Proceedings of ITC/ISHW2007

A Comparison of Intermittency in Neutral Fluids andMagnetically Confined Plasmas
Milan RAJKOVIĆ1) , Miloš ŠKORIĆ2) and Mitsuo KONO3)
1)

Institute of N uclear Sciences V inča, Belgrade, Serbia
N ational Institute f or F usion Science, T oki 509 − 5292, Japan,
3)
Chuo U niversity, Hachioji, T okyo, Japan

2)

Intermittency in the scrape-oﬀ layer (SOL) region of magnetically confined plasma and velocity
intermittency of helium gas are compared from the aspect of global and local multifractal properties.
The analysis reveals subtle diﬀerences in the global aspect between turbulent fluctuations of confinedplasmas and neutral fluid as well as diﬀerences among magnetically confined devices themselves
especially with respect to energy cascade mechanisms in the corresponding systems. Analysis of local
multifractal characteristics reveals even more pronounced diﬀerences.
Keywords: turbulence, plasma edge turbulence, intermittency, multifractal

1 Introduction
Velocity field statistics are an essential tool and themajor source of information in neutral fluid turbulence. The theory of local velocity field structure ofincompressible hydrodynamic turbulence, introduced
by Kolmogorov in 1941 and usually denoted as K41,
is based on the assumptions of homogeneity, isotropy,
and the existence of an inertial range. The inertial
range in the kinetic energy spectrum is a range in wave
number, k˜1/l where l is the separation length between two points (eddie size), which is located between
typical large scales and small dissipative scales and istherefore independent of external drive or output dissipation. The spectrum in the inertial range does notdepend on viscosity and is characterized by the power−2/3 −5/3k,
law dependence of the form E(k) = CK h iwhere CK is the Kolmogorov constant and h i is theaverage rate of energy dissipation per unit mass.
The influence of the external magnetic field on
turbulent flow of an incompressible fluid leads to much
more complicated behavior which modifies turbulent
motions. Even more dramatic eﬀects are observed in
the case of compressible fluids. Hence, significant differences exist between turbulent flows of neutral fluids and confined plasma. In particular, nonlinearities
in plasma turbulence are more numerous having different spectral cascade directions in addition to theE × B nonlinearity, leading to more complex fluctuating characteristics. One of the most important differences is that time and space measurements lead todiﬀerent information on the structure of turbulence[1]. Driving mechanisms and damping characteristics
are reflected in the temporal aspect of fluctuations
author’s e-mail: milanr@vin.bg.ac.yu

while measurements at diﬀerent spatial locations provide information on spatial structures for various scale
lengths. For the case of neutral fluids, time records ofturbulent velocity at a single spatial location obtainedwith the use of a hot-wire or laser Doppler anemometer, are usually interpreted via Taylor’s frozen flow
hypotheses, as one-dimensional spatial cuts throughthe flow. However, this approach that generates information about temporal measurements from spatial
ones and vice versa, is not applicable in the case ofplasma turbulence. Specifically, turbulence in the caseof neutral fluids is generated at a certain spatial position and carried by the flow past the probe location so that recordings at diﬀerent times at a fixedlocation are equivalent to simultaneous recordings at
diﬀerent spatial locations along the flow. However,in plasma turbulence due to specific nature of nonlinearities, turbulence is created and damped at thesame spatial position where measurements are taken
so that spatial and temporal informations are interwoven. For the same reason the inertial range [2], may
exist only locally in space or in time, and the extent
of this range changes along the temporal scale as well
as along space, for example along poloidal direction.
In this study we present diﬀerences between these
two types of turbulences from the aspect of their global
and local multifractal properties. In particular we areinterested in the intermittency phenomenon in the twocases, and in the case of confined plasma we study theintermittency in the scrape-oﬀ layer (SOL) and also
compare three diﬀerent confinement regimes, the Lmode, dithering H-mode and the H-mode. The quantity of interest in the case of neutral fluid turbulence isone-component fluid velocity while in the plasma case
it is the ion saturation current fluctuations of recipro-

