Proceedings of ITC/ISHW2007

Transport Dynamics and Multi-Scale Coupling of Turbulence
S. Inagaki1) , N. Tamura2) , Y. Nagashima1), T. Tokuzawaa2) , T. Yamada1) , K. Ida2) , T. Maruta3) , S.
Kubo2), K. Terasaka3) , T. Shimozuma2), Y. Nagayama2) , K. Kawahata2) , A. Komori2), S. Shinohara3) ,A. Fujisawa2) , M. Yagi1) , Y. Kawai1) , S.-I. Itoh1), K. Itoh2) and the LHD experimental group
1)

2)

RIAM, Kyushu University, 6-1 Kasuga-Koen, Kasuga 816-8580, Japan
National Institute for Fusion Science,322-6 Oroshi-cho, Toki 509-5292, Japan
3)
IGSES, Kyushu University, 6-1 Kasuga-Koen, Kasuga 816-8580, Japan

Correlation between electron heat flux and electron temperature gradient is obtained in cold pulse experiments. Evidence for a long distance correlation is found in plasma, where the nonlocal temperature rise isobserved. Multi-scale coupling of density fluctuation is analysed by using envelope method. Existence of the
low frequency (≤ 2 kHz) modulator of the density fluctuation is suggested in a nonlocal plasma.
Keywords: dynamics, nonlocal, correlation, fluctuation, LHD

1

Introduction

radh35108

ρ=0.1202Nonlocal Transport in LHD

Some local transport models based on temperaturegradient-driven turbulence indicate that the heat transport
is non-linear and thus the electron heat diffusivity, χe , hasa dependence on electron temperature, T e , and/or electron
temperate gradient, ∇T e [1]. To study the non-linearity ofheat transport, transient transport analysis is recognized
as a very powerful tool because it can yield ∂qe /∂∇T e
and ∂qe /∂T e, here qe is the electron heat flux [2]. The
transient experiments (inducing cold and/or heat pulses)
have been performed and many non-linear transport models have been proposed and tested[3]. However, fast propagation of a temperature perturbation were observed, forexample, the ballistic propagation observed in TJ-II[4], theauthor’s e-mail: inagaki@riam.kyushu-u.ac.jp

exp

-0.05

δTe (keV)

Turbulence transport is higher than the neoclassical levelin the edge to intermediate region in helical devices (heliotron, stellarators, heliac and others) as well as in tokamaks. The turbulence causes not only quantitative featuresof transport but also qualitative ones. Transport dynamics reveals such qualitative features of transport. Varioustransient transport phenomena observed in helical plasmas
indicate complexities of turbulent transport. Recent theoretical works on drift wave turbulence show drift waves
are strongly coupled with micro-scale, meso-scale andmacro-scale waves. These non-linear couplings form turbulent structures, which have strong influences on transport. Thus, the dynamics and the multi-scale coupling ofdrift waves are now the most important issues to be clarified in transport studies. A comprehensive understanding of transport dynamics through first principle turbulencemodels is strongly required for achieving predictive capability of turbulence transport.

sim χ∝∇Tsim χpb7

ρ=0.32

0

-0.05
-0.1
ρ=0.520-0.1
TESPEL

-0.2

0

0.01

Time (s)Fig. 1 Time evolution of δT e at three different normalized radii.
The simulation results based on models of χ = χpb andχe ∝ ∇T e α , α = 7 are also shown.

heat pulse propagation caused by L-H transition [5] andheating power switching [6]. If χe has a dependence likeχe ∝ ∇T e α , α ∼ 50 is required to explain these transient
responses. Fast cold pulse propagation is also observed in
the Large Helical Device (LHD) [7]. Figure 1 shows typical time evolution of a cold pulse induced by a tracer encapsulated solid pellet (TESPEL) injection in the low density (1 × 1019 m−3 ) NBI plasma on LHD (a major radiusat the magnetic axis of Rax = 3.5/3.6 m, an averaged minor radius of a = 0.6 m and a magnetic field at axis of up
to 2.83T) [8]. The simulation results based on models ofχ = χpb and χe ∝ ∇T e α are also shown, here χpb is a heatdiffusivity determined from the power balance analysis inProceedings of ITC/ISHW2007#49708(a) δTe (keV)

