Proceedings of ITC/ISHW2007

Measurements of micro-turbulence in high beta and high density
regimes of LHD and comparison with resistive g-mode scaling
Clive MICHAEL1) , Kenji TANAKA1) , Leonid VYACHESLAVOV2) , Andrei SANIN2) ,Tsuyoshi AKIYAMA1) , Yoshiro NARUSHIMA1) , Kiyomasa WATANABE1) , Hisamichi FUNABA1) ,Sadayoshi MURAKAMI3) Katsumi IDA1) , Mikiro YOSHINUMA1) , Ryuichi SAKAMOTO1) ,Kazuo KAWAHATA1) , and Shigeki OKAJIMA4)
1)

National Institute of Fusion Science, Toki 509-0293, Japan 2) Budker Institute for Nuclear Physics, Novisibirsk, Russia 3)
Department of Nuclear Engineering, Kyoto Univ., Kyoto 606-8501, Japan 4) Chubu University, Kasugai, Japan
November 26, 2007
Density fluctuations are analyzed in high volume average beta and high core density discharges in the Large
Helical Device (LHD) using a 2D phase contrast imaging system and far infra-red interferometer. Though boththese regimes share similarly high beta gradients, the physical origin of the fluctuations appears to be different.
In high volume average beta plasmas, both large and ion-gyro scale density fluctuation levels are compared withthe growth rate of resistive interchange modes and a clear correlation is found most particularly in the edge as
expected due to having the strongest magnetic hill in this region. In high core density plasmas with internal
diffusion barrier, resistive interchange is not a candidate for fluctuations near the barrier because magnetic well
exists there. However, fluctuations near the core have a bursty character in time and are destabilized when thetemperature profile is transiently hollow immediately after pellet injection and are quiescent and much reduced
when it becomes peaked, consistent with a slab ion temperature gradient mode. In configurations with outward
shifted magnetic axis, where core plasma pressure and density are greater, fluctuation bursts occur less frequently
suggesting anomalous transport is lower.
Keywords: high beta, density fluctuation, phase contrast imaging, MHD, resistive interchange, super dense coreregime, ion temperature gradient turbulence

1 Introduction
The Large Helical Device (LHD) has set many impressive
results with respect to high performance plasma operation,
including operation at low field (B ∼ 0.425T), high volume average beta (up to 5%) [1], as well as at high field(B > 2T), high central beta with high central density produced through an internal diffusion barrier (IDB) [2]. Suchhigh performance regimes are invaluable for fusion reactor operation; high volume average beta for reduction ofmagnetic field requirement, and high central density operation for production of another possible route to fusion
through low temperature, high density operation. It is essential to gain an understanding of limiting factors influencing the performance of these plasmas for a route to a fusion reactor, in particular energy transport, which degrades
the performance (such as beta gradient, density gradient,etc) either in a soft manner (gradually increasing with performance) or a hard manner (increasing suddenly above a
performance threshold). Transport studies have been carried out on these regimes [3], it is mostly found that transport exceeds the neoclassical level significantly and that it
is dominated by fluctuation-driven processes. Moreover,high density operation is not well studied yet. Therefore
this study focuses on fluctuation behavior in high perforauthor’s e-mail: clive@nifs.ac.jp

