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