Exp (ρ=0.22)

0.5
0Exp (ρ=0.70)

0Sim (ρ=0.22)(b) δ∇Te (keV/m)

Sim×10-2 ρ=0.22
Exp-4
5 ρ=0.22

Sim×10Exp

0(c)-5 δqe/n (keVm/s)
2.78

2.8

2.82

2.84

Time (s)Fig. 2 Time evolution of (a) δT e at two different normalized
radii, (b) δ∇T e at ρ = 0.22 and (c) δqe at ρ = 0.22. Simulation result based on the power balance χ model arealso shown. Simulated results are much smaller than experimental observations, and thus, simulation results are
multiplied by ten in this figure.

stationary state [9]. In simulation, the electron temperature perturbation, δT e , is calculated from the perturbation
equation written as
3 ∂δT ene= −∇ · δqe .
(1)
2∂t
The core T e begins to decrease before the diffusive transport effect (calculated by χpb ) reaches this region. To
explain this prompt response of the core T e by the nonlinearity of χe , a strong ∇T e dependence of χe (α = 7) isrequired. On the other hand, α = 1 − 3/2 is required to
explain the stationary transport property both in tokamaks
and helical devices [9]. A new perspective to turbulencetransport (i.e. non-locality) has been introduced to explain
this discrepancy.
In addition, the response to edge cold pulses often
have reversed polarity, with the core T e increasing in
response to edge cooling in many tokamaks and LHD[10, 11, 12, 13, 14]. A significant core T e rise has been
observed by TESPEL injection in LHD as shown in Fig. 2(for more details see [13]). Typical parameters in this experiment are as following: line averaged density 1 − 2 ×
1019 m−3 , central electron temperature up to 1.5 keV, central ion temperature of 1-2 keV, plasma β value of 0.1%,
absorbed ECH power of 0.8 MW, deposited neutral beam

power of 2 MW. Characteristic time and spacial scales are
ion Larmor radius ρi ∼ 1 mm, c/ωp ∼ 2 mm, typical spatial√
scale of meso-scale structure aρi ∼ 30 mm, a/cs ∼ 1µs,
globalenergy confinement time of 50 ms, meso-scale timep(a/ρi )(a/cs) ∼ 30µs, where ωp is the plasma frequency, cis the speed of light and cs is the ion sound velocity. The
relationship between heat flux and temperature gradient isessential to understand heat transport. Thus, time evolutions of the core ∇T e perturbation and the heat flux perturbation (normalized by density) are also shown in Fig. 2.
Multi-channel heterodyne radiometer is used to track thesmall T e perturbations at different normalized radii [15].
The absolute error of ∇T e determined by the absolute calibration of ECE is 20% and the relative errors of the ∇T e
determined by the noise levels of ECE is only 1%. The
electron heat flux perturbation can be determined from theperturbation equation written as and it can be written as,Z ρ
1
3 ∂δT e(ρ, t)δqe (ρ, t) = −
ne
dV,
(2)
S (ρ) 0 2∂twhere S is the surface area of the closed flux surface, V isthe volume. The power balance simulation indicates smallchanges in the temperature gradient and heat flux in thecore region and they are quite different from experimental
observations. In addition, the local transport assumption
can not explain the sign of change in the heat flux. In spiteof the smooth change in the temperature gradient, abrupt
reductions in the heat flux (heat flux jump) are observed
after several ms from the TESPEL injection.
To determine the nonlocal correlation between theheat flux and the temperature gradient, two-point crosscorrelations between the core and the edge are calculated. The cross-correlation is defined as C f,g (ρref , ρ, τ) =p
h f (ρref , t)g(ρ, t + τ)i/ h f 2 (ρref , t)ihg2 (ρ, t)i, here hi means
RT
temporal average, defined as hh(t)i = (T )−1 0 h(t)dt.
Figure 3 indicates time evolution of cross correlations,Cδqe /ne ,−δ∇T e (ρδqe /ne , ρδ∇T e , τ), between core to core (local:ρδqe /ne = 0.19, ρδ∇T e = 0.19), core to edge (non-local:ρδqe /ne = 0.19, ρδ∇T e = 0.58), edge to core (non-local:ρδqe /ne = 0.58, ρδ∇T e = 0.19) and edge to edge (local:ρδqe /ne = 0.58, ρδ∇T e = 0.58). In the paradigm for local
transport, a strong correlation between qe (ρ) and ∇T e (ρ) is
trivial and the relation of δqe = −ne χpb ∇δT e is indeed satisfied in a simulation. On the other hand, experimental result
indicates that core (ρ = 0.19) heat flux is strongly coupledwith edge (ρ = 0.58) temperature gradient for a short timelag, and therefore, a presence of the long distance/nonlocal correlation between heat flux and temperature gradient is clarified. The strong negative non-local correlationis reduced with a increase in the time lag. The correlation
time, which is defined as peak width of time lag at halfheight, is 8 ms. This time is much longer than a/cs andp(a/ρi )(a/cs). The local core to core correlation indicatesa gradual reduction similar to that predicted by local transport when τ ≥ 15 ms. Subsequently, the non-local correlation is dominant over the local correlation. The edge

