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, Taiwan2 Plasma and Space Science Center, National Cheng Kung University, Tainan, Taiwan3 Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA, USA4 School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA5 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(| |, )P E ( ) ( )E E t E t

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

max| |

0

( ) ( ) ( ) ,E

q q

qS E E P E d E

q

qS ~)(

• 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).

qqH q)(

q

1q q 1

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.

0 0( , ) ( ) ( )s ssP E P E

1s H 1q q

Structure Functions of Electric Field Fluctuations

Indication of multiple physical regimes of time scales

log Sq vs. log τ not a straight line

Regimes 1 2 3 4

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.

q

H(q) is not a constant, varying considerably.

Indications of multifractal behaviorWith traditional structure function analysis:

0 0( , ) ( ) ( )s ssP E P E

0( ) ( , )ssP Y P E where 0( ) sY E

Single-parameter scaling does not apply well to the multifractal electric field fluctuations.

Apply single-parameter scaling formula ( ms):0 5

0.69s 0.9s

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.

| |E

q

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

( )0( ) s YY E

( ) ( )0 0( , ) ( ) ( )s Y s Y

sP E P E

( )0( ) ( , )s Y P E

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

[ , ]low highY Y Y

0

0

( )

( )

' ( ) ( ) ,

shigh

slow

Yq

q

Y

S E P E d E

'' ( ) ~ q

qS

'q qs

Example: Regime 1 Y1 = [0.8, 1.2]

s1 = 0.80 from this plot

With increased resolution, s1 = 0.804

1'q qs

Validity of the solution

Note: negative q is applicable

1'q qs

Plot of scaling parameter s1 for different ranges of Y1

In principle, s1=s1(Y1) a continuous spectrum; but for practical purpose, statistics reaches limitation as Y-ranges keep decreasing

Considerable variation of s1 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 2Developing 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 1 Regime 2

Regime 3 Regime 4

Regime 1 Regime 2 Regime 4

Resemblance in shape between s(Y) and H(q)

Resemblance in shape between s(Y) and H(q)

max| |

0

( ) ( ) ,E

q

qS E P E d E

q

qS ~)( ( ) qH q q

q increases fractal property at larger |δE| is emphasized

0

0

( )

( )

' ( ) ( ) ,

shigh

slow

Yq

q

Y

S E P E d E

'' ( ) ~ q

qS 'qs q

for each Y-range

Y increases |δE| increases

Exception: Regime 3 Reason:

Significant decrease in s(Y) over a small range of Y

( )0( )s YE 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

| |E'q qs

( )0( ) s YY E

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)

s1

s2

s3

s4

Except for highly unstable turbulence, a generally decreasing trend for si at given Yglobal 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 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