mance discharges from measurements based on 2D CO2
laser phase contrast imaging and FIR interferometer, andthe role that these fluctuations play on transport.
These two regimes are studied separately, and someattempt is made to unify the results of these regimes, giventhat they both share high beta (pressure) gradients, which
are free energy sources for fluctuations. However, the important processes in each regime appear to be different. Inhigh volume average beta, the main confinement degradation comes from the edge where resistive interchange (or gmode) turbulence appears to be excited as a result of themagnetic hill common to Stellarators and RFPs. In thehigh core density discharges [4], resistive g is not a candidate for turbulence around the internal diffusion barrier because magnetic well exists in the inner region. In this case
it appears that the opposite sign of density and temperature profiles, produced transiently after pellet injection, appears to strongly drive unstable modes in a burst like fashion characteristic of a very hard critical threshold. This ischaracteristic of a slab ITG mode instability; though other
MHD effects such as a ballooning mode may play a role. It
appears that outward shifted configurations have less hollow temperature profiles resulting in lower fluctuation level
therefore achieving higher central pressure.
This paper is organized as follows. The density fluctuation diagnostic systems are first discussed in Sec 2; in-Proceedings of ITC/ISHW2007
cluding the technique for spatial localization from lineintegrated measurements. In Sec. 3, we introduce measurements in high volume average beta discharges, including theory, experiments and results of dependence of fluctuation (of both large scale and ion gyro-scale) level andposition on beta, as well as a comparison of fluctuation
characteristics with thermal conductivity obtained frompower balance. In Sec. 4, we present the dynamical evolution of core turbulence level in dense core discharges forplasmas in configurations with vacuum magnetic axis position Rax = 3.65m, 3.75m, and compare with differencesof pressure rise. Finally, in Sec. 5, we summarize andcompare these regimes.

2 Density fluctuation measurementswith CO2 laser phase contrast imaging and FIR interferometer
For density fluctuation measurements we use the CO2 laser
2D phase contrast imaging diagnostic for diagnosis of ion
gyro-scale fluctuations (with 1 < k < 10cm−1 ), and the FIRinterferometer [5] for diagnostic large scale fluctuations(with k < 1cm−1 ). Phase counters of the FIR have sufficient precision (1/60 fringe) and bandwidth ( f < 50kHz)
to diagnose fluctuations at high beta, though at low beta,
the fluctuation level is comparable to the noise. Note that
the PCI diagnostic does not admit k = 0 components because of its optical arrangement, while the FIR interferometer does. Both the PCI and FIR systems only directly
measure the line integrated density fluctuation amplitude,
rather than the local value. However, with the 2D PCI diagnostic, some moderate spatial resolution is possible as described below. Routinely the fluctuation components near
the edge are much stronger than any core components, so
for diagnosis of edge resistive g modes, line-integral values
are sufficient, but for detailed analysis, and for analysis of
turbulence near the core of high core density discharges,
a proper analysis to resolve fluctuations along the line ofsight is necessary to analyze changes in core fluctuations
as distinct from edge fluctuations.
The 2D phase contrast imaging diagnostic system on
LHD employs a 2D imaging principle to split the lineintegrated fluctuation signal into contributions from different layers along the line of sight, according to the “magnetic shear” principle [6, 7, 8]. The sightline is verticalpassing at R = 3.603m, so penetrates from the edge tothe core, depending on the magnetic axis position of theplasma (Rax ), implying core and edge fluctuation components (from both top and bottom) can be separated. The
system cannot, however, fundamentally recover the local
fluctuation amplitude because of line-integration effects
unless the spectrum is isotropic, which it generally is not[?]. The line-integrated Ñ signal is split into contributions
along the line of sight and is related to the local amplitude ñ

according to Ñ 2 (ρ) = ñ2 (ρ)lz lres , where lz represents the ratio of fluctuation power propagating exactly perpendicular
to the probing beam to the total fluctuation power, and lresis an instrumental resolution which is increases with measured fluctuation wavelength. For typically measurements,
the peak wavelength is such that the instrumental width isaround half a radius, or less near the edge because the average wavelength is generally smaller than in the core.

