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Sunny W. Y. Tam 1,2 , Tom Chang 3 , Paul M. Kintner 4 , and Eric M. Klatt 5

ROMA (Rank-Ordered Multifractal Analysis) for Intermittent Fluctuations with Global Crossover Behavior. Sunny W. Y. Tam 1,2 , Tom Chang 3 , Paul M. Kintner 4 , and Eric M. Klatt 5. 1 Institute of Space, Astrophysical and Plasma Sciences, National Cheng Kung University, Tainan, Taiwan

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Sunny W. Y. Tam 1,2 , Tom Chang 3 , Paul M. Kintner 4 , and Eric M. Klatt 5

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  1. ROMA (Rank-Ordered Multifractal Analysis) for Intermittent Fluctuations with Global Crossover Behavior Sunny W. Y. Tam1,2, Tom Chang3, Paul M. Kintner4, and Eric M. Klatt5 1 Institute of Space, Astrophysical and Plasma Sciences, National Cheng Kung University, Tainan, Taiwan 2 Plasma and Space Science Center, National Cheng Kung University, Tainan, Taiwan 3 Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA, USA 4 School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA 5 Applied Physics Laboratory, Johns Hopkins University, Laurel, MD, USA

  2. Outline • Data • Electric field in the auroral zone • Multifractal Analyses and Scaling Behavior • Traditional Structure Function Analysis • ROMA (Rank-Ordered Multifractal Analysis) • Individual Regimes • ROMA for Nonlinear Crossover Behavior • Across Regimes of Time Scales • Summary

  3. Electric Field Data • SIERRA sounding rocket in the nighttime auroral zone • Time series of an electric field component perpendicular to the magnetic field • Consider E measured between 550 km altitude and the apogee (735 km) of SIERRA • Typically observed broadband extremely low-frequency (BB-ELF) electric field fluctuations • Subset of the observed electric field fluctuations found to be intermittent in nature [Tam et al., 2005]

  4. The broadband power spectrum signature of the BB-ELF fluctuations has been suggested as the manifestation of intermittent turbulence; origin of intermittent fluctuations interpreted as the result of sporadic mixing and/or interactions of localized pseudo-coherent structures [Chang, 2001; Chang et al., 2004] • Pseudo-coherent structures (c.f. nearly 2D oblique potential structures based on MHD simulations by Seyler [1990]) nearly non-propagating, measurements due to Doppler-shifted spatial fluctuations, mixed with small fractions of propagating waves • Time scales τin data can be interpreted as spatial scalesΔ=Uτ(horizontal speed of rocket,U ≈1.5 km/s)

  5. Multifractal Analyses and Scaling Behavior • Traditional Structure Function Analysis • ROMA (Rank-Ordered Multifractal Analysis)[Chang and Wu, 2008] • ROMA forNonlinear Crossover Behavior [Tam et al., 2010] • Double rank-ordering

  6. Common procedures for the methods: • Generate Probability Distribution Function (PDF) for different values of , where

  7. Traditional Structure Function Analysis • Define the structure function of the moment order q at the time scale : • q is required to be non-negative to avoid divergence of Sq • One looks for the scaling behavior

  8. If the “fractal dimension” is proportional to q, i.e. , all the fractal properties can be characterized by a single number monofractal • The Hurst exponent is constant if the fluctuations are monofractal; multifractals are indicated by non-constant H(q).

  9. Single-Parameter Scaling • Monofractal condition can be satisfied by a one-parameter scaling with the parameter s[Chang et al., 1973]: One can show that • For monofractal fluctuations, the single-parameter scaling is able to provide a clear description of how the strength of the fluctuations varies with the time scale.

  10. Indication of multiple physical regimes of time scales Structure Functions of Electric Field Fluctuations log Sq vs. log τnot a straight line

  11. Regimes 1 2 3 4 Rank-Order the time regimes intoi =1 to 4 Assume adjacent regimes roughly have a common time scale: Regime 1: 5 – 80 ms (kinetic) Regime 2: 80 – 160 ms (crossover) Regime 3: 160 – 320 ms (crossover) Regime 4: 320 ms and longer (MHD) Slope ζq Study the multifractal characteristics of each regime separately Consider only Regime 1 in detail as an example.

  12. With traditional structure function analysis: Indications of multifractal behavior For the electric field fluctuations, the plot of vs. q is not exactly a straight line. H(q) is not a constant, varying considerably.

  13. Apply single-parameter scaling formula ( ms): where Single-parameter scaling does not apply well to the multifractal electric field fluctuations.

