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

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 Ps : P(δX, τ) τs=Ps(δX/τs)=Ps(Y), with Y= δX/τs (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) τs(Y)=Ps(Y) with Y= δX/τs(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.

ROMA •The key in ROMA is then to find s(Y) and Ps(Y) from P(δX, τ) with the following scaling: P(δX, τ) τs(Y)=Ps(Y) with Y= δX/τs(Y) (2)• Existence of s(Y) and Ps(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 Ps(Y): (a) using (2) directly. [Given s →Y] (b) using ranked-ordered structure functions for the small range Y1<Y<Y2:

with α=Y1 τs, β=Y2 τs. Search for s such that Sm ~ τ sm and s(Y)=s. [Given Y →s]• Consistency check: From s(Y) and Ps(Y), P(δX, τ) can be calculated from the scaling relation (2) and can be checked with the data/model results.

Xd),X()X(),X(

PmmS

ROMA •3D fluid turbulence from the JHU turbulence database •2D MHD simulations•Solar wind data

•Finding s(Y) and Ps(Y) using method (a) and Consistency check

3D fluid turbulence flowfrom the JHU turbulence database cluster

turbulence.pha.jhu.edu

• Forced isotropic turbulence:• Direct numerical simulation (DNS) using

1,0243 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)

3D fluid turbulence flow

• There are 10244 data points in the data set. • Here we use only 5 x 10242 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)| = |vx(r+δix)-vx(r)| or |vy(r+δiy)-vy(r)| and δ= (16,…, 160) Δ; Δ=grid spacing=2π/1024.

• According to Kolmogorov (K41), S3(δv||,δ) ~ δ, meaning 3 s=1 and s=1/3.

3D fluid turbulence flow

PDF(δv||,δ) on 5 z-planes: left panel with δ=32Δ and Right panel with δ=96Δ.

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. 1dX),X(PDF

S=0.2: pdfs collapse at Y~38.5 and Ps~1.16 10-2

Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δgreen:48 Δ; black: 64 Δ.

S=0.3: PDFs collapse at Y~25 with Ps~1.8 10-2, and Y~144 with Ps~2.2 10-5.

Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δgreen:48 Δ; black: 64 Δ.

S=1/3: PDFs collapse at Y~20 with Ps~2.2 10-2, and Y~80 with Ps~8. 10-4.

Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δgreen:48 Δ; black: 64 Δ.

S=0.35 PDFs collapse at Y~17.5 with Ps~2.45 10-2, and Y~62 with Ps~1.98 10-3.

Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δgreen:48 Δ; black: 64 Δ.

S=0.4 PDFs collapse at Y~0 with Ps~3.9 10-2, and Y~32 with Ps~1.1 10-2.

Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δgreen:48 Δ; black: 64 Δ.

S=0.5 PDFs collapse at Y~15 with Ps~3 10-2.

Left: blue δ=32Δ; red δ=96Δ right: blue: δ=32Δ; red δ=96Δgreen:48 Δ; black: 64 Δ.

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

Consistency check 1: Given s(Y) and Ps(Y), one can compute PDF through the scaling relations: P(δX, δ)=Ps(Y)/τs(Y) and δX= τs(Y) Y. The results are consistent with the raw PDF from the simulation.

Computed PDFs by markers; raw PDFs by solid curvesRed 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.

Consistency check 2: δ=24, 48, 96, 128∆

Computed PDFs by markers; raw PDFs by solid curvesRed circles: δ=24∆; green squares: δ=48∆;Magenta diamonds: δ=96∆; blue triangles: δ=128∆

Consistency check 3: δ=16, 48, 96, 160∆

Computed PDFs by markers; raw PDFs by solid curvesRed circles: δ=16∆; green squares: δ=48∆;Magenta diamonds: δ=96∆; blue triangles: δ=160∆

Consistency check 4: sensitivity to changes in s(Y) and Ps(Y)

Y, s(Y), f(Y)Red circle: 40, 0.38, 7.1 10-3Green right triangle: 40, 0.36, 7.1 10-3Blue left triangle: 40, 0.40, 7.1 10-3Magenta up triangle:40, 0.38, 7.8 10-3Black down triangle:40, 0.38, 6.3 10-3

Black curves are PDFat δ=32, 48, 64, 96Δ

δ=32Δ

δ=96Δ

Consistency check 5: sensitivity to changes in s(Y) and Ps(Y)

Y, s(Y), f(Y)Red circle: 5, 0.398, 3.75 10-3Green right triangle: 5, 0.44, 3.75 10-3Blue left triangle: 5, 0.36, 3.75 10-3Magenta up triangle:5, 0.398, 4.05 10-3Black down triangle:5, 0.398, 3.45 10-3

Black curves are PDFat δ=32, 48, 64, 96Δ

δ=32Δ

δ=96Δ

δ=32Δ

δ=96Δ

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=96Δs

Ps

3D fluid turbulence: fluctuations of v2

Ps(Y)

3D fluid turbulence: fluctuations of v2

Computed PDFs by markers; raw PDFs by solid curvesRed circles: δ=16∆; green squares: δ=32∆;Magenta diamonds: δ=64∆; blue triangles: δ=96∆

2D MHD simulations: fluctuations of B2

sPs

2D MHD simulations: fluctuations of B2

Computed PDFs by markers; raw PDFs from simulations by solid curvesRed circles: δ=32∆; green squares: δ=48∆; blue triangles: δ=96∆ Left panel for a large range of δB2; right panel is an expanded view.

Solar wind data: fluctuations of B2 Scaled PDF [solar wind data: Chang, Wu, and Podesta, AIP Conf Proc, 1039, 75 (2008)]

From solar wind dataPs(Y) used in the calculation

Solar wind data: fluctuations of B2

ROMA spectrum from datas(Y) used in the calculation

Solar wind data: fluctuations of B2

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.

Solar wind data: fluctuations of B2

Using s=0.44 (monofractal) and the same Ps(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

Conclusion

ROMA is robust in the three cases studied here: 3D turbulent flows,

2D MHD simulation and solar wind data.