J. Plasma Fusion Res. SERIES, Vol. 5 (2002) 418423

Effect of Rotating Helical Magnetic Fieldon the Turbulence Fractal
BUDAEV Viatcheslav, KIKUCHI Yusuke, TOYODA Mitsuhiro, UESUGI Yoshihikoand TAKAMURA Shuichi

Department of Energy Engineering and Science, Graduate School of Engineering,
Nagoya University, Nagoya 464-8603, Japan(Received: 11 December 2001 / Accepted: 19 June 2002)

Abstract
The turbulence characteristics related to the fractal structure and turbulent transport in the edge
plasma of tokamak HYBTOK-II with variation of Rotating Helical Field (RHF) frequency have beenstudied. It was shown that the turbulence has a multifractal structure. The effect on turbulence fractal
structure and transport characteristics depends on the RHF frequency. The intermittency parameter as ameasure of multifractality level, and turbulent flux are increased with the RHF near the resonant surfaces
and in the SOL. The intermittent bursts formed by fluctuations with a time scale of 40-100 pr, are more
intensive in the RHF experiment.

Keywords:

1. Introduction
The control of edge plasma turbulence andVarying the rotation frequency in external coils for theturbulence driven transport is one of the key issues toachieve a steady-state operation of tokamak. To improvethe edge plasma conditions the concepts of dynamic

resonant modes of drift wave turbulence in edge plasma

ergodic divertor [1] and poloidally rotating helicalthe fractal structure in tokamak HYBTOK-II withmagnetic field (RHF) 12-41 to induce the plasma rotation

variation of the RHF frequency.

RHF, a selective control (driving or damping) ofis expected. In this work, we have investigated the edge
plasma turbulence and transport characteristics related toat the edge have been proposed. Recently a generalapproach [5] to controlling chaos in physical systems
has been proposed it is based on the existence of
unstable periodic orbits within the strange attractor. In
the context of chaos control, it was proposed the delayed

2. Experimental Setup
HYBTOK-II has the major and minor radii of 40
cm and 12.8 cm, the plasma current of Ir= Q.9 kA, thetoroidal magnetic field of Bt= 0.27 T and operates atthe edge electron density about 2 x 1018 m-3 and theelectron temperature Z" about 20 eY. The RHF iscreated by two sets of local helical coils which areinstalled outside the vacuum vessel at eight toroidal
sections among 16 with the poloidal mode number of zr= 6 and the toroidal mode number of n = | [8]. These

feedback control with periodical modulation [6]. Usingsuch approach, experiment on spatiotemporal open-loop

synchronization of drift wave turbulence in amagnetized cylindrical plasma was carried out by phase-shifted sinusoidal driver [7]. The RHF on a small
tokamak HYBTOK-II provides a system to drive arotation with phase controlling by external exciter.

@2O02 by The Japan Society of Plasma
C o r re s p onding autho r' s e - mail : budaev @ e e s. nagoya -u. a c.j p418

Science and Nuclear Fusion ResearchBudaev V. et al., Effect of Rotating Helical Magnetic Field on the Turbulence Fractal Structure in the Tokamak Edge Plasma

two coils are powered by inverter power supplies

the fluctuations in Q(r), 6,Q) = 4Q)18,. They are used toindependently with the phase difference of 90 degrees,determine the radial turbulent flux as f,(r) = ft(t)i),(t).
The radial profile of plasma potential Vo estimated fromthe floating potential V1 and electron temperature 2", Vo= Vr * 37", is used to estimate the radial electric field d.
The poloidal E, x I drift velocity is estimated from oo =EtB.. We have analyzed the date over the radial rangeof 6.5 < r < 12.5 cm, through a sequence of repetitive
discharges in tokamak. The data were collected during 7
ms at quasi-steady state phase with a sampling rate ofand the RHF frequency was of 5-30 kHz in thisexperiment. The rotation was in the electron
diamagnetic drift direction. From the measurement ofpoloidal magnetic field, the resonance surface for themain mode of mln = 6/1 was estimated to be around r =8 cm , the magnetic island width of 0.8 cm [8].
Sideband components mln = 5ll andTll are resonant
near the surfaces r * 7 cm and r = 9.5 cm, respectively
with the same order of the magnetic island widths. The
movable Langmuir probe is installed at a midplane of

1.0 MHz.

the tokamak. It is composed of 4 tungsten tips with 0.5mm in diameter and 0.5 mm in length. The difference

3. Effect of the RHF on the Edge Turbulence

between the floating potentials on the two tips

Radial profiles of the averaged n, E* and Ze are notchanged significantly with application of the RHF. Itwas observed that the RHF has effect on the structure ofthe edge plasma fluctuations. To characterize this effectwe use the spectral, statistics and multifractal analysis ofthe time-series signals.