Proceedings of ITC/ISHW2007
cating Langmuir probe installed at the edge of magnetic confinement devices. We study intermittency
properties of the MAST spherical tokamak (L-mode,dithering H-mode and H-mode) and the Tore Supratokamak with limiter configuration (L-mode). Intermittency of liquid helium recorded in a specially designed cryogenic apparatus [3], is considered in thecase of neutral fluid turbulence. The specific designof the apparatus enabled access to a wide range ofReynolds numbers at constant geometry.
The important issue in the multifractal spectra analysis of intermittent plasma turbulence is thechoice of relevant measure. In neutral fluid turbulence, in addition to velocity, enstrophy and energy
dissipation represent quantities of particular interest
although they cannot be constructed in their entirety
from a single point velocity time-series. To overcome
the diﬃculty surrogate dissipation
surr (x) = Cν

µ∂vx
∂x

¶2,

Fig. 1 Fig. 1 LDS of 6861 L-mode of the MAST device(1)

where C is a constant, sometimes taken equal to 15 isusually used . Using Taylor’s frozen flow hypothesis
which is naturally justified in neutral fluid turbulence,expression (1) becomessurr (t) ∼ νµ∂vx
∂t

¶2.

(2)

In the plasma case, based on the arguments explained
in detail in [4], we devise two measures yielding identical multifractal spectraand¡¯ dn ¯ ­¯ dn ¯®¢2
¯n ¯ − ¯n ¯D= c · ¡¯ dt ¯ ­¯ dt ¯®¢2 E ,¯n dn ¯ − ¯n dn ¯dtdt³¡¢2 D¡ dn ¢2 E´2− n dt
n dndtÀ,= c · ¿³
¡ dn ¢2 D¡ dn ¢2 E´2n dt − n dt

(3)
Fig. 2 Fig. 2 LDS of 9031 L-mode of the MAST device(4)

where c is a constant.

2 Large Deviation Spectra
The datasets analyzed here consist of measurementsof the ion saturation current (ISAT ) performed by themoveable Langmuir probe located at the outboard
midplane on MAST device [5], at a sampling rateof 1 MHz. Discharge 6861 is high density L-mode
plasma and 9031 represents a dithering H-mode withheating power close to the threshold for L-H transition with intermittent high frequency edge localized
modes (ELMs), while 5738 is an H-mode with ELMs

Fig. 3 LDS of the H-mode of the MAST device

Proceedings of ITC/ISHW2007
to the MAST turbulence.

3 Local Turbulence properties
Local properties of turbulence are modeled with fractal Brownian motion [7]. Since fBm is a self-similar
process it provides unique parameter values for H andσ in the restricted temporal domain in which turbulent signal is self-similar (the Hurst exponent H determines the correlation distance for the incrementsof the process and the quantity σ 2 quantifies the absolute level of correlations). For this purpose a rangeof frequencies over which the power law
Fig. 4 LDS of the L-mode of Torre Supra.

Fig. 5 LDS for the neutral fluid turbulence, Re=328.

at 400 Hz. The corresponding Large Deviation Spectra (LDS) are presented in Figs. 1, 2 and 3. LDS [6]reveal diﬀerences between each confinement regime aswell as between the two devices. The most striking
feature of the MAST spectra (Figs. 1,2 and 3) is theirdeparture from a pure bell-shape and concavity andis a good example where LDS provide more information than Legendre spectra, which are strictly concave
although they may be asymmetrical. Their shape reflects existence of several multiplicative laws underlying the cascade processes so that there is a lumpingof measures whose supports are disjoint. It is evident
that the L-mode of MAST has more complex multifractal structure in the sense that there are more αvalues at which the irregularity of the spectrum occurs(i.e. more phase changes) than in the case of L/Hmode or H-mode of MAST. The most striking featurein the Tore Supra L-mode spectra is nonexistence orvery mild lumping of measures with no superpositionof measures. In Fig. 5 we show the LDS for the caseof neutral fluid turbulence.The LDS reveal multifractal similarity of neutral fluid turbulence and the ToreSupra L-mode, while a striking diﬀerence with respect

PBH ∝ σ2 |ω|−(2H+1) .