3 High volume average beta
For high volume average beta plasmas, turbulent transportinduced by resistive interchange modes excited, driven bybeta gradient in the magnetic hill region in the edge is expected to be a limiting mechanism on the attainable plasma
beta in helical devices. In LHD, these modes have beenshown to have a “soft” character since their occurrence
does not produce a catastrophic effect on plasma operationand since plasma operation has been extended well beyond
the Mercier instability boundaries [9]. This is in contrast
to pressure (beta) driven instabilites in the edge of Tokamak devices, such as ELMS, which grow very quickly beyond the instability threshold producing a sudden crash.
Though the physical mechanism is different, such as apeeling-balooing modes, since the Tokamak does not havemagnetic hill because the toroidal current is internal to theplasma. Low order mode numbers are routinely identified
and have been compared successfully to predictions of resistive interchange modes [10]. However, theoretically, resistive interchange can be excited up to high order mode
numbers, up to the dissipative gyro-scale. Non-adiabatic
response due to resistive effects produces an anomalousflux. This is in contrast to a turbulent transport inducedby drift wave type instabilities (such as ion temperature
gradient (ITG) and trapped electron mode (TEM)) whose
drive mechanisms do not depend so directly on beta, rather
on functions of normalized density (Ln−1 ) and temperature(LT−1 ) gradients such as η = LT−1 /Ln−1 (although they are
known to be stabilized at higher beta [11]). Because drift
and resistive interchange instabilities coexist in the same
spatial scale, distinction between them becomes difficult.
However, they can be distinguished possibly in (1) coherence with magnetic fluctuations, since ITG/TEM areelectro-static, while resistive-g has a magnetic component.
In reality, however this may be difficult to detect for ion
gyro-scale fluctuations, since magnetic probes external tothe plasma are most sensitive to large scale fluctuations andthe non-locality of magnetic probe measurements would
tend to reduce the coherence of small scale fluctuations,and (2) phase velocity; drift waves propagate poloidally ataround the drift velocity plus ExB velocity while resistive
interchange turbulence propagates at only the ExB velocity[12].
Previous work on LHD [13] has looked at the role ofresistive interchange turbulence and its relationship to theProceedings of ITC/ISHW2007

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Fig. 1 Density, temperature and magnetic field strengths covered over in the parameter scans, together with contoursof β, νei and Larmor radius.

Theory of resistive interchange turbulence

A simple analytic theory verified by numerical simulation
was developed by [15, 16, 17]. The growth rate is given as:
2/3 R20 κn 2πdρ −1−1/3 
m(1)
γ=S τhp∇β2dιwhere, the magnetic Reynolds number S = τR /τhp ∝T 3/2 n−1/2 , the resistive skin time τR = r2 µ0 /η, and the√poloidal Alfvén time τhp = R0 µ0 mi n/B, with η ∝ T 3/2 n0
being the Spitzer resistivity. Therefore, the scaling of thegrowth rate is not simply given by β = ne T e /B2 , it is givenby:
1/6 −2/3 2/3 −2/3γ ∝ n1/6
B κn L pe (ne T e )(2)Under this theory, the thermal conductivity is:χ = γW 2 ,(3)where W is the radial width of the mode [16]. This theory does not clearly guide which is quantity is relevant
to compare with density fluctuation level. On one hand,
Eq. (3) is derived from mixing length theory, for whichñ/n = λ/Ln where λ is the wavelength of the fluctuationand Ln is the density scale length, therefore being independent of growth rate γ. The γ dependence in Eq. (3)comes from the associated velocity fluctuation amplitude.
There are many criticisms to the use of linear growth rates
for assessing fluctuation level when another physical process is important for saturation. However, in one study,
non-linear and linear simulations have been shown to agree[18]. Therefore, for this case, we compare fluctuation levels ñ/n with γ, however the obtained increasing trend goesagainst simple mixing length theory.

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Te centre (keV)electron heat transport coefficient, and it was shown thatat high beta, a transport increases more strongly with beta
than predicted by a simple gyro-Bohm type model (incorporating drift wave instabilities); and that a model describing resistive interchange turbulence accurately reproduced
the beta tendency of electron heat conductivity. This established (1) the role of high beta/ resistive g on energy transport. Here, however, we plan to show the other possible
links, (2) role of high beta on fluctuations and (3) role offluctuations on energy transport. This is the first such workof a comparison of density fluctuations and energy transport; previously particle transport coefficients were compared with the fluctuation level in low beta high field discharges [14].

