Transcript Relationship of the Reflectivity Factor to other Meteorological Quantities Z D  j j Vc Precipitation content (W): The mass of condensed water substance (water or ice) present.

Relationship of the Reflectivity Factor to other
Meteorological Quantities
Z
6
D
 j
j
Vc
Precipitation content (W): The mass of condensed water substance
(water or ice) present in the form of precipitation-sized particles
(detectable with radar), per unit volume.
W
 m j
j
Vc


6
3
D
 j
j
Vc
Where:
mj is the contribution to the total
mass from each raindrop j
Precipitation Content
Basic units: kg/m3
Simple interpretation:
Mass of water in a unit volume
Extreme values:
0.1 gram/m3 in light drizzle
10 gram/m3 in rain in hurricane eyewall
Example:
A distribution of 1000 1-mm raindrops per cubic
meter would have a precipitation content of about
0.5 grams/m3 .
Z   D 6j
W   D3j
j
Illustration of inequality
j
Problem:


3
j D   j D 


2
Consider two drops 1 mm and 2 mm
6


3
j D   j D 


2
6
Therefore: There is no exact
Relationship between precipitation
content and radar reflectivity
16  26  65
1  2   9
3
3 2
2
 81
Nevertheless, precipitation contents can be qualitatively related
to the radar reflectivity factor, and radar scientists have sought
empirical relationships of the type:
W 
Z  Z R  
 W0 
b
where ZR is the value of Z
when W = W0
Relationship of the Reflectivity Factor to other
Meteorological Quantities
Z
6
D
 j
j
Vc
Precipitation rate (R): The volume of precipitation passing
downward through a horizontal surface, per unit area, per unit time.
R
 rj
j
Vc


6
3
D
 j wj
j
Vc
Where:
rj is the contribution to the
rainfall rate from each raindrop j
wj is the fall velocity of each drop j
Precipitation Rate
Basic units: m3/(m2sec) = m/s
Standard units: mm/hr
Simple interpretation:
Depth of accumulated rainfall on a runoff-free surface
Extreme values:
0.1 mm/hr in light drizzle
1000 mm/hr in a hurricane eyewall
Example:
A distribution of 1000 1-mm raindrops per cubic
meter, falling at their terminal fall speed of 4 m/s in
the absence of vertical motion, would give a
precipitation rate of 2.1  10-6 m/s or about 7.5 mm/hr.

R
Dw

6
3
j
j
j
What is the fall velocity of a raindrop?
Vc
For drops with diameters between 02 mm (most drops) the fall velocity
is proportional to diameter
R   D4
j
so what is the relationship
to the radar reflectivity?
Terminal velocity of raindrops
In still air (Foote and duTroit 1969)
Problem:
Illustration of inequality
1.5


4
j D   j D 


6
Consider two drops 1 mm and 2 mm
1.5


4
j D   j D 


6
Therefore: There is no exact
Relationship between rainfall
Rate and radar reflectivity
16  26  65
1
4
 24

1.5
 17  70.09
1 .5
Nevertheless, rainfall rates are qualitatively related to the radar
reflectivity factor, and radar scientists have sought empirical
b
relationships of the type:
 R
Z  Z R  
 R0 
where ZR is the value of Z
when R = R0
Relationship of Z to Precipitation Rate
Methods of determining Z-R relationships
1. The direct method: Values of Z and R are measured by a radar
and raingages. The data are compared using correlation statistics
and a Z-R relationship is determined from a best fit.
Relationship of Z to Precipitation Rate
Methods of determining Z-R relationships
2. The indirect method: Values of Z and R are calculated from
the same measured raindrop size distribution.
Methods to measure raindrop size distributions
Mechanical: stained filter paper: Uses water stains in filter paper
to estimate raindrop sizes (used originally by Marshall and Palmer)
Impact disdrometer: Uses raindrop’s momentum when striking
surface to estimate its size.
Ground Based Optical disdrometers
Airborne Optical disdrometers
Determine drop sizes by shadows
recorded on optical arrays
Foil impactors: determine drop sizes from impact craters
Foil impactors
Example of raindrop images collected with an airborne optical array
spectrometer in a shower in Hawaii with the largest raindrop
ever recorded in nature (courtesy Ken Beard)
Typical measured raindrop size distributions
To estimate Z and R, exponential approximations to
raindrop size distributions are often developed
The Marshall-Palmer Distribution
Developed from raindrop samples collected in Canada on powdered
sugar filter paper in 1948 by radar pioneers Marshall and Palmer
The Marshall-Palmer Distribution
nD  n0 exp D
n0  0.08cm4  8 106 m4
 R
   R  
 R0 
c
R0  1mm/ hr  R  41cm1
The Marshall-Palmer distribution stood as the standard for
many decades although many subsequent studies showed that it
was not universally applicable.
The exponential distribution has properties that make it useful
because it is easy to relate the drop size distribution to rainfall
rate, precipitation content, and radar reflectivity
General properties of an exponential size distribution
nD  n0 exp D