Fractal Structure(poloidally separated by 2 mm) is used to determine thepoloidal electric field Eo neglecting the electron
temperature fluctuations. The density fluctuation fr(r) is
deduced from the ion saturation current on the third tip,and the fluctuating radial drift velocity is deduced fromRHF off

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81012r, cmFig. 1 (a) Time traces of density n and driven radial particle flux f,. Bursty character is observed to change with the RHF
at r= 8.1 cm. (b) Power spectra S(f) = l,{(f }|'? of the density fluctuations. (c) Probabiliry distribution function of thedensity fluctuations normalized by standard deviation on(gray - RHF off, solid line - RHF on) demonstrates a deviation from the Gaussian (dashed line). (d) Radial dependence of the pdf's flatness (solid line- RHF off, x - 5 kHz. tr 15 kHz) illustrates an effect of the RHF on the pdfs tails.
419

Budaev V. et al., Effect of Rotating Helical Magnetic Field on the Turbulence Fractal Structure in the Tokamak Edge Plasma

dissipation scales in three-dimension turbulence is theintermittency. The energy dissipation rate lt = vl2(duildx, + du,ldx;)z is considered, u; is the i component of thevelocity, v is the viscosity. To estimate energy
dissipation rate we can consider a quadratic form [9] ofthe density fluctuation squared as a measure of energy.
To describe the multifractal character of the plasma
turbulence the scaling of the measures should beconsidered over different time scales. We construct thefollowing measures from thetime series of density data

The signals of the density r? and radial particle fluxl. (Fig. l(a)) possess a high frequency part and thepeaks of total intensity (referred as the bursts) caused by

the intermittent structure. There exist fluctuations ineach burst and maxima are separated by a time greaterthan the auto-correlation time. The characteristic timescale of the bursts is of 40-100 ps. This bursts appear tocontain a non-negligible amount of the turbulence
intensity with respect to the background turbulence.
Figure 1 demonstrates that the RHF effects on the bursty
behaviour of density and flux f,: intensity of the bursts
is increased with the RHF, at the same time theaveraged values are not changed significantly. Several{ni= n(t), i = 1,2,..., N},(r, -0t,>)'lri =scaling ranges with respect to the frequency are(1)#,t (,, -(n,))'registered in the frequency spectra (Fig. l(b)). We notethat no lf behavior of the spectra and no significantchange with the RHF in the high frequency range.

To calculate the measure over different time scales,
it is averaged over subblocks of data of a length T, p(Ti)The analysis of fluctuation statistics is an= i2

instrument to characterize the structures in the signal.
For this, probability distribution function (pdf), obtained
from a histogram of the signal, is studied. A deviationfrom a Gaussian (fully random process) pdf could be
due to existence of coherent events. This is called anintermittency. The deviation of the pdf from a Gaussian
is characterized by third (skewness) and fourth (flatness)order moments of the pdf that are of 0 and 3 for a
Gaussian respectively. For density fluctuations, thepositively skewed pdf's change with the RHF near theresonance surfaces (Fig. 1(c)). A deviation from the best
fit by a Gaussian (plotted by dashed line for thereference in Fig. l(c)) is exhibited by the flatness (Fig.

tt i + k . The moments of the measure averaged

over the number of block, i, (tt(T,Dq), scale as a power(a)oo;fi>.