Proceedings of ITC/ISHW2007#4970840ρδq=0.19

τ (ms)

30
0.3

20

0.1
-0.1

0.5

10

0.1

(b)

0

-5

-0.3

0

-0.5
-0.9

0
0

-0.5
0.2

0.2

0.4
0.6
ρ∇δT

-0.7

0.8

1

Fig. 4 Contour map of cross correlation between δqe at ρ = 0.19
and δ∇T e at different normalized radii.

heat flux is not coupled with core temperature gradient.
The local edge to edge correlation indicates strong negative coupling, i.e. a decrease in the flux with an increase in
the gradient (negative heat diffusivity). The negative heatdiffusivity is often observed in the transition phase. Somesort of transition, thus, may take place in the nonlocal T erise phenomena.
To demonstrate a presence of the long distance correlation in turbulent transport, the cross correlations between
the core heat flux and the temperature gradient at different
normalized radii is shown in fig. 4. This diagram indicates the temperature gradient at ρ = 0.54−0.61 has strong
negative correlation (≤ −0.8) with the core heat flux. The
obtained correlation length (0.6a) is obviously larger than
the so-called meso-scales and thus the global scales. A
presence of the long distance/nonlocal correlation between
heat flux and temperature gradient is clarified.
The nonlocal T e rise takes place in the low-ne andhigh-T e (low collisionality) regime in LHD just as TFTR
scaling predicts.[13, 14, 11]. The critical density for nonlocal T e rise is 1.5 × 1019 m−3 in LHD. This value is 2-3
times larger than that for eITB formation (0.6 × 1019 m−3 ).

0-3-0.1 -0.2

0 19
m -3

Fig. 3 Time evolution of cross correlations between core to core( ρδqe /ne = 0.19, ρδ∇T e = 0.19), core to edge ( ρδqe /ne =0.19, ρδ∇T e = 0.58), edge to core ( ρδqe /ne = 0.58, ρδ∇T e =0.19) and edge to edge ( ρδqe /ne = 0.58, ρδ∇T e = 0.58).

19

ne=2x10 m

x1

40

τ (ms)

45

20

-3

-1
0

0

19

-0.5

0.74x10 me

ρ=0.190.

ρδq =0.58

e

0

(a)

-3

e

0.5

19

e

0.5

e

Cδq /n ,-δ∇Teeρδ∇T =0.58

1.1x10 mρδq =0.19

δqe/ne (keVm/s)ρδ∇T =0.19

δqe/ne (keVm/s)

1

-5

-10

δ∇Te (keV/m)-15

Fig. 5 Flux-gradient Lissajous diagrams at ρ = 0.19 in (a) the
local transport condition (density is above critical value)and (b) the nonlocal condition (density is below criticalvalue). The arrows denote the direction of variation.

