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Looking for deterministic behavior from chaos
GyuWon LEEASP/RALNCAR
What are we looking at?
Movie (rain)
Drop size distributions?
(Ex) Frequency distribution of drops falling on a plate for a minute.
D [mm]
Nt(D
)N
umbe
r of d
rops
[#]
D [mm]N
umbe
r de
nsity
[m
-3m
m-1]
Distribution function of a discrete random variable
Distribution function of a continuous random variable
Can this distribution be compared with different measurements?
Distribution should be normalized with a sampling volume and diameter interval
N(D): Drop size distribution
Integral parameters of DSDs
max
min
max
min
)()(D
DDii
ni
D
D
nn
i
DDNDdDDNDM
n-th moments of DSDs, Mn
01.5
2
3.5~4.467.3
63
0
6
#
MenergykineticRain
MextinctionOptical
MKMR
MZMLWC
MdropsofTotal
DP
L
max
min
)(D
D
nn dDDNDM
Moments of DSDs
max
min
)()( 3D
DdDDNDDvR
~ M3.67
M6 ~
Accurate estimation of R is related to a better description of DSDs !
Application: Variability of DSDs vs. rain estimate
Current observational tools
1. Impact disdrometer
Filter paper Joss-Waldvogel disdrometer
filter dusted with powdered gentian violet dye
(From Ph.D. thesis of W. McK. Palmer)
Current observational tools
2. Optical disdrometerOptical Spectro Pluviometre 2-dim Video disdrometer Parsivel
Hydrometer Velocity and Size Detector
Current observational tools
3. Radar-based “disdrometer”Micro rain radar (MRR)Precipitation Occurrence Sensor System (POSS)
Pludix (PLUviometro-DIsdrometro
in X band)
Functional fits to measurementsEx) M-P drop size distribtuions: Marshall and Palmer (1948)
A = 1 mm/hB = 2.8 mm/hC = 6.3 mm/hD = 23 mm/h
][1.4
108
)(
121.0
1330
0
mmR
mmmN
eNDN D
Measurements with filter papers during summer of 1946
Paradigm shift
- DSDs in moment space
- Physical constraint: Scaling law
New paradigm: 1. DSDs in moment space
Number density vs. Diameter Moment vs. Moment order
max
min
max
min
)()(D
DDii
ni
D
D
nn
i
DDNDdDDNDM
New paradigm: 1. DSDs in moment space
Microphysical parameterization in numerical weather prediction - Bin models are too expensive to run them in real time
Application aspects - Radar hydrology:
Measure Z or polarimetric parameters (integral values of DSDs), then estimate R (again, integral value)
Thus, we need to transform from one integral value to another integral value or vice versa.
Self-similarity or invariance of line, square, cube as a function of scale (or size)
lL
N
lL
N
Dimension
1
2
3
Mathematically, Power law relationship: y(x)=axb
If x is scaled (x), then y(x)=a bxb=C y(x)y(x) maintains the same functional relationship.
lL
N3
Scaled down by
Ex) mass at various scales
m(L) = kL 3
m(l) = kN-1 L3 = N-1 m(L)
N L
l
Scaling exponent
New paradigm: 2.Scaling law
Scaling exponent, fractal dimension, or self-similarity dimension
lL
N
N L
l
A -dimensional self-similar object can be divided into N smaller copies of itself each of which is scaled down by a factor l.
generator
Self-similarity or invariance of line, square, cube as a function of scale (or size)
New paradigm: 2. Scaling law
Determination of a scaling exponent ()
log N log L
l
Scaling exponent: slope of the number of self-similar parts versus scaling factor in log-log coordinates.
