Proceedings of ITC/ISHW2007

Zonal Flows and Fields Generated by Turbulence in CHS
A. Fujisawa, K. Itoh, A. Shimizu, H. Nakano, S. Ohshima, K. Matsuoka, S. Okamuara,
CHS-group
National Institute for Fusion Science, Oroshi-cho, Toki, Japan 509-5292
The paper reports two major progresses made in heavy ion beam probe (HIBP) measurements carried out in
CHS. One is to prove the nonlinear couplings between zonal flows and turbulence by using the modern nonlinear
data analyzing techniques. The other is the identification of zonal magnetic field, generated by the turbulence,
utilizing the potential ability of the HIBP. The measurements of magnetic field fluctuations successfully provide thefluctuation spectrum, and the first experimental evidence of the turbulence dynamo, which is a long-standing
historical candidate for the geomagnetism and analogous phenomena, that the turbulence generates a structured
magnetic field.
Keywords: zonal flows, turbulence, zonal magnetic field, heavy ion beam probe, nonlinear interaction,
mesoscale structure, dynamo, wavelet bicoherence analysis

1. Inroduction
Turbulence is a ubiquitous phenomenon observed in
the universe. It has been recognized that the turbulence is
not the process of randomization, but also the one to
produce a structure. For example, it is well known that the
Rossby wave turbulence should generate the Jovian belt or
zonal flows [1], and that turbulence should be the cause forthe geomagnetic (dipole) field. In 1970s, it was pointed outthat the drift wave turbulence should obey the same
equation as the Rossby wave turbulence [2], therefore, it isexpected that the zonal flows should exist in toroidal
plasmas.
In the research of magnetic confinement, recently, theimportance of the zonal flows has been recognized since
the saturation level of drift wave turbulence, whichgoverns the transport processes, should be determined by
the interaction between zonal flows and drift waves [3].
After the evidence to hint the zonal flows and their roles intransports was presented in theoretical and simulation
works [4-6], the existence of zonal flows was
experimentally proven in a number of devices [7-19] (forreview [20]).
After the identification of the zonal flows [4], theCHS experiments have made a great progress in thephysics of zonal flows and turbulence. The investigation
has been made on the interaction or nonlinear coupling
between zonal flows and turbulence [21-24]. Besides, wehave developed a potential ability of the heavy ion beamprobe (HIBP), which means the direct measurement ofmagnetic field fluctuations of plasma interior from thehorizontal secondary beam movement, and successfully
obtained a spectrum of magnetic field fluctuations in CHS
e-mail: fujisawa@nifs.ac.jp

plasmas [25].
The paper presents the direct quantitative evidence toshow the coupling between zonal flows and turbulencewith an advanced analysis technique, the wavelet
bicoherence, and presents the existence of zonal magnetic
field, as well as zonal flows, with recently developed
technique of the magnetic field fluctuation measurementwith HIBP. The discovery of the zonal magnetic fieldshould be the first evidence to show that the turbulence
really generates the structured magnetic field, which is
associated with a historical physics problem, i.e., dynamo
problem.

2. Experimental Set-up
CHS is a toroidal helical device of which the major
and averaged minor radii are R=1 m and a=0.2 m,
respectively. The device is equipped with two HIBPs tomeasure plasma potential in different toroidal sections
apart by 90 degree. Each HIBP has three channels toobserve the adjacent spatial points of the plasma. The
local electric field can be directly measured by making a
difference between potentials of neighboring twochannels.
It has been known for HIBPs, in addition to theelectrostatic potential, that a vector potential component
can be measured from the beam movement on thedetector; particularly in an axisymmetric magnetic
configuration, a formula is derived to relate themovement to the toroidal component of the vector
potential [26]. In a real geometry, the fluctuation of amagnetic field component, or derivative of vector
potential, can be directly measured by taking theProceedings of ITC/ISHW2007

1

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GAM

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(c)
turb

q

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55
t (ms)

A (V)

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P (V / kHz)(a) 150f(kHz)

