1. Application of rank-ordered multifractal analysis (ROMA) to intermittent fluctuations in 3D turbulent flows, 2D MHD simulation and solar wind data Cheng-chin Wu and Tom Chang

2. ROMA a generic fluctuating temporal X(t) a scale dependent difference series δX(t,τ)=X(t+ τ)-X(t) time lag τ the probability distribution functions (PDFs) P(δX, τ) of δX(t,τ) for different time lag values τ. If the fluctuating event X(t) is monofractal –- self-similar, the PDFs would scale (collapse) onto one scaling function P s : P(δX, τ) τ s =P s (δX/τ s )=P s (Y), with Y= δX/τ s (1) where s, a constant, is the scaling exponent. Data and model (MHD and fluid) results indicate turbulent flows are generally not monofractal and are multifractal. Chang and Wu [2008] proposed ROMA for multifractal fluctuations with the following scaling: P(δX, τ) τ s(Y) =Ps(Y) with Y= δX/τ s(Y) (2) where the scaling exponent s(Y) is a function of Y. Data and model (MHD and fluid) results indicate the applicability of ROMA. Some will be discussed in the talk.

3. ROMA The key in ROMA is then to find s(Y) and P s (Y) from P(δX, τ) with the following scaling: P(δX, τ) τ s(Y) =Ps(Y) with Y= δX/τ s(Y) (2) Existence of s(Y) and P s (Y) is not trivial: a function P(δX, τ) of two variables is replaced by two functions of a single variable. Two methods of finding s(Y) and P s (Y): (a) using (2) directly. [Given s →Y] (b) using ranked-ordered structure functions for the small range Y 1 <Y<Y 2 : with α=Y 1 τ s, β=Y 2 τ s. Search for s such that S m ~ τ sm and s(Y)=s. [Given Y →s] Consistency check: From s(Y) and P s (Y), P(δX, τ) can be calculated from the scaling relation (2) and can be checked with the data/model results.

4. ROMA 3D fluid turbulence from the JHU turbulence database 2D MHD simulations Solar wind data Finding s(Y) and P s (Y) using method (a) and Consistency check

5. 3D fluid turbulence flow from the JHU turbulence database cluster turbulence.pha.jhu.edu Forced isotropic turbulence: Direct numerical simulation (DNS) using 1,024 3 nodes. Domain: (2π) 3 Navier-Stokes (with explicit viscosity terms) is solved using pseudo-spectral method. Energy is injected by keeping constant the total energy in shells shuch that |k| is less or equal to 2. There is one dataset ("coarse") with 1024 timesteps available, for time t between 0 and 2.048. There is another dataset ("fine") that stores every single time-step of the DNS for t between 0.0002 and 0.0198)

6. 3D fluid turbulence flow There are 1024 4 data points in the data set. Here we use only 5 x 1024 2 values of velocity fields, which consists of values on 5 z-planes: (t, z) = (1, 0), (0, 9 Δ), (2, 99 Δ), (0.5, 499 Δ), and (1.5, 499 Δ) with Δ=grid spacing=2π/1024. fluctuating field δX(r,δ)=|δv || (r,δ)|=|[v(r+δi)-v(r)]·i|, with i unit vector. In the calculation: |δv || (r,δ)| = |v x (r+δi x )-v x (r)| or |v y (r+δi y )-v y (r)| and δ= (16,…, 160) Δ; Δ=grid spacing=2π/1024. According to Kolmogorov (K41), S 3 (δv ||,δ) ~ δ, meaning 3 s=1 and s=1/3.

7. 3D fluid turbulence flow PDF( δv ||,δ) on 5 z-planes: left panel with δ=32Δ and Right panel with δ=96Δ.

8. 3D fluid turbulence flow PDF(δv ||,δ) average over 5 z-planes: blue with δ=32Δ and red with δ=96Δ. Normalization: Note the cross over of PDFs.

9. S=0.2: pdfs collapse at Y~38.5 and P s ~1.16 10 -2 Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δ green:48 Δ; black: 64 Δ.

10. S=0.3: PDFs collapse at Y~25 with P s ~1.8 10 -2, and Y~144 with P s ~2.2 10 -5. Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δ green:48 Δ; black: 64 Δ.

11. S=1/3: PDFs collapse at Y~20 with P s ~2.2 10 -2, and Y~80 with P s ~8. 10 -4. Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δ green:48 Δ; black: 64 Δ.