L
Ex) Length around snow crystal:
Length (l)=k N (l) =k (L/l)
log N
log (L/l)
= 1.26 Log(L/l) log(N)log (1) log(3)log (31) log(3x41)log (32) log(3x42)log (3k-1) log(3x4k-1)
New paradigm: 2. Scaling law
- Examples of known power laws:Vol D3, Area D2
P 5/3 (power spectrum)LWC D3, vD Db, Z D6, LWC=aRb, A=aRb KDP Db, Z=aRb , R=aZh
bKDPc
Examples of known power laws
Implicitly, we have been using properties of scaling objects when studying of DSDs !!!!
New paradigm: Scaling law
Scaling of DSDs with moments
New paradigm: DSDs in moment space + Scaling law
In DSDs, similarity of shape of DSDs with various moments (or rainfall intensities R)
After scaling, we may obtain a general scaled DSD that is independent of moments (or rainfall intensities R).
Self-similarity as a function of length scale.
• DSDs can be expressed as:
DpNDN T)( NT: Expected concentration of drops
p: probability distribution function
Resulting scaling law formalism
)()( –ii DMgMDN
(n) (n 1)
Hypothesis: Power-law between the moments of DSDs
Mn C1,n M i(n ) dDDNDM n
n )(
Self-consistency constraints: for n=i1)1( i 111,1 )(1 dxxxgC i
i
When Mi=R (M3.67):
N(D)R1 4.67g(DR )167.4
167.3
1167.3,1 )(1 dxxxgC
Scaling normalized DSDs (single-moment)
Determination of scaling exponent and general DSD g(x)
Scaling exponent:
Mn C1,n Mi(n) C1,n Mi
(n1)
Slope of γ(n) vs. n+1 (or n)
General DSD g(x):
N(D)Mi [1 (i1) ] vs. DMi
(x1)
N(D)Mi1 (i1) g(DMi
– )
Single-moment scaling DSDs
Single-moment scalingDouble-moment scaling
ij
in
jij
nj
i,nn MMCM
2
) Mh(DMMMN(D) ijj
iji
ji
i
jij
j
i
1111
)()( –)1(1 i
ii DMgMDN
(n) (n 1)
Mn C1,n Mi(n) C1,n Mi
1(n i)
C2,n h(x2)x2ndx2
C1,n g(x1)x1ndx11)1( i
No’ : Generalized characteristic
number concentrationDm
’ : Generalized characteristic diameter
Double-moment scaling DSDs
)/(1'
)/()1()/()1('0
/ ijijm
jiij
ijji
MMD
MMN
Double-moment scaling DSDs
) Mh(DMMMN(D) ijj
iji
ji
i
jij
j
i
1111
44
53
4*0
34
)4(
4
/
M
MN
MMDm
i=3, j=4
Testud et al. (2001)Sekhon and Srivastava (1971)
i=3, j=6
WZ
WN
W
ZDm
3/4
0
3/11
Waldvogel (1974)
)β(nαi,nn MCM 1
1
)()( –ii DMgMDN
Single-moment scaling
Scaling :
Mn C2,n Mi
j n
j i M j
n i
j i
C2,n h(x2)x2ndx2
regression :
Mn CMia M j
b
Observed DSDs follow the scaling law ?
DoubleSingle
Advantage in scaling DSDs
Measured DSDs
Application: Derivation of R-Z relationship
)()( DRgRDN
0
633.216 ))(()( dxxgxRdDDDNZ
- Exponent of R-Z is linearly related to the scaling exponent
- Coefficient of R-Z is 6-th moment of average g(x)
Application: Derivation of R-Z relationship
67.3
6
67.3
33.2
j
j
jj RaMZ
5.15.0'0 RNaZ
RDaZ m
33.2'
)/()( ''0 mDDhNDN
- Exponent and coefficient of R-Z is determined by the relationship between R and No’ (or, Dm’).
Summary
- Traditionally, functional fits have been used to describe DSDs.
- We have tried to describe DSDs in moment space with physical constraint (scaling law) - This leads to single- and double-moment scaling normalized DSDs
- The new formalization can be easily used in microphysical parameterization in numerical models and remote sensing application