3. Coupling between Zonal Flows and Turbulence
The target plasmas for the experiments introduced
below are produced with electron cyclotron resonanceheating of ~200 kW. The magnetic field strength of thedischarges is B=0.88 T, and the density is kept constant
approximately at ne~ 5x1012cm-3. The HIBP measurement
is performed at ρ~0.35, where the amplitude of zonal
flow tends to be large, which gives a better condition toinvestigate the interaction between zonal flows andturbulence.
In this discharge condition, electron and ion
temperatures Te~ 0.5 keV, Ti~0.1 keV, (i.e., incollisionless regime), ion Larmor radius ρi~0.1 cm, timescale of micro-instabilities ω*/2πR~50 kHz with kperp
ρi~0.3 and energy confinement time τE~2 ms (orcharacteristic frequency of global confinement τE-1 ~0.1
kHz), where kperp is the wavenumber and ω* is the drift
frequency defined as kperpTe/eBLn with Ln being acharacteristic length of density gradient. Here thepressure-gradient-driven microinstabilities are associatedwith electromagnetic field fluctuations.
The fundamental statistical properties of electricfield fluctuation are obtained from the traditional analysis
using the Fast Fourier Transformation (FFT). Figure 1
shows the spectrum of electric field fluctuation at ρ~0.35,
with coherence between electric field fluctuations at two
toroidal positions. The spectrum can be divided into four
ranges, i) a lower frequency range less than f<~ 1.5 kHzwith a quite high coherence of ~0.7, ii) an inertia range off<10 kHz, iii) the range of 10<f<30 kHz with a sharp

peak, and iv) the background turbulence range with abroad-band peak around ~50 kHz. The fluctuation in f~2
kHz to show a long-distance correlation is regarded as thezonal flow. Besides the sharp peak at 19 kHz, there canbe seen the other two peaks (at least) at 38 and 57 kHz.
These three modes have a long-distance characteristic,since the coherence between the potential fluctuation andthat of electric field at the other toroidal position gives aquite high value for three peaks (>0.5). Therefore, thepeaks are conjectured as the GAMs. The detailed analysisof the coherent modes is available in ref. [18].
The intermittent characteristics of fluctuation can beextracted using a wavelet analysis. Figure 2a shows anexample of the evolution of the zonal flow Z(t) and thewavelet power spectrum of electric field fluctuation
W(f,t) at ρ~0.35. The used wavelet analysis [27] has thenatural correspondence with the traditional Fouriertransformation. The zonal flow component is filtered out
from electric field fluctuation using a numerical low-pas
filter (see details in ref. [7]). Note that the positive valuemeans the ion diamagnetic direction. The zonal flow
amplitude (the blue dashed line) shows a slow change in
confinement time scale.
Figure 2b shows three plots of normalized
fluctuation power as a function of the normalized zonalZ(t) (V)difference between the beam movements on the detectors
from the neighboring ionization points. The details of themethod are described in ref. [25].

0.20noise level50

f (kHz)100

noise level

0.01
0.1

noise level (coherence)

1

10
f (kHz)0100

Fig. 1 A spectrum of electric field fluctuation. The
measurement is carried out at ρ~ 0.35 (solid line).
The coherence between electric field fluctuations
at two toroidal locations (dashed line). The
coherence is plotted in the low frequency range
where the result is above the noise level (in ref.
[22])

Fig. 2 (a) Evolutions of wavelet spectrum of electric field
fluctuations as a function of frequency, andevolution of zonal flow with low-pass filtered. The
unit of the color bar is in V 2/kHz. (b) Fluctuation
power fractions in three frequency regimes as afunction of zonal flow power fraction. (c)Conditional averaged spectra of wavelet power inthe local maxima and minima of the zonal flow.
The grey dashed line represents the conditional
averaged spectrum when |Z(t)|~ 0 (in ref. [22])Proceedings of ITC/ISHW2007(a)phase A(b)phase C(c)phase E(d)0.1

phase Af2(kHz)f2(KHZ)

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f2(kHz)-50-100
90 phase C 100
(e)
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100

f1(kHz)100

Fig. 3 Diagrams of wavelet squared bicoherence for three phases of zonal flow. The zonal flow is divided into fivephases in its magnitude as is shown in Fig. 2a for the conditional integral of the wavelet bicoherence. (a)Bicohrence diagram of the phase A (maximum), (b) that of the phase C (zero) and (c) that of the phase E(minimum). (d) An expanded view of the diagram for the phase A, and (e) another for the phase C [in ref.
[23]].

f1+f 2=-0.5

0.5

b 2(f1,f2)(a)A
noise level

0
0.1

(b)

E
B

C
1

10
f (kHz)D
100

noise level

500

50
E

2

! (f)