The relations between the heat flux and the temperature
gradient are determined both above and below the criticalcondition and shown in fig. 5. The heat transport above
the critical condition is considered to be dominated by local processes. As shown in Fig. 5(a), the obtained data lie
on the straight line. In other words, local heat flux is proportional to local temperature gradient. On the other hand,
Lissajous diagrams (Fig. 5(b)) reveal hysteresis in heat flux
versus temperature gradient. After the TESPEL injection,
the heat flux changes abruptly and discontinuously. This
flux jump is not accompanied by a change in the local temperature gradient, therefore transport is non-diffusive andnon-local. The area of the hysteresis loop increases with adecrease in the density. The density is close to the criticalvalue for the eITB formation, the nonlocal T e rise triggers
the eITB formation.

Proceedings of ITC/ISHW2007#49732

ρ=0.41010

ρ=0.61(b) δ∇T (keV/m)
e

-3
-4

10

-6

-7

-8

2.83-2.84s
2.85-2.86s

-9

-10

10
(c) reflect (a.u.)

2.84

10

4

5

10

Fig. 7 Change of power spectral density of reflectmetry signal
at ρ = 0.45. Note the high pass filter (≥ 1 kHz) is applied
to the reflectmetry signal.

0
-2

3

10

f (Hz)

ρ=0.452

Ablation
2.85

2.86

Time (s)Fig. 6 Time evolution of (a) δT e at ρ = 0.41 and 0.61, (b) averaged temperature gradient in the region of ρ = 0.43−0.55
and (c) X-mode reflectmetry signal at ρ = 0.45.

3

10

10

ρ=0.43-0.55

4

PSD (a.u.)

0.5 (a) δT (keV)eEnvelope Analysis

The nonlocal mechanism in plasma considered to be implicated in a structure, which is generated by interactionsof turbulence over long distances. Models assuming theexistence of the fluctuations with long radial correlation
length on the order of the minor radius has successfully
explained several features of nonlocal transport dynamics[16]. However, the experimental observation for the structures with a long correlation length has not been observed.
For further study of nonlocal transport, experimental observations of spatiotemporal structure of turbulence is essential. Langmuir probe diagnostics are most suitable tofulfill a high temporal and spatial resolution measurementof fluctuations. The low temperature plasma, therefore, isvery useful to study structures in turbulent plasma. The
drift wave turbulence is successfully excited in the LMD-U
plasmas [17]. The LMD-U has a linear magnetic configuration. In other words, geometrical effects due to configuration can be simplified. Thus, thermodynamical and statistical features of plasma turbulence will be emphasizedin the turbulent structure formation. Thereby, results obtained in LMD-U experiments have a commonality beyond
configuration differences. The many analysis tools are developed and tested in LMD-U. The non-linear coupling offluctuation is identified by using bi-spectral analysis. Envelope analysis is also one of new tools.
In toroidal plasmas, the most common methods tomeasure fluctuations are those for density fluctuations suchas conventional reflectometry. However, it is difficult to detect a certain type of fluctuation (e.g. zonal flows) by simply using the density fluctuation diagnostics due to the lowlevel of density fluctuation component. Recent progressin the fluctuation diagnostics (e.g. HIBP[18]) has clarified non-linear coupling of drift wave turbulence and structures. For example, the turbulent density fluctuations are
modulated by the zonal flows through the parametric modulational instability[19]. Thus, density fluctuations havesome information about structures which can modulatethem [20]. The multi-channel reflectmetry, thereby, hasa capability to measure a correlation length of structure.
Two channel X-mode reflectmetry was worked on the low
density plasma in LHD. Figure 6 shows typical time evolution of a refrectmetry signal during the nonlocal T e rise.
After a TESPEL injection, T e at ρ = 0.61 drops and T e
at ρ = 0.41 rises, hence, temperature gradient begins toincrease. The refrectmetry signal (≥ 10kHz) at ρ = 0.45
increases after TESPEL injection as shown in Fig. 7. Thisenhancement of density fluctuation may be driven by anincrease in pressure gradient at ρ = 0.45. In order to deduce the existence of modulator, the envelope analysis isapplied. Figure 8 shows an example of signal envelope andpower spectrum densities of the envelope. An envelope iscalculated from a high-pass filtered signal (≤100kHz) byusing of Hilbert transform. The refrectmetry signal data instationary state before TESPEL injection are used to makeensemble averaging for reduction of statistical error. It isfound that the envelope is modulated. The power spectrum
density at ρ = 0.45 appears an existence of the modulator
in the low frequency region (≤2kHz). The geodesic acoustic mode (GAM) frequency is higher (∼10kHz) than thatof the modulator, and thus GAM is ruled out as a candidate of the modulator. There is no significant MHD mode

