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Analyzing Drop Size Distributions: From Chaos to Deterministic Models

This study explores the relationship between drop size distributions (DSDs) and rainfall estimation. By examining the statistical properties of drop sizes collected using various instruments, including optical and radar disdrometers, we investigate how DSDs can be modeled as both discrete and continuous random variables. The paper emphasizes the scaling laws that govern the self-similarity of drop sizes and their moments. Improving our understanding of DSDs can enhance precipitation estimation accuracy and promote more efficient numerical weather prediction models.

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Analyzing Drop Size Distributions: From Chaos to Deterministic Models

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  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

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