A
2

! (f)

flow power, qzonal(=q(0.25,1.5 kHz)). The regression
analysis shows that the turbulence power qturb(30,200)
should be strongly dependent on the zonal flow power as
qturb~0.9-1.6 qzonal with their correlation coefficient being
γ=0.74, while no significant relation γ=-0.01 is found forthe fluctuation power of the coherent mode qGAM(10,30).
On the other hand, the fluctuation neighboring to thezonal flow shows a positive correlation (γ=0.49), andfollows the zonal flow concomitantly like a tail as
qtail(1.5,10)~0.63 qzonal. Accordingly, the zonal flow power
fraction increases as the turbulence power fraction
decreases [22].
In comparison between zonal flow and the wavelet
fluctuation powers (Fig. 2a), the colored pattern seems tobe synchronous with the tide of the zonal flow. The phasedependency can be examined by taking the conditional
averages of the wavelet spectra for the phase of zonal
flow. Figure 2c shows the conditional averaged wavelet
spectra for the maxima and minima of the zonal flow. It
is obvious that the fluctuation power around ~ 50 kHz
shows an increase (an decrease) around a local minimum(a maximum) of the zonal flow. In addition, the FFTanalysis indicates that the modulation frequency of W(f,t)
all over the frequencies should coincide with the zonal
flow frequency. The result suggests that the fluctuation
characteristics should be varied with the phase of zonal
flow, as well as with its amplitude [22].
The wavelet bicoherence analysis [27] could be
useful in order to reveal the hidden linkage between thezonal flow and fluctuations. The analysis is extended toadopt a conditional average in consideration for the

E

0

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D

B
C
noise level
40 f (kHz)

B

C

0
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A

10
f (kHz)80

D100

Fig. 4 (a) The squared bicoherece along the line of
f1+f2=-0.5 kHz for the five phases of the zonal
flow. (b) Summed squared bicoherence for thefive phases of the zonal flow. The grean and redsolid (dashed) lines represent the zonal flow
phases of A and B (E and D)), respectively. The
blue solid line indicates the phase C. The inset
is an expanded view of the frequency range from
10 kHz to 90 kHz (in ref. [23]).

dependency of the fluctuations on the zonal flow phase[23]. The conditional bicoherence is evaluated on theelectric field fluctuation for five phases of the zonal flow

Proceedings of ITC/ISHW2007

4. Identification of Zonal Magnetic Field
As is similar to the previous case, the target plasma
is also produced with electron cyclotron resonanceheating of ~200 kW, to avoid a large-scale plasma motion

caused by magnetohydrodynamic (MHD) instabilities
that interfere with the generation of magnetic fields in themeso-scale range. The plasma parameters in theexperiment are; magnetic field strength B = 0.88 T anddensity ne~5x1012 cm-3. The normalized plasma pressure,β-value, is ~0.2%.
Collisionless (or electron) and collisional skindepths are estimated as 2.4 mm and 12 mm at 0.5 kHz,
respectively. The magnetic Prandtl number is evaluatedas µν/η~102 using the experimental viscosity of ν ~χ~ 10(a) 1000

1
Magenetic Field FluctuationP~fcoherence

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noise level(coherence)

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B
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D
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70
t (ms)

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(c)
B (G)

(see Fig. 2b); A) Z(τ)>0.8 V, B) 0.8> Z(τ)>0.25 V,C)0.25> Z(τ)>-0.25 V, D) -0.25> Z(τ)>-0.8 V, and E)
Z(τ)<-0.8 V.
The number of used ensembles isapproximately the same as ~20000 for every case.
Figure 3 shows three bicoherence diagrams for thephases, A, C and E. The results indicate clear changes in
the coupling between waves according to the phase of thezonal flow. The most important feature is that the
couplings become stronger along the lines of f1+f2~0.5,-0.5 kHz at the phases A (maximum) or E (minimum).
The expanded views of a region (90<|f|<100 kHz) clearly
demonstrate that the couplings on the lines of f1+f2~0.5
and -0.5 kHz becomes stronger in the phase A. Figure 4ashows the squared bicoherence as a function of f1along
the line of f1+f2=-0.5 kHz for every phase of the zonal
flow. Obviously, the coupling becomes stronger in therange of f>~2 kHz as the absolute value of the zonal flow|Z(t)| increases. However, the coupling of the lower
frequency of f<~2 kHz is rather high and almost the samefor every phase.
As a consequence of degraded independency ofensembles in lower frequency, the statistical noise levelof the wavelet bicoherence analysis is usually evaluatedaccording to ref. [27]. The noise level is regarded as,
however, a statistically sufficient condition to show thesignificance of couplings, since the phase of the zonal
flow lying in the frequency around f3~0.5 kHz is
conditioned for the evaluation. This finding supports theimage that the waves with adjacent frequencies interacts
collectively with each other to produce the zonal flow
through the processes of such as the modulational
instabilities.
The results confirm that the coupling becomesstronger when the zonal flow increases in the absolutevalue. In two phases, A and E, the level of the summed
bicoherence in the frequency range of the zonal flow(f<~1 kHz) is significantly large to surpass the maximum
noise level, or a statistically sufficient condition for the
existence of the coupling. As for the couplings between
turbulent waves, as is shown in the inset of Fig. 4b, thetotal bicoherence in the range of f>~30 kHz is larger as
the zonal flow is in the opposite direction to the bulk flow.
The couplings between turbulent waves are varied withthe sign of the zonal flow to cause the modulation of the
turbulence power spectrum (Fig. 2c). The corresponding
increase in the bicoherence can be seen particularly in theregions around (f1,f2)~(50,-80) kHz, and (f1,f2)~ (80,-50)
kHz in the bicoherence diagram of the phase E (Fig. 3c).