Fluctuation measurements and consistency
with resistive g

For this study, density, temperature and magnetic fieldstrength were systematically varied in a series of discharges. At certain times during the flat-top of each discharge when the plasma was at equilibrium, fluctuation andparameter profiles were stored in a small database to beanalyzed. The range of central densities and electron temperatures, and magnetic field strengths spanned are shownin Fig. (1), together with lines of constant β, νei (related
to magnetic Reynolds number S ), and gyro-radius. It canbe seen that the temperature does not change strongly withdensity. The volume average βvol reached only up to 3% inthese discharges as the helical pitch configuration parameter was γ = 1.25 rather than γ ∼ 1.20 at which the recent very high beta shots were obtained [1]. The value βvolwas calculated from measured density and electron temperature values, applying a single correction factor for alldischarges to match the average diamagnetic β.
The βvol dependence of the line integrated fluctuationlevels from the FIR and PCI systems as described in Sec.
(2) are plotted in Fig. (2). It is clear that for both systems,
there is a strong increasing trend of fluctuation level with
β. The increase with βvol appears to be most significant forβvol > 1%, consistent with the findings that χexp /χgmod = 1
for βvol > 1% in [13], where χexp is the experimentallyderived thermal conductivity and χgmod is the resistive gmode growth conductivity as in Eq. (3). For the 2D PCI(with k > 1cm−1 ), the fluctuation level appears to double roughly as beta goes from ∼ 0.3% to 3%, while for
the FIR, which measures k < 1cm−1 , the increase is muchlarger. This may be attributed to the fact that resistive interchange turbulence can drive both large scale and ion gyroscale fluctuations, and has a strong beta dependence, while
the level of electrostatic drift wave turbulence (ITG/TEM),
which does not depend so explicitly on beta, is strong at
the ion gyro-scale but non-existent at large scales. In both
diagnostics there also appears to be a large scatter at highβ. This may be due to (1) the loss of detail due to lineintegration effects, and (2) the resistive g mode growth rate
has additional dependence than simply βvol . The motivates
us to look more in detail at the comparison of the fluctuation level with the resistive g mode growth rate scaling
formula as written in Eq. (2).
A local analysis is performed on one shot from theProceedings of ITC/ISHW2007
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4κnFig. 2 βvol dependence of line integrated fluctuation level forboth (a) ion gyro scale structures with k > 1cm−1 from
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the higher density discharge has about twice the fluctuationlevel, and is largest between ρ = 0.8 − 1.0 where resistiveg is located. The scaling of fluctuation level here is clearly
attributable to stronger density, despite having similar βvol .
The position of the strongest fluctuation peak as afunction of βvol is plotted in Fig. (7), separated out into
parts above and below the mid-plane, as discussed before.
As βvol is increased, the region of magnetic well extends
towards the edge, as indicted by the shaded region, meaning that the resistive g peaks must be localized further towards the edge at higher beta. This seems to be consistent
with the measured positions, indicated by dots, more particularly for components on the bottom. For components
on the top, there appears to be a concentration of peaks

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database at βvol = 2.5% considering profiles of density,
temperature and pressure, density fluctuation amplitudeand normal curvature κn , as plotted in Fig. (3). Profilesof fluctuation amplitude both above and below the midplane are shown, considering the Shafranov shift as determined from Thomson scattering. The mismatch of thesetwo profiles may be attributable to either (1) because different directional wave-vector components are measured
on the top compared with the bottom, or because (2) fluctuation structure is not symmetric on a flux surface, possibly due to ballooning structure. The spatial resolution isaround ∆ρ = 0.2 at the edge and around ∆ρ = 0.5 towards
ρ = 0.3, because the average k is higher towards the edge
and since ∆ρ ∝ 1/k. The strongest peak appears aroundρ = 1.05, and appears to correspond to a peak in the pressure gradient profile, plotted in Fig. (3c), demonstratingthat pressure gradient may be the free energy source at theedge. However, the normal curvature is positive only outside ρ = 0.75 so the the fluctuation power inside this radius
must be driven by a different mechanism other than resistive interchange.
Taking the component at ρ = 0.9, we compare the local ñ/n with the local g-mode growth rate in Fig (4). It canbe seen that there is a much clearer relationship than in Fig.
(2), particularly at higher γ. The cause of this better agreement is the increased density dependence arising from theS dependence in Eq. (2). This density dependence is highlighted in Fig. (5), where line integrated Ñ/N is comparedwith both hnT i/B2 ∼ βvol and hn(nT )i/B2 (where hi denotes volume averaging). The χ2 residual of points conforming to a smooth curve is a factor of two better in thecase of the stronger density dependence (b). (Note, however that the g-mode growth rate γg6 from Eq. (2) has a B−4
scaling as compared with the B−2 dependence here.)
The increased density dependence is also highlighted
by comparing 2 discharges with similar βvol , but significantly different density. The profiles of fluctuation levelñ/n, n, T and are compared in Fig. (6). It is clear that

