Proceedings of ITC/ISHW2007

Scaling Laws of Intermittent Plasma Turbulencein Edge of Fusion Devices
1,22

V.P. Budaev , S. Masuzaki , T. Morisaki2, N. Ohno3, N. Asakura4 , S.Takamura5 , H. Yamada2, A. Komori2
1

Nuclear Fusion Institute, RRC Kurchatov Institute, 123182, Kurchatov Sq.1, Moscow, Russia
National Institute for Fusion Science, Oroshi 322-6, Toki, Gifu 509-5292, Japan3
Eco Topia Science Institute, Nagoya University, Nagoya 464-8603, Japan
4
JAEA, Naka Fusion Research Establishment Naka-machi, Naka-gun, Ibaraki-ken, Japan.
5Faculty of Engineering, Aichi Institute of Technology , 1247 Yachigusa, Yagusa-cho, Toyota, Japan(Received day month year / Accepted day month year)
2The high-order structure functions have been analyzed to characterize the edge plasma intermittency in fusion devices.
The scaling properties of edge turbulence have shown a strong deviation from a prediction of the Kolmogorov’s K41
model. The turbulent fluctuations demonstrate a generalized scale invariance and log-Poisson statistics.
Keywords: edge plasma turbulence, intermittency, log-Poisson model, extended self-similarity, multifractal statistics
1. Inroduction
Incompressible hydrodynamic turbulence is describedby Kolmogorov theory [1] (K41) which considers ahierarchical energy cascade and the energy scaling E(k)~k-5/3. At large Reynolds numbers Re>1, statisticalquasi-equilibrium of fluctuations is established in theprocess of energy transfer from large turbulent eddies
towards the smaller ones in the inertial range l (η << l << L,
where L is the largest eddy/ structure scale, η - dissipation
Kolmogorov’s scale). Investigation of self-similarturbulence properties resulted in multiplicative hierarchicalcascade models [2]: the log-normal model K62, theβ-model, and most favorable the log-Poisson model [3, 4].
To quantify whether boundary conditions influence thestatistical properties in turbulence it was proposed
multifractal formalism (see [2]). Intermittency leads to a
local breaking of the turbulence homogeneity, when 'active'regions coexist with quasi-laminar ones. The intermittent
fluctuations
demonstrate
non-Gaussianstatistics,
self-similarity and multifractality. The intermittency isobserved in turbulent hydrodynamic fluids [3] and inturbulent magnetized plasmas (see e.g [5]).
The spectrum of magnetohydrodynamic (MHD)
turbulence was first addressed by Iroshnikov andKraichnan (IK) who considered the turbulent energy
cascade affected by a magnetic field (see [6,7]). The IK
model yields E(k)~k-3/2 resulted from Alfven decorrelation
effect. The validity of the two phenomenologies (K41 andIK) in MHD turbulence and plasma confined in fusion
devices is still under a discussion. Numerical andexperimental data indicate that in MHD turbulence theauthor’s e-mail: budaev@mail.ru

energy transfer occurs predominantly in thefield-perpendicular direction [6]. This raises a question
whether anisotropy is crucial for the energy cascade, andwhether it changes the spectrum of turbulence. The
two-dimensional direct numerical simulations (DNS)
support the IK picture, while three-dimensional
simulations and recent analytical results suggest K41
energy spectra (see [6]). A phenomenology of“intermediate” turbulence by Goldreich and Sridhar [7](GS95) postulates a balance between K41 and IK energy
cascades and accounts for the local anisotropy induced bya magnetic field. The GS95 assumes one-dimension
filaments as the most dissipative structures, the same as inhydrodynamic turbulence. Other MHD turbulence models
assume the singular structure shape of a current sheet [6,7].
There are numerous experimental observations of
the magnetized plasma turbulence that share a lot offeatures of neutral fluid turbulence including many
scales, the cascades, strong mixing, non-linear scalings
and so on. Despite equations described neutral fluids
and plasmas are different they have the same type ofscale invariance (dilatation symmetry, namely, x→λx,t→λ1-ht, υ→λhυ). This common dilatation symmetry is
responsible for a common scaling property.
Most favorable log-Poisson model of intermittent
turbulence consider anisotropic stochastic cascade and thegeneralized self-similarity implying the long-range
correlations, which drive an anomalous transport, e.g.
superdiffusion. An experimental analysis of the statistical
moments in the frame of the log-Poisson model suggests adescription of the transport processes in the real turbulent