0
-30
45

50

t (ms)

55

60

Fig. 5 Magnetic field fluctuation spectrum and existenceof mean field. (a) Power spectrum (poloidal
component), and coherence between themagnetic field at two toroidal locations. The
fluctuation less than ~1 kHz shows a quite highcoherence, which means generation of global
magnetic field of azimuthal symmetry. The
spectrum is calculated for the stationary period of~50 ms for the discharge duration of ~100ms withfrequency resolution of 0.24 kHz and the Nyquist
frequency of 250 kHz. (b) The waveforms ofzonal fields at the same radial but different
toroidal locations. The positive sign means the
direction of vacuum field direction. (c) The otherwaveforms of zonal fields at the different
positions in both toroidal and radial directions.
The difference in radial direction is 1 cm (in ref.
[25])

Proceedings of ITC/ISHW2007
These observations in Figs. 5 and 6 show, therefore, theexistence of the mean magnetic field with radially zonal
structure symmetric around the magnetic axis. Thismagnetic field resistively damps away, if there is no
driving source, on a time scale of τR=µ0η-1 kr-2~ 50 µs. It
cannot be sustained by the external circuit throughinductive coupling, because the direction changes with(a)2

! t(ms)

0.5
1

0.0
0

-0.5

(b)

C(!r,!t)m2/s and resistivity of η~10-7 Ωm, where χ represents thethermal diffusivity evaluated as the global average fromthe confinement time.
The measurement was done at the radial position ofrobs=12+-.5 cm, where the signal-to-noise ratio is maximal
for the HIBP measurement. In this measurement, it is
known from a trajectory calculation that the horizontal
beam movement reflects the poloidal magnetic field. A
spectrum of the magnetic field fluctuation is shown in Fig.
4, with coherence between two toroidal locations. In the
low frequency range, the contamination is sufficiently
small, while the electric field contamination may be
dominant around ~50 kHz and above. The coherence ofthe frequency lower than 1 kHz is quite high (~0.7),
which is an average of ~70 temporal windows fromidentical shots; a higher coherence value is obtained in an
appropriate period of a single shot. This means that the
probing beams are coherently swung by the magnetic
field at two toroidal positions. It is unambiguously
demonstrated that a mean magnetic field fluctuation withlong-distance correlation does exist.
The magnetic field fluctuations in that frequency
range are visualized using a low-pass filter [7,22].
Figures 5b and 5c demonstrate two examples of evolutionof the magnetic field fluctuation in the frequency range(0.3 -1 kHz). The zonal field amplitude is evaluated to be~ 30 G at maximum. The accuracy of the absolute
magnetic field value is about 50% owing to uncertainty inthe absolute measurement of beam location, although therelative motion of the beam is more precisely measured
with accuracy of ~0.1%. The in-phase movement of two
beams on the same magnetic flux surface (Fig. 5b)
suggests that the fluctuation is symmetric or
homogeneous around the magnetic axis under theassumption of poloidal symmetry. In contrast, the otherwaveforms in anti-phase behavior (Fig. 5c), when theposition is different in radial direction by 1 cm from theother, suggests that the fluctuation should have a finite
radial structure.
A constant phase relation between the signals fromtwo different radial positions, as is shown in Figs 5b and5c, allows us to estimate the spatio-temporalcharacteristics of the mean zonal field in the radial
direction by calculating the correlation function of low
pass-filteredmagnetic
field
fluctuations
withspatio-temporal separation of Δr=r’-r and Δt=t’-t. Figure
6a shows the cross-correlation obtained by altering anobserved position r’, shot by shot with fixing the other at
r (=12 cm). Figure 6b illustrates the spatial structure ofthree times at Δt = 0, 1 and 2 ms. These correlation
diagrams show a quasi-sinusoidal structure in the radial
direction with a characteristic radial wavelength of λr~1
cm, while memory of the structure is lost in ~ 2 ms.