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Fig. 4 Comparison of local fluctuation amplitude ñ/n at ρ = 0.9
and resistive g driving growth rate γg at the same ρ.

Proceedings of ITC/ISHW2007

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Fig. 7 Position ρ of strongest peak of ñ(ρ) on (a) bottom and (b)top sides of the midplane, as a function of βvol , comparedwith changes in the magnetic structure including well/hillboundary and rational surfaces#67683@1;be=0.878;dpar=3.82;fl/n= 920.
#67657@2;be=0.874;dpar=1.89;fl/n= 539.

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around the ι = 1/2 rational surface, approximately where
the magnetic shear is close to zero. This type of MHD
activity close to rational surfaces has also recently been reported in [19].
In summary, all these results appear to demonstrate
the consistency between measured fluctuation amplitudes
and positions and the expectation with resistive g mode.
The largest discrepancies are from the existence of fluctuations inside the region of magnetic well which may besome drift wave component. Hybridization of resistive gand drift-waves is also possible [17]. This distinction could
be more rigourously made on the basis of the difference
between the poloidal phase velocity and to the ExB velocity, or the coherence between density and magnetic fluctuations.

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Comparison of fluctuation amplitude withpower balance χ at high β

The role of fluctuations on confinement itself is very important to confirm the original assertion that transport is

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dominated by anomalous processes; and to check that themeasured fluctuations are important for confinement, evenregardless of the role of what is the driving mechanismof the fluctuations (resistive g or drift wave). For the setof discharges analyzed above, power balance analysis was
carried out to determine the “effective” thermal conductivity χ = (χe + χi )/2 as per the procedure described in [13],based on the FIT code [20] which computes the power deposition from NBI by calculating fast particle orbits andtheir interactions with bulk plasma. The PROCTR code[21] is then used to analyze the diffusivity. For the comparison we choose to compare the edge local fluctuationlevel with the edge χ (ρ = 0.95). The results, for a selected
subset of shots in the previous section are plotted in Fig.
(8). It is clear that there is is an increasing trend of fluctuation level and conductivity. However, there are significant
outliers; these may be due to the fact that turbulence saturates to a level depending on the power input, rather thanon the conductivity itself (dependent on gradient), because
the gradient steepens to a value such that the heat flux balances the input power in steady state. [*** put this figure
in too***].