Proceedings of ITC/ISHW2007
plasmas. Despite the large amount of experimental datathat has been obtained in fusion devices, ourunderstanding of the turbulence and diffusive transport
process in magnetized plasmas is still rather limited. In
this work we focus on the scale invariance property andscaling laws of the edge plasma turbulence in fusion
devices and test analytical fits.
2. Experimental data
We analyze Langmuir probe signals of ion saturation
current Isat that is essentially plasma density. On Large
Helical Device [9] Isat was measured by 16 graphite
dome-type electrodes (diameter of 1 mm separated by 6mm) embedded in the divertor plate and by reciprocatingprobe. On JT-60U tokamak measurements (L-mode) in
the SOL has been done by reciprocating Mach probes [10]installed at the low field side (LFS) mid-plane and justbelow the X-point .On T-10 tokamak [5] (R/r= 1.5 m / 0.4m, Ip=200÷220 kA, B=2.2÷2.4 T) Langmuir probe(tungsten tips of 0.5 mm in a diameter and 3 mm in alength) Isat was measured at SOL in a steady state ofrepetitive ohmic discharges with no MHD activity. In thelinear divertor plasma simulator NAGDIS-II (ne~ 1020
m-3 , Te~ 1–3 eV , B= 0.25 T ) attached and detachedplasmas were investigated [11,5]. All signals are digitizedwith sampling rate 0.5 MHz in JT-60U and LHD and 1
MHz in other devices, the sample number 105 - 106.
Signals have the intermittent behaviour typically observedin edge of fusion devices [5]. Power spectra exhibit acomplicated frequency dependence without a trivial
power-law behavior [5].
3. Generalized scale invariance
The classical approach [12] for an exploration of
statistical features is an analysis of statistical moments(structure functions) Sq(τ)= 〈|X(t+τ)-X(t)|q〉, where 〈…〉means an ensemble average of the time-dependent signal
X(t). The structure function technique is equivalent to thedetailed investigation of the probability distributionfunction of the turbulent fluctuation. The Kolmogorov
theory К41 of isotropic turbulence infers Gaussianstatistics for fluctuations. It predicts the structure function
scaling Sq(τ)~τζ(q), ζ(q)=q/3 , in the inertial range.
Experimental
investigations
ofthedevelopedhydrodynamic turbulence demonstrate a departure of thehigh-order structure function scalings (q > 3) from theKolmogorov К41 prediction due to the intermittency. The
experimental structure functions typically shows a powerlaw only over a limited (inertial) range η << l << L (of ~10µs in fig.1). Whereas the generalized self-similarity isregistered in a broadened range extended to the dissipation
scales [13,2]. A generalized scale invariance (extended
self-similarity - ESS) was proposed in [13], and thenconsidered in the log-Poisson model of turbulence [3,4].
Hidden statistical symmetries of the Navier-Stockes

equations, hierarchy of moments, multifractality are behindthe property of the ESS. The ESS infers a scaling Sq(l)~S3(l) ζ(q)/ ζ(3) for the extended range l ≥ 5η. All data fromfusion devices, that we analyzed, demonstrate such ESS
property(linear behavior in fig.2) over the time scales ~1Fig.1. The structure functions Sq(τ) of high orders (q=2÷8from bottom to top) vs. time scale τ. (a) LHD divertor probe#10 (b) LFS SOL JT-60U,shot#44421

Fig.2. ESS plot of the structure function Sq(τ) of high orders(q=2÷8 from bottom to top) from the third-order one. (a) LHD
divertor probe, (b) LFS SOL JT-60U,distance from separatrix
42mm (c) SOL Т-10

msec substantially longer than an inertial range in fig.1.
The ESS corresponds to the considering of the scaling in aturbulent cascade not with respect to the usual distance, butwith respect to an effective scale defined by the third ordermoment of the velocity field.
4. The log-Poisson model of the turbulence
The scaling of the third-order moment can be deduced
analytically (ζ(3) =1), therefore scaling of ζ(q)/ ζ(3) can beanalyzed in experiments to improve the precision of thescaling estimation, especially at moderate Reynolds
numbers assessed in experiments. It allows to obtain moreaccurate values of ζ(q) by using a property of the ESS
plotting Sq as a function of S3, fig. 2. We treat experimental
scalings in the frame of log-Poisson turbulence model [4]predicted a scaling:q⎤q∆ ⎡3⎢1 −β ⎥
3 1 −β⎣⎦ζ ( q ) = (1 − ∆ ) +(1).

It is based on the hypotheses of a “hidden symmetry”
and a hierarchical structure of the moments of the energy
dissipation. The logarithm of energy dissipation obeys thePoisson statistics (so-called the log-Poisson statistics)
characterized by special scale-covariance properties. AProceedings of ITC/ISHW2007
hidden symmetry can be interpreted as a generalized
scale covariance and β is a characteristic of theintermittency of the energy dissipation (β=1 for
non-intermittent fully developed turbulence). Thequantity εl∞ (associated with the most intermittent
dissipative structures) has a divergent scaling εl∞ ~ l-∆, as l
→ 0, where ∆ is a parameter depending on thedimension of the dissipative structure. In an isotropic 3D
turbulence ∆= β=2/3 which is obtained if the mostdissipative structures are filaments. The ESS property isinvolved in the log-Poisson model. We use the wavelet
transform modulus maxima method (WTMM) [5] toestimate ζ(q) from experimental signals. In fig.3, theThe log-Poisson model of 3D turbulence wasmodified in MHD case to account for the IK
phenomenology [7]. To test IK model we plot in fig.4
a deviation of relative exponents ζ(q)/ ζ(4) from IK
scaling q/4. The scaling of data in the vicinity of X-point
in JT-60U is close to the IK indicating strong MHD
turbulence property. The data from SOL are deviatedstrongly from IK scaling (see [6]). At the same time they
are not fitted by the scaling for MHD log-Poisson model(see definition in [7]). It can be interpreted that IK
phenomenology (two-dimension strong anisotropy) is notavailable for a treatment of the SOL plasma turbulence.
0.05