++*1o
trtrtr!fit3frnx 17 kHz
O offxox*oa1(d)).

oa

In understanding of plasma turbulence we often
compare the dynamics of plasma with neutral fluidturbulence. For the fully developed turbulence,
Kolmogorov deduced the self-similarity of the turbulent
velocity fluctuations in a simple model and defined theT,psDlix

inertial range [12]. Plasma has more nonlinear
instabilities and dissipation ranges. Hence, it probably
has a very limited or nonexistent inertial range.
Experimental evidence of the intermittency in the"u.oXN

> N'o'o^7ll ' o 'o Ltl" x'9>o4.."olll xllF oD>xo.4

plasma fluctuations suggests a different scaling of thefluctuations in different time scale ranges. The structureof this process is more complicated than a single fractal
structure. To clarify the features of the self-similarity
one needs a variety of measures of plasma dynamics.
We follow the methods developed for the multifractal
analysis of fluid turbulence [9]. Multifractal analysis is a
powerful tool to understand a deviation from pure selfsimilarity. A characteristic value related to the energy

0.2(c)

0Lb81012r. cmFig. 2 (a) Moments of measure. Nontrivial function of gillustrates a multifractality in the range of timescales I- 1-300 ps. (b) Intermittency parameter
C(f) depending on time scales L at r= 7.1 cm. (c)Radial behaviour of C (solid line - RHF off, x - 17kHz,D-30kHz)
420

Budaev V. et al., Effect of Rotating Helical Magnetic Field on the Turbulence Fractal Structure in the Tokamak Edge Plasma
511

6t1

7t1

LCFSSOL

211N 11<+i13111012r.cmFig. 3 The intermittency parameter Con the RHF frequency-radius plane. Intermittency is effected by the RHF in the layers(magneticislands) aroundaresonantsurfaces mln=611 (r=8cm) ,711 (r=9.5cm),51'l(r-7cm) shownby
white dashed lines, and in the SOL. Last Closed Flux Surface (LCFS) shown by bold dashed line.

of the time scales f, (U{f iSn| = (TlI$-^Q'd. The intermit-characteristic time scale of the intermittent events. Itwas observed that the intermittency parameter may
change with radius and the frequency of the RHF. Fig.
2(c) illustrates radial profile of the intermittency parameter for different frequencies of the RHF. We note thatthe intermittency parameter in experiments with theRHF exceeds the parameter estimated for the "off' case(Fig. 2(c)). Increasing of the intermittency parameter inexperiment with the RHF was observed in the SOL andseveral radial domains. This regions are close to thetency is defined as C(f) = dL(T,q)/d4 lq=r. For amonofractal (pure self-similar) structure C(7) = g a14
the scaling exf'onent A(?la) is a linear function of q. For
multifractal 0 < C(f) < I and A(?iq) is non-linear function of 4. We have studied a range of time scales of Z =1-1000 ps. In Fig. 2(a) is plotted the typical moments ofthe measure. For a monofractal structure all curves
should collapse into a single curve. In experiment, it isobserved the nontrivial function of q in the range oftime scales Z - l-300 ps. It illustrates a multifractal
character of the turbulence. In the mesoscale range T >300 ps a monofractal behavior was observed. To calculate C(Z) the scaling exponent A(7la) is processed.
An error in ther estimation of C is about l0 Vo in the experiments. Intermittent structures are characterized byintermittency parameter C > 0. Typical behaviour ofC(I) has a maximum in the range Z - 10-100 ps asshown in Fig. 2(b). This maximal value of C(Z), that isresonant surface positions: mln = 5ll (- 7 cm), 6/1 (- 8cm),7/1 (- 9.5 cm) [8]. Layers corresponded to themagnetic islands around resonant surfaces are shown inFig. 3 for the reference. Exceeding of the intermittency
parameter over the "off" case may change with the RHF
frequency. The intermittency radial profile behaviour isnot repeated with variation in the RHF frequency:
maxima in the radial profile of the intermittency are ob-served in different radial domains. This non-trivial
variation of the intermittency radial profile for different
frequencies of the RHF is shown in Fig. 3. It is observed
several domains on the/- r plane where intermittency

in the range ol' C - 0.2-l in the experiment, is referred
as intermittenc:y parameter of the signal. The time scale

Z corresponderl to the maximum in C(Z) is referred as a421

Budaev V. et al., Effect of Rotating Helical Magnetic Field on the Turbulence Fractal Structure in the Tokamak Edge Plasma

