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. D [mm] N t (D) Number of drops[#] D [mm] Number density [m -3 mm -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

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

5. Moments of DSDs ~ M 3.67 M 6 ~ Accurate estimation of R is related to a better description of DSDs ! Application: Variability of DSDs vs. rain estimate

6. 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)

7. Current observational tools 2. Optical disdrometer Optical Spectro Pluviometre 2-dim Video disdrometer Parsivel 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) A = 1 mm/h B = 2.8 mm/h C = 6.3 mm/h D = 23 mm/h Measurements with filter papers during summer of 1946

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

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

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. Self-similarity or invariance of line, square, cube as a function of scale (or size) Dimension 1 2 3 Mathematically, Power law relationship: y(x)=ax b If x is scaled (x ), then y(x )=a b x b =C y(x) y(x) maintains the same functional relationship. Scaled down by Ex) mass at various scales m(L) = kL 3 m(l) = kN -1 L 3 = N -1 m(L) Scaling exponent New paradigm: 2.Scaling law

14. Scaling exponent, fractal dimension, or self-similarity dimension 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

15. Determination of a scaling exponent (  ) 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 (3 1 ) log(3x4 1 ) log (3 2 ) log(3x4 2 ) log (3 k-1 ) log(3x4 k-1 ) New paradigm: 2. Scaling law

16. - Examples of known power laws: Vol  D 3, Area  D 2 P  5/3 (power spectrum) LWC  D 3, v D  D b, Z  D 6, LWC=aR b, A=aR b K DP  D b, Z=aR b, R=aZ h b K DP c Examples of known power laws  Implicitly, we have been using properties of scaling objects when studying of DSDs !!!! New paradigm: Scaling law

17. 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.

18. DSDs can be expressed as: N T : Expected concentration of drops p: probability distribution function Resulting scaling law formalism Hypothesis: Power-law between the moments of DSDs Self-consistency constraints: for n=i When M i =R (M 3.67 ): Scaling normalized DSDs (single-moment)

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

20. Single-moment scalingDouble-moment scaling N o ’ : Generalized characteristic number concentration D m ’ : Generalized characteristic diameter Double-moment scaling DSDs

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

22. Observed DSDs follow the scaling law ?

23. DoubleSingle Advantage in scaling DSDs Measured DSDs

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

27. 4. New paradigm: Normalized DSDs Waldvogel (1974) Sekhon and Srivastava (1971)

28. - Single-parameter normalization: Sempere-Torres et al.(1994) - Double-parameter normalization: Testud et al. (2001) R = any moment of DSDs (usually rainfall intensity) ,  : Scaling (normalization) exponents D m : volume weighted mean diameter N o * : intercept parameter 4. New paradigm: Normalized DSDs