ζq/ζ4-q/4

0-0.1-0.2-0.3
0Iroshnikov-Kraichnan
MHD log-Poisson
SOL LFS JT-60U
X-point JT-60U
LHD SOL
T-10 SOL
NAGDIS-II
24q

68Fig.4. Deviation of scaling from Iroshnikov-Kraichnan scaling
q/4 (solid line). MHD IK log-Poisson scaling (dashed line).
Scaling in the vicinity of X-point JT-60U is close to IK.

5. Transport scaling laws

Fig.3. (a) Structure function scaling vs. order q. Kolmogorov
К41 (a dashed line) and log-Poisson (a solid line, β=∆=2/3)
models and (b) the same for a departure of the scaling fromthe K41.

scalings ζ(q)/ ζ(3) are shown in the same plot with thescalings predicted by the K41 and the log-Poisson models.
The scalings are anomalously deviated from the K41
scaling, fig.3b. Each experimental scaling could be fittedby (1) with adjusted parameters ∆ and β. A solving of
non-linear least-squares problem of fitting to (1) gives
indexes in the range ∆ =0.15 ÷ 0.8, β=0.25 ÷0.7. Somesignals have non-intermittent behaviour (β=1). Theobserved range of ∆ (between 1/3 and 2/3) can beinterpreted [14] that the most intermittent dissipativestructures are one-dimensional filament structures inthese cases. Such dissipative structures have most likely
not a trivial geometrical topology but a fractal one.

Fig.5 Diffusion scaling index (a) JT-60U LFS SOL (shot#44421) (b) SOL LHD high β shots vs. vertical coordinate z,
B=0.425 T

The statistical description of transport processes infusion plasmas is an alternative approach to thetraditional characterization of a transport based on thecomputation of effective transport diffusion coefficients.

Proceedings of ITC/ISHW2007
The log-Poisson model could be used to estimate atransport scaling based on the self-similarity indexes βand ∆ (1) that responsible for percolation effect in theturbulence. In a simplified approach [14], the diffusion
scaling depends on the structure function scaling asD ∝ τК(-1), the exponent K(q) relates with the scaling of
the high-order structure function ζ(q) as K(q)=q-ζ(3q). A
displacement of particles across a magnetic field withtime τ is scaled as 〈δx2〉∝Dτ∝τα with an exponent α ∝ 1+K(-1). This index was estimated from experimental

connection length Lc reaches the ergodic layer
surrounding the core plasma region. Lc varies from lessthan a few meters to over a few kilometers (see black linein fig.6a,b ). Ion particle flux to the divertor plates [9] andmultifractality level (deviation from Gaussian statistics,
see a definition in [5]) follows the deposition profile ofthe magnetic field lines, fig.6a. The probe connected to
the field line with a large Lc has a large ion particle flux.
In LHD, an increasing of α>1 was observed in domainsthat are characterized by short connection length (probe #7 in fig.6b). Scalings illustrate specific behaviour of #7
probe location compare with others that are closer to K41
scaling, fig.6c. This property is kept even at discharge
evolution at super-dense core (SDC).
In conclusion, the statistical properties of theintermittent turbulence show a striking empirical similarity
in the SOL plasma region of fusion devices. Scalings of thestructure functions strongly deviate from the Kolmogorov’s
K41 theory prediction. The anomalous behavior of scaling
is similar in the SOL plasma of helical device, tokamaks,
and linear machine. Experimental scalings are close to thelog-Poisson model. One-dimension filament structures arelikely the most intermittent dissipative structures. The
similar behavior of the scalings has been observed in theedge of fusion devices with different magnetic topology
and heating. It supports a view that the edge plasma
turbulence displays universality. By using self-similarity
indexes, transport scaling indexes are estimated frompercolation property of the turbulence with non-trivial
self-similarity. The results of our study improve ourunderstanding of intermittent turbulence in the edge offusion devices.
Acknowledgements. This work was supported by the NIFS
budged NIFS06ULPP511, JT-60U budged,the Rosatom RF
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Fig.6 LHD divertor probes data, shot #68995 with SDC.
(a) Multifractality level, (b) Diffusion scaling index α,(c) Structure function scalings, deviation from K41, cf. fig3a.
Magnetic connection length Lc (logarithm of magnitude,a.u.)in black line, and plasma storage energy evolution (magenta)

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