HPF(>100kHz)

Proceedings of ITC/ISHW2007

(a)

0.4
0.2
0.0
-0.2
2.701

envelope

for their excellent support. This work is partly supported by a grant-in-aid for Specially-Promoted research(16002005) and by the collaboration programs of NIFS
(NIFS07KOAP017).

raw

References

Time (s)

2.702

-7

10

(b)

PSD (a.u.)

ρ=0.45
0.9

-8

10

p∝f

256 ensemble averaging

-9

10

0.5

2

10

3

10

4

10

5

10

f (Hz)
Fig. 8 (a) Time evolution of high-pass filtered signal (≤100kHz)
at ρ = 0.45 and its envelope, (b) Power spectrum densityof envelopes at two different normalized radii.

observed with the magnetic measurement system. In theoretical works, fluctuation of the long wave length mode excited from background micro fluctuation is predicted as thecandidate[21]. On the other hand, there is no clear modulator in the power spectrum density at ρ = 0.9 and no strong
coherence between envelopes at ρ = 0.45 and ρ = 0.9. The
core heat flux is correlated to the edge temperature gradient around ρ = 0.6, thus, reflectmetry around ρ = 0.6
is required to discuss the correlation length of a observed
modulator. The multi-channel reflectmetry and coherence
and bi-spectral analysis will be the object of future work.

4

Summary

In summary, we investigate the dynamical structure of
LHD cold pulse experiments providing experimental evidence for the presence of nonlocal coupling. We havefound that 1) a correlation structure which have, indeed,
the global spatiotemporal scales exists in turbulent plasma,2) existence of turbulent density fluctuation modulatorswith low frequency.

Acknowledgments
We thank Professor O. Motojima for his continuous encouragements. We are grateful to the technical group[1] F. Ryter et al , Phys. Rev. Lett. 86 (2001) 5498
[2] N. J. Lopes Cardozo, Plasma Phys. Control. Fusion 37(1995) 799[3] S. Inagaki et al , Nucl. Fusion 46 (2006) 133[4] B. P. van Milligen et al , Nucl. Fusion 42 (2002) 787[5] V. V. Parail et al , Nucl. Fusion 37 (1997) 481[6] U. Stroth et al , Plasma Phys. Control. Fusion 38 (1996)
1087[7] O. Motojima et al , Nucl. Fusion 43 (2003) 1674
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[11] M. W. Kissick et al , Nucl. Fusion 38 (1998) 821[12] P. Mantica et al , Phys. Rev. Lett. 82 (1999) 5048
[13] S. Inagaki et al , Plasma Phys. Control. Fusion 48 (2006)
A251[14] N. Tamura et al , Nucl. Fusion 47 (2007) 449[15] K. Kawahata et al , Rev. Sci. Instrum. 74 (2003) 1449
[16] T. Iwasaki et al , J. Phys. Soc. Japan 68 (1999) 478[17] K. Terasaka et al , Plasma Fusion Res. 2 (2007) 031[18] A. Fujisawa et al , Phys. Rev. Lett. 93 (2004) 165002[19] K. Itoh P. H. Diamond, S.-I. Itoh and T. S. Hahm, Plasma
Phys. Control. Fusion 47 (2005) R5[20] Y. Nagashima et al , Plasma Phys. Control. Fusion 49(2007) 1611
[21] S.-I. Itoh and K. Itoh, Plasma Phys. Control. Fusion 43(2001) 1055