12. S=0.35 PDFs collapse at Y~17.5 with P s ~2.45 10 -2, and Y~62 with P s ~1.98 10 -3. Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δ green:48 Δ; black: 64 Δ.

13. S=0.4 PDFs collapse at Y~0 with P s ~3.9 10 -2, and Y~32 with P s ~1.1 10 -2. Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δ green:48 Δ; black: 64 Δ.

14. S=0.5 PDFs collapse at Y~15 with P s ~3 10 -2. Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δ green:48 Δ; black: 64 Δ.

15. Summary: blue + and green * indicate obtained s(Y) and P s (Y). s(Y) and P s (Y) given by red curves are used in the consistence check.

16. Consistency check 1: Given s(Y) and P s (Y), one can compute PDF through the scaling relations: P(δX, δ)=P s (Y)/τ s(Y) and δX= τ s(Y) Y. The results are consistent with the raw PDF from the simulation. Computed PDFs by markers; raw PDFs by solid curves Red circles: δ=32∆; green squares: δ=48∆; Magenta diamonds: δ=64∆; blue triangles: δ=96∆ Left panel for the whole range of δv || ; right panel is an expanded view.

17. Consistency check 2: δ=24, 48, 96, 128∆ Computed PDFs by markers; raw PDFs by solid curves Red circles: δ=24∆; green squares: δ=48∆; Magenta diamonds: δ=96∆; blue triangles: δ=128∆

18. Consistency check 3: δ=16, 48, 96, 160∆ Computed PDFs by markers; raw PDFs by solid curves Red circles: δ=16∆; green squares: δ=48∆; Magenta diamonds: δ=96∆; blue triangles: δ=160∆

19. Consistency check 4: sensitivity to changes in s(Y) and P s (Y) Y, s(Y), f(Y) Red circle: 40, 0.38, 7.1 10-3 Green right triangle: 40, 0.36, 7.1 10-3 Blue left triangle: 40, 0.40, 7.1 10-3 Magenta up triangle:40, 0.38, 7.8 10-3 Black down triangle:40, 0.38, 6.3 10-3 Black curves are PDF at δ=32, 48, 64, 96Δ δ=32Δ δ=96Δ

20. Consistency check 5: sensitivity to changes in s(Y) and P s (Y) Y, s(Y), f(Y) Red circle: 5, 0.398, 3.75 10-3 Green right triangle: 5, 0.44, 3.75 10-3 Blue left triangle: 5, 0.36, 3.75 10-3 Magenta up triangle:5, 0.398, 4.05 10-3 Black down triangle:5, 0.398, 3.45 10-3 Black curves are PDF at δ=32, 48, 64, 96Δ δ=32Δ δ=96Δ δ=32Δ δ=96Δ

21. Red circles: δ=16∆; green squares: δ=48∆; Magenta diamonds: δ=96∆; blue triangles: δ=160∆ Consistency check 6: PDF for 0.4 < s(Y) < 0.8 δ=96Δ s PsPs

22. 3D fluid turbulence: fluctuations of v 2 P s (Y)

23. 3D fluid turbulence: fluctuations of v 2 Computed PDFs by markers; raw PDFs by solid curves Red circles: δ=16∆; green squares: δ=32∆; Magenta diamonds: δ=64∆; blue triangles: δ=96∆

24. 2D MHD simulations: fluctuations of B 2 s PsPs

25. Computed PDFs by markers; raw PDFs from simulations by solid curves Red circles: δ=32∆; green squares: δ=48∆; blue triangles: δ=96∆ Left panel for a large range of δB 2 ; right panel is an expanded view.

26. Solar wind data: fluctuations of B 2 Scaled PDF [solar wind data: Chang, Wu, and Podesta, AIP Conf Proc, 1039, 75 (2008)] From solar wind data P s (Y) used in the calculation

27. Solar wind data: fluctuations of B 2 ROMA spectrum from data s(Y) used in the calculation

28. Solar wind data: fluctuations of B 2 Computed PDFs from the scaling relations are shown in the front; data are shown in the back. Green (o): τ=1000s; blue (x): 96s; red(+): 9s.

29. Solar wind data: fluctuations of B 2 Using s=0.44 (monofractal) and the same P s (Y), computed PDFs from the scaling relations are shown in the front; data are shown in the back. Green (o): τ=1000s; blue (x): 96s; red(+): 9s. S=0.44

30. Conclusion ROMA is robust in the three cases studied here: 3D turbulent flows, 2D MHD simulation and solar wind data.