-1
1!t=0ms!t=1ms!t=2ms
0

-1 -2

-1

0
!r (cm)1

2

Fig. 6 Radial structure of zonal field. (a) Correlation
function of space and time, C(Δr, Δt), around theradial position of r=12 cm. (b) Correlation
functions with three different time delays; Δt=0,1
and 2 ms. These correlation diagrams show
radial structure of zonal field, which has aquasi-sinusoidal
radialstructurewithacharacteristic radial wavelength (~1cm) in
mesoscale. On the other hand, the temporalcorrelation indicates that the memory of themesoscale structure fades away with oscillation in~2 ms (in ref. [25])radius. It therefore must be sustained by the internal
plasma dynamics.
The wavelet bicoherence analysis is applied, as issimilar to the case of zonal flow, to quantify the nonlinear
coupling strength between the zonal magnetic field andturbulence. The results showed the intermittent coupling
between the zonal field and the background turbulence[25]. In this analysis, the contamination of electric field
fluctuation in the turbulence regime may disturb theabsolute quantification of the coupling between the zonal
field and the pure magnetic field turbulence. In theprevious measurement [7], the electrostatic mode
conjectured as Geodesic Acoustic Mode (GAM) was
clearly identified as a sharp peak at ~ 20 kHz in theProceedings of ITC/ISHW2007
electric field spectrum. However, the corresponding peakis completely absent in the magnetic field spectrum. Thismeans that the electric field contamination can be muchsmaller than the level of the upper boundary shown in Fig.
5. The degree of the contamination could be evaluatedmore precisely in a future analysis with taking into
account the detailed fluctuation properties.
Finally, the presented observations verify theexistence of zonal magnetic field, as well as zonal flow,
generated from the turbulence, probably throughmodulational instabilities. From the analogy to the zonal
flows in the Jupiter, future observations might find zonal
magnetic field structure in a rotating star. A residual
macroscopic field could be expected even in the averageover the whole zonal structure, in an inhomogenous
background. In addition, the zonal field could serve as aseed leading to global instabilities to cause a macroscopic
field or dynamo. Therefore, the discovery could be a step
toward general understanding the dynamo problem,
stimulating a question if the mesoscopic zonal field canbe developed into macroscopic structure.

5. Summary
In summary, we have shown two recent findings forturbulence and mesoscale structure, i.e., zonal flow andzonal field. The first one is the direct measurement onelectric field fluctuation combined with the wavelet
bicoherence analysis, which succeeds to find anintermittent coupling structure between fluctuations, e.g.,
the zonal flow and turbulence, turbulence and turbulenceand so on. The most important finding is that thenonlinear coupling between turbulence changes according
to the phase of the zonal flow. The couplings offluctuations and turbulence to the zonal flow turn visible
in the maximum or minimum phases of the zonal flows.
Finally, the presented observations verify theexistence of zonal magnetic field, as well as zonal flow,
generated from the turbulence, probably throughmodulational instabilities. A residual macroscopic field
could be expected even in the average over the whole
zonal structure, in an inhomogenous background. In
addition, the zonal field could serve as a seed leading to
global instabilities to cause a neoclassical tearing mode intoroidal plasmas, or macroscopic field or dynamo in theuniverse. Therefore, the discovery could be a step toward
general understanding the dynamo problem, stimulating aquestion if the mesoscopic zonal field can be developed
into macroscopic structure.

Acknowledgments
The author thanks Profs. S. Okamura, K. Matsuoka andthe members of CHS experimental group for theircooperation in the experiments, Profs K. Itoh, S-I. Itoh, O.
Motojima for their continuous supports. This work is
partlysupportedbytheGrant-in-Aidsfor
Specially-Promoted Research (16002005) and Scientific
Research (18360447).

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