NT ,   N ( D)dD 
Total concentration of droplets
0
Rainfall rate
R 

6

3
w
D
t
 N ( D)dD 
Precipitation content
W 
w 
6
0
4b
6
wt  aDb
3
D
 N ( D)dD 
n0  w 4
6
7 
Z    D N ( D)dD  n0 7

0

Radar reflectivity
an0 4  b
0
where the fall velocity
n0

6
4
Drop distributions do not extend to infinite size – the integration
must be truncated at the maximum droplet diameter Dm
Effect of such a truncation:
Dm
F
 Xn( D)dD
0

 Xn( D)dD
0
D0 is mean diameter
Calculation of Z from a measured drop size distribution
Note which drops
contribute most to
the radar reflectivity
Z = 1.7  105 mm6/m3
52.3 dBZ
General form of Z/R Relationships
 R
Z  Z R  
 R0 
Z 
R  R0  
 Z0 
b
1
b
Radar scientists have tried to determine Z-R relationships because
of the potential usefulness of radar determined rainfall for
FLASH FLOOD
NOWCASTING
WATER MANAGEMENT
AGRICULTURE
(irrigation needs/drought impacts)
There have been hundreds of Z-R relationships published – here
are just a few between 1947 and 1960 – there have been 4 more
decades of new Z-R relationships to add to this table since!
Z-R relationships are dependent on the type of rainfall
(convective, stratiform, mixed), the season (summer, winter), the
location (tropics, continental, oceanic, mid-latitudes), cloud type
etc.
For the NEXRAD radars , the NWS currently uses five different
Z-R relationships and can switch between these depending upon
the type of weather event expected.





Default WSR-88D (Z= 300R1.4)
Rosenfeld tropical (Z=250R1.2)
Marshall/Palmer (Z=200R1.6)
East Cool Season (Z=200R2.0)
West Cool Season (Z=75R2.0)
The single largest problem in applying Z-R relationships has
been accounting for effects of the radar bright band
The bright band: The melting level, where large snowflakes
become water coated, but have not yet collapsed into small
raindrops.
Wet snowflakes scatter energy very effectively back to the radar
Stratiform area
Convection
Altitude (km)
BB
Distance (km)
Reflectivity factor (dBZ)
The bright band appears as a ring on PPI displays where the
radar beam crosses the melting level
An extreme example of bright band contamination of
precipitation estimation – radar estimates 6 inches of
rain in a winter storm on January 31, 2002!
Other problems:
1. Estimating R from Z in regions of storms that are mixed phase
(e.g., hail vs. rain)
2. Regions affected by ground clutter or blocking (particularly a
problem for estimating rainfall during flash floods in mountainous
regions)
SNOW
Few attempts have been made to develop Z-S relationships
1. Snow density varies significantly from storm to storm and
within storms
2. Scattering by ice is non-Rayleigh (not spheres) and so the
relationship between mass and Z is even less certain
3. Radars calibrated for rain (Z determined from K for rain, not
ice, even in winter)
Measurements have been made of the size distributions of
snowflakes and related to precipitation rates (melted equivalent), and
Z-S relationships have been proposed but these relationships have
largely been ignored in practice
Hail
Very few attempts have been made to quantity hailfall from
thunderstorms. Most work focuses on trying to identify whether
hail is reaching the surface. This work is now focused on studies
using polarization radar technology, which we will examine later
in the course.