  14. Drawbacks of Tradition Structure Function Analysis on Multifractal Fluctuations • Different parts of the PDF are emphasized by different moment order (larger q for larger ) and have different fractal properties (non-constant H), but characterizes only the average fractal properties over the entire PDF. • Negative q is ill-defined.

  15. Rank-Ordered Multifractal Analysis (ROMA) for Individual Regimes • Technique introduced by Chang and Wu [2008] • Technique retains the spirit of structure function analysis and single-parameter scaling • Divide (Rank-Order) the domain of (Note: s=s(Y)) into separate ranges and, for each range, look for one-parameter scaling • Scaling function and scale invariant Y

  16. To solve for s(Y), the scaling parameter s for the range : • construct the range-limited structure functions with prescribed s • Look for the scaling behavior • The solution s will satisfy

  17. Example: Regime 1Y1 = [0.8, 1.2] s1 = 0.80 from this plot With increased resolution, s1 = 0.804

  18. Validity of the solution Note: negative q is applicable

  19. Plot of scaling parameter s1 for different ranges of Y1 Considerable variation of s1 multifractal In principle, s1=s1(Y1) a continuous spectrum; but for practical purpose, statistics reaches limitation as Y-ranges keep decreasing

  20. Traditional single-parameter scaling Comparison of the scaling by the two multifractal analyzing techniques ROMA

  21. Regime 1 Persistency (s > 0.5): probably due to kinetic effects Rapidly changing s: indication of possible developing instability and turbulence Slowly changing s: More stable and developed turbulent state

  22. Regime 2 Developing turbulence at small Y seems to be of a mixture of persistent (s > 0.5) and anti-persistent (s > 0.5) nature Effects beyond the kinetic range play a non-negligible role Turbulence settled down to more stable and developed state Persistent probably because kinetic effects are still more dominant than those of MHD

  23. Regime 3 Similar to Regimes 1 and 2, developing turbulence at small Y Highly unstable turbulence compared with the other 2 regimes, indicated by the wide range of s and the range of Y where s exhibits such large fluctuations

  24. Regime 4 Anti-persistency (s < 0.5) Monotonically decreasing s beyond a certain Y Same features in the original ROMA calculations for results of 2D MHD simulations [Chang and Wu, 2008] Signature of developing MHD turbulence?

  25. Scaling Functions Regime 1 Regime 2 Regime 3 Regime 4

  26. Regime 1 Regime 2 Regime 4 Resemblance in shape between s(Y) and H(q)

  27. qincreases fractal property at larger |δE|is emphasized for each Y-range Y increases |δE| increases Resemblance in shape between s(Y) and H(q)

  28. Reason: Significant decrease in s(Y) over a small range of Y • a narrow range in the domain • of |δE| corresponds to a wide • range in the domain of Y • Narrow range of |δE| • emphasized by H(q) actually • characterizes the average • fractal behavior at a wide • range of Y • s(Y) is a more accurate • description than H(q) Exception: Regime 3

  29. Advantages of ROMA • Fractal properties at different and • is known at each range of Y • Scaling behavior • s is found for each range of Y; scale invariance is determined: • Negative q • Applicable except for the range that includes Y = 0

  30. ROMA Across Regimes of Time Scales • Assume that crossover ranges of time scales between contiguous time regimes are narrow • Because regimes are contiguous and scaling with the time scales is power law in nature, Yi can be mapped onto Yi-1 , and so on. Eventually, all the Yi can be mapped onto one global scaling variable Yglobal • Correspondingly, the scaling functions of all the regimes can be mapped to a global scaling function Ps1(Yglobal)

  31. s1 s2 s3 s4 Except for highly unstable turbulence, a generally decreasing trend for si at given Yglobal as igoes from 1 to 4,with the regimes crossing over from kinetic to MHD.

  32. Global Scaling Functions Regime 1 – 4

  33. Summary • Traditional structure function analysis vs. ROMA for time (or spatial) series of fluctuations • Both methods indicate multifractal nature of the electric field fluctuations in the auroral zone • ROMA has the advantages of providing clearer information regarding the fractal properties and scaling behavior of the fluctuations • ROMA is extended to apply to fluctuations with multiple regimes in time scale • Double rank-ordered parameters: regime index i and power-law scaling variable Yi • Determine global scaling function and global scaling variable across different regimes • Scaling parameter s generally decreases as the regimes cross over from kinetic to MHD • Collapse of PDF at all time scales of all regimes

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