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 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

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

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] Electric Field Data

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 τ in data can be interpreted as spatial scales Δ=Uτ ( horizontal speed of rocket, U ≈1.5 km/s)

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

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

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 S q One looks for the scaling behavior

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).

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.

Structure Functions of Electric Field Fluctuations Indication of multiple physical regimes of time scales log S q vs. log τ not a straight line

Regimes1234 Consider only Regime 1 in detail as an example. 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 ζ q Rank-Order the time regimes into i =1 to 4 Study the multifractal characteristics of each regime separately

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

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

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.

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

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

Example: Regime 1 Y 1 = [0.8, 1.2] s 1 = 0.80 from this plot With increased resolution, s 1 = 0.804

Validity of the solution Note: negative q is applicable

Plot of scaling parameter s 1 for different ranges of Y 1 In principle, s 1 =s 1 (Y 1 ) a continuous spectrum; but for practical purpose, statistics reaches limitation as Y-ranges keep decreasing Considerable variation of s 1 multifractal

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

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

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

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

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?

Scaling Functions Regime 1Regime 2 Regime 3Regime 4

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

q increases fractal property at larger |δE| is emphasized for each Y- range Y increases |δE| increases

Exception: Regime 3Reason: 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)

Advantages of ROMA 1.Fractal properties at different and is known at each range of Y 2.Scaling behavior s is found for each range of Y; scale invariance is determined: 3.Negative q Applicable except for the range that includes Y = 0

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, Y i can be mapped onto Y i-1, and so on. Eventually, all the Y i can be mapped onto one global scaling variable Y global Correspondingly, the scaling functions of all the regimes can be mapped to a global scaling function P s1 (Y global )

s1s1 s2s2 s3s3 s4s4 Except for highly unstable turbulence, a generally decreasing trend for s i at given Y global as i goes from 1 to 4, with the regimes crossing over from kinetic to MHD.

Global Scaling Functions Regime 1 – 4

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 Y i –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