum diameter set to 4.0 cm
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Radar model: T-matrix model for individual particle scattering properties
(size, shape, dielectric) and Mueller matrix method for computing radar
observables (Zdr) for simulated particle size distributions, radar elevation angle
and particle canting angle (e.g., Vivekanandan et al. 1991, JAM)
– C-band (5.3 cm)
– 0° elevation angle
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Monodisperse PSD: Zdr vs. STD canting angle
Solid: axis ratio=0.8
Dash: axis ratio = 0.6
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Wide range of Zdr possible!
As expected, |Zdr| is larger for
smaller axis ratio (more
oblate)
Major axis horizontal (vertical)
produces positive (negative
Zdr)
Increasing standard deviation
of canting angle decreases Zdr
significantly (tumbling)
At C-band, resonance affects
Zdr for the simulated effective
sizes and shapes (i.e. including
fall mode and dielectric)
To be consistent with
observations at C-band (Zdr=38 dB), monodisperse wet hail
would need to be fairly oblate
(0.6), in a stable fall mode
(small STD canting angle) with
major axis horizontal and sizes
between 1.1-1.8 cm.
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Hail with water torus
doesn’t tumble (more
later)
Diameter
0.5 – 1.0 cm
1.1 – 1.6 cm
1.7 – 2.2 cm
Anderson (2010)
Major axis horizontal
Major axis vertical
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Solid: axis ratio=0.8
Dash: axis ratio = 0.6
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Similar trends of Zdr with
shape, canting angle and
orientation as before
Difference in size range
where D0 causes
maximum positive Zdr.
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Exponential PSD: Zdr vs. STD canting angle
Median
Volume
Diameter
(D0)
0.5 – 1.0 cm
5.5 dB for D0=0.6 cm
where there is peak in
resonance
As D0 increases, |Zdr|
actually decreases
Resonant peaks seen in
mondisperse are
smoothed out by
exponential PSD. Seen
earlier with Depue et al.
(2007)
Anderson (2010)
1.1 – 1.6 cm
1.7 – 2.2 cm
Major axis horizontal
Major axis vertical
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Resonance peak in Zdr for monodisperse
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µ sensitivity study for
Gamma PSD
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Increasing µ approaches
monodisperse
Major axis horizontal
axis ratio = 0.6
Wet (0.5 mm)
Stable (not canting much, 5°)
µ= 4 to 20
As you increase µ to
increasing positive values,
PSD becomes more narrow
and peaked (more like
monodisperse)
Ziegler et al. (1983) found
hail gamma PSD’s with
large +µ up to 9
So could get peak of Zdr=6
dB at D0 = 1 cm for gamma
PSD with µ=9
Anderson (2010)
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