One of them is a reconstruction of magnetic structurenear resonant surfaces: magnetic islands have been
observed to grow in the RHF experiments [8]. The
hierarchy of the magnetic islands is involved in thedynamics of the process, modification of the hierarchy isreflected in the fractal structure with higher level ofmultifractality. This mechanism is not expected todepend directly on the RHF frequency and can notexplain the observed fractal modification in the Scrapeoff-Layer (SOL) where no magnetic structure
modification was observed with the RHF.
Another mechanism is a delayed synchronization
control discussed in the Introduction. This mechanism
has a frequency selective nature and it is supposed toplay a role in the modification of the turbulence
structure in the RHF experiment. The poloidal drift
velocity uo is estimated from E, to change its directionand amplitude across the plasma column (Fig . (c)).
Supposing that up profile reflects the behaviour of thedrift modes phase velocity, it offers the key tounderstand the process. Estimated poloidal phasevelocity of the RHF becomes close to the poloidal drift
velocity in the SOL and in the radial layers at r = 6.5-8
cm (Fig. 4(c)), variation in the RHF frequency may
select a radial domain for a resonance with the drift
modes. Frequency dependence of the intermittency
modification seems to reflect a delayed synchronization
control of resonant drift wave modes in these radialdomains. This mechanism modifies the set of thenonlinear coupled drift modes involved in the process
reflecting in the reconstruction of fractal structure. We
note that this discussed resonance effect may be more
complicated than a resonance of a single mode with theRHF, because of the poloidal phase velocity of drift
modes at given r is expected to be distributed overbroad range of the values. It is needed the measurementsof the poloidal phase velocity distribution for thedetailed analysis ofthe control effect. In conclusion, theturbulence characteristics related to the fractal structure
and turbulent transport in the edge plasma of tokamak
HYBTOK-II with a variation of the RHF frequency

r'cm

Fig. 4 (a) The radial profiles of turbulence driven particle
flux (time-averaged), (b) phase shift between thefluctuations in density and poloidal electric field.
(c) E, x I poloidal drift velocity uo (circles) and thephase velocities of the rotating helical magnetic
field with frequency of 5, 10, 20 kHz (lines). Positive direction of the velocity in the electron diamagnetic drift one,parameter growths to high level C - O.7-1. The effectdepends on the RHF frequency. Frequency range of l02O kHz seems to be more effective one to reconstruct
the multifractal structure near the resonant surfaces andin the SOL. We note that this frequency bandwidth corresponds to the characteristic time scale of 40-100 ps ofthe bursts observed in the fluctuations (Fig. 1(a)).
Figure 4(a) illustrates a radial profile of theaveraged turbulent flux, f. =+J ft(t)6,(t)dt, resulted fromthe level of fluctuations in fi(r) and D.(t), and coherency
between them. The increasing of the flux in the SOL
and r = 7-8 cm is mainly resulted from the change inthe correlation between density and poloidal electricfield fluctuations: cross-phase shift (A@,rJ defined fromhave been studied. It was shown that the turbulence has

Fourier analysis of fi.(t) and D,(r), varies with the RHF
frequency (Fig a@)). The change in the flux seems to
response from a reconstruction of the fractal structure ina multifractal structure. The intermittent bursts formedby fluctuations with a time scale of 40-100 ps, are more

intensive in the RHF experiment. The effect on edge
turbulence fractal structure and transport characteristics
depends on the RHF frequency. The intermittency
parameter as a measure of multifractality level, andturbulent flux are increased with the RHF near theresonant surfaces and in the SOL. A delaved

the RHF experiments.

4. Discussion
The observed effect of the fractal structurereconstruction can be resulted from several mechanisms.
422

Budaev V. et al., Effect of Rotating Helical Magnetic Field on the Turbulence Fractal Structure in the Tokamak Edee Plasmasynchronization control ofresonant drift wave modes by[5] E. Ott, C. Grebogi and J.A. Yorke, Phys. Rev. Lett.

the RHF is considered as a candidate mechanism toexplain a deoendence of the effect on the RHF
frequency. Detailed measurements are needed to clarify
the selective character of this effect.
This work has been partly supported (V.B) by JSPS
(Japan Society for the Promotion of Science ).

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