Looking for deterministic behavior from chaos

1. Looking for deterministic behavior from chaos GyuWon LEE ASP/RAL NCAR

2. What are we looking at? Movie (rain)

3. Drop size distributions? (Ex) Frequency distribution of drops falling on a plate for a minute. N(D): Drop size distribution Nt(D) Number of drops[#] Number density [m-3mm-1] D [mm] D [mm] Can this distribution be compared with different measurements? Distribution should be normalized with a sampling volume and diameter interval Distribution function of a discrete random variable Distribution function of a continuous random variable

4. Integral parameters of DSDs n-th moments of DSDs, Mn

5. ~ M3.67 M6 ~ Application: Variability of DSDs vs. rain estimate Moments of DSDs Accurate estimation of R is related to a better description of DSDs !

6. Current observational tools 1. Impact disdrometer Filter paper Joss-Waldvogel disdrometer (From Ph.D. thesis of W. McK. Palmer) filter dusted with powdered gentian violet dye

7. Current observational tools 2. Optical disdrometer Parsivel 2-dim Video disdrometer Optical Spectro Pluviometre Hydrometer Velocity and Size Detector

8. Current observational tools 3. Radar-based “disdrometer” Micro rain radar (MRR) Precipitation Occurrence Sensor System (POSS) Pludix (PLUviometro-DIsdrometro in X band)

9. Functional fits to measurements Ex) M-P drop size distribtuions:Marshall and Palmer (1948) Measurements with filter papers during summer of 1946 A = 1 mm/h B = 2.8 mm/h C = 6.3 mm/h D = 23 mm/h

10. Paradigm shift • DSDs in moment space • - Physical constraint: Scaling law

11. Moment vs. Moment order New paradigm: 1. DSDs in moment space Number density vs. Diameter

12. 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.

13. Scaling exponent New paradigm: 2.Scaling law Self-similarity or invariance of line, square, cube as a function of scale (or size) Dimension 1 2 3 Scaled down by Ex) mass at various scales m(L)= kL3 m(l) = kN-1L3 = N-1m(L) 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.

14. generator A -dimensional self-similar object can be divided into N smaller copies of itself each of which is scaled down by a factor l. Scaling exponent, fractal dimension, or self-similarity dimension New paradigm: 2. Scaling law Self-similarity or invariance of line, square, cube as a function of scale (or size)

15. L Ex) Length around snow crystal: Length (l)=k N (l) =k (L/l) Log(L/l) log(N) log (1) log(3) log (31) log(3x41) log (32) log(3x42) log (3k-1) log(3x4k-1)  = 1.26 log N log (L/l) New paradigm: 2. Scaling law Determination of a scaling exponent () Scaling exponent: slope of the number of self-similar parts versus scaling factor in log-log coordinates.

16. New paradigm: Scaling law Examples of known power laws - 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=aZhbKDPc  Implicitly, we have been using properties of scaling objects when studying of DSDs !!!!

17. New paradigm: DSDs in moment space + Scaling law Scaling of DSDs with moments Self-similarity as a function of length scale. • 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).

18. Scaling normalized DSDs (single-moment) Hypothesis: Power-law between the moments of DSDs Resulting scaling law formalism Self-consistency constraints: for n=i When Mi=R (M3.67): NT: Expected concentration of drops p: probability distribution function • DSDs can be expressed as:

19. Single-moment scaling DSDs General DSD g(x): Determination of scaling exponent  and general DSD g(x) Scaling exponent: Slope of γ(n) vs. n+1 (or n)

20. Double-moment scaling DSDs Double-moment scaling Single-moment scaling No’ : Generalized characteristic number concentration Dm’ : Generalized characteristic diameter

21. Single-moment scaling i=3, j=6 i=3, j=4 Testud et al. (2001) Sekhon and Srivastava (1971) Waldvogel (1974) Double-moment scaling DSDs

22. Observed DSDs follow the scaling law ?

23. Advantage in scaling DSDs Measured DSDs Single Double

24. Application: Derivation of R-Z relationship - Exponent of R-Z is linearly related to the scaling exponent - Coefficient of R-Z is 6-th moment of average g(x)

25. Application: Derivation of R-Z relationship - Exponent and coefficient of R-Z is determined by the relationship between R and No’ (or, Dm’).

26. 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