(5)

pertains is detected and then the parameters from thesame expression are evaluated. Wavelet scale spectra
are used for this purpose because they provide timescale decomposition that is compliant with power law
processes, independent of their stationarity. Parameters of the power-law model, σ and H are functions oftime and model is applied only over a subset of scales
known as the inertial range. Usually multifractal data,
besides variations in σ and H, show variations in theinertial range itself.
The main steps in estimation of local turbulence
properties are the following:
1. Partitioning of data into segments of equal
temporal extent within which turbulent signal is approximately stationary. A special filtering procedure
is devised in order to remove dependence of the estimated parameters on segmentation.
2. Wavelet decomposition of the data and evaluation of the scale spectra within each segment.
3. Determination of the inertial range of the scale
spectra and evaluation of the power law parameters
based on the fBm model. The turbulent data corresponding to the inertial range are assumed to be statistically well represented by fractal Brownian motion.
Local turbulence parameters σ and H are therefore determined from the scale spectra corresponding to theinertial range. The extent of the inertial range varies
from segment to segment and is rarely equal to thesegment size. Since inertial range exists over specificscales (or equivalently over corresponding time range)
evaluated parameters have local character.
In Figs.6 and 7 we present temporal variations of
the Hurst exponent for the MAST L-mode and theL-mode of Tore Supra. The same quantities are presented in Fig. 8 for the case of neutral fluid turbulence.
A brief comparison of Figs. 6, 7 and 8 shows that
there are considerable diﬀerences in values of the local
Hurst exponent and variance at unit lag for the twomagnetic confinement devices and for the neutral fluidProceedings of ITC/ISHW2007

Fig. 6 Parameters of the fBm model, Hurst exponent andthe variance at unit lag for the L-mode in MAST.
Note random variations of each parameter reflecting multifractal character of the plasma density
fluctuations.

Fig. 7 Parameters of the fBm model, Hurst exponent andthe variance at unit lag for the L-mode of Torre
Supra. Note very small variations of both quantity.

Fig. 8 Parameters of the fBm model, Hurst exponent andthe variance at unit lag for the neutral fluid case,
Re=328.

turbulence. Namely, these local fractal quantities exhibit purely stochastic variation for the case of MAST
L-mode while for the case of the L-mode of Tore Suprathese variations are very small and deterministic-like.
On the other hand stochastic variations are evident
in the case of neutral fluid turbulence indicating its
multifractal character, however the values of H and σ
are very much diﬀerent in two cases (and in all cases
presented here). Hence, in spite of similarity of largedeviation spectra for the L-mode of Tore Supra andthe neutral fluid turbulence, their local (multi)fractal
characteristics are very diﬀerent. Briefly summarizing
the multifractal features of each type of turbulence,
several characteristics are immediately evident:
1. Large deviation spectra reveal considerable differences between diﬀerent regimes and diﬀerent devices of magnetically confined plasma. These disparities indicate diﬀerent cascade mechanisms of energy
transfer among coherent structures.
2. Local features of turbulence in all cases exhibit
unique variability and range of values with the possibility of multifractional behavior (not purely multifractal) in certain devices instead of the multifractal
properties which dominate both the magnetically confined plasma turbulence and neutral fluid turbulence.
3. Neutral fluid turbulence has very well defined
inertial range which is well captured in the wavelet
scale spectra irrespective of the segmentation of theturbulent signal. On the other hand scale spectra reveal that in the plasma case inertial ranges are hardto detect and to a large extent depend on the segmentation of the signal. These properties pertain to all
regimes in spite of particular features of each. Moreover the extent of inertial ranges is diﬀerent for allcases considered.
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