ntil/n

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Rax=3.75m; #68996
ntil/nRecently, pellet-fueled high density plasmas were achieved
with central density approaching 1021 m−3 [2, 4]. These
plasmas are characterized by having a high density corewith a “diffusion barrier” around mid radius, correlatingwith the position of zero magnetic shear. Though theseplasmas are made at high field (B > 2T), the central beta
approaches that of the high volume average beta plasmas,
and so the beta gradients in the diffusion barrier region arevery high; therefore they may have some similar characteristics to high volume average beta plasmas. However,the diffusion barrier is always in the region of magnetic
well, meaning that resistive interchange turbulence is not a
candidate to explain turbulence.
It has recently been shown [4] that the maximum attainable central density and stored energy increases as themagnetic configuration is changed to move the magnetic
axis further outward. For formation of the diffusion barrier, and improved performance at outward shifted configurations, turbulent transport play a significant role. In a
separate study [22], the temporal evolution of turbulence(from 2D PCI) and particle particle transport studied after multiple pellet injection with in the configuration withvacuum Rax = 3.6m. In this analysis, fluctuation andplasma properties are compared between Rax = 3.65m andRax = 3.75m, to confirm whether fluctuation propertiesare reduced in line with improved confinement. Because
the 2D PCI sightline passes around R = 3.603m, as themagnetic axis moves outwards, the system is no longer
sensitive to core fluctuation properties. In particular, forthe highest performance around Rax = 3.9m, fluctuations
around the diffusion barrier cannot be measured; however,they can be measured sufficiently up to Rax = 3.75m forwhich the sightline penetrates down to ρ = 0.4.
The time evolution of the central pressure, computedfrom Thomson scattering, and fluctuation level ñ/n at ρ =0.4 are compared for both configurations in Fig. (9). Afterpellet injection, the temperature is reduced considerably,
and the density falls, while the temperature rise so that thepressure rises. It can be seen that at Rax = 3.75m, the pressure has risen more quickly and to a higher value than in
the case of Rax = 3.65m. The fluctuation behavior changes
strongly in time after pellet injection. In both configurations, there appears to be strong bursts in time, up to a time
when the bursts no longer occur and the fluctuation level isdramatically reduced. However, it is easy to notice that in
the Rax = 3.75m case, bursts persist over a shorter periodof time, and there appears to be a “quiet” period between
the bursts meaning that the time average fluctuation level,
and hence induced transport, should be smaller. This correlates with the stored energy being higher in this case, so
it demonstrates that the reason for the higher pressure may
be due to reduced fluctuation level despite stronger density/pressure gradient.
One possible reason for the reduced fluctuation level

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Proceedings of ITC/ISHW2007

Fig. 9 Comparison of the time evolution of the central pressure,computed from Thomson scattering, and fluctuation levelñ/n at ρ = 0.4 for configurations with (a) Rax = 3.65m
and (b) Rax = 3.75m

when the pressure is higher is due to the difference of thehollowness of the temperature profile, which is produced
transiently after pellet injection. Contour plots of the electron temperature profile evolution are shown for both casesin Fig. (10). The temperature profile is more hollow immediately after pellet injection in the inward shifted case(Rax = 3.65). For Rax = 3.65m, the temperature profile
switches from hollow to peaked At t − t0 = 0.25s, aroundabout the same time as the bursts of fluctuations disappear. The same is true for Rax = 3.75m, where the profile switches from hollow to peaked and fluctuations disappear at around t − t0 = 0.18s. This indicates that the
opposite sign of temperature and density gradient is unstable. Such an instability threshold is characteristic of a slab
ITG mode but this does not discount other possible drive
mechanisms such as ballooning modes. Because the caseof Rax = 3.75m has a less hollow temperature profile, it iscloser to marginal stability and so fluctuation bursts do notoccur so frequently.
The burst-like nature of fluctuations observed in dense
core plasmas is similar in nature to ELM activity in theedge of Tokamak devices. In both cases, these occur when
the pressure gradient is very high, but in the case here,
it seems that pressure gradient does not drive the mode,
rather some simple ITG-like mode. The bursts may be dueto the profiles being near a marginal stability threshold,and that the instability has a very distinct hard threshold.
Such ITG instability thresholds have been demonstrated inTokamak transport before.
[1] K. Y. Watanabe. Confinement study on the reactor relevanthigh beta lhd plasmas. In these proceedings, (2007).
[2] N. Ohyabu et al. Phys Rev Lett 97, 055002/1 (2006)[3] K Tanaka et al. Nucl Fusion 46, 110 (2006)[4] R. Sakamoto these proceedings (2007)[5] K. Kawahata et al. Fusion Eng & Des 34-35, 393 (1997)[6] A.L. Sanin et al. Rev Sci Instrum 75, 3439 (2004)

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t-t0 (s)

0.80

0.60

0.50

0.0
-0.2

0.4

0

0.70

0.5

0

0.3

0.40

0.10
0.20

0.4
0
0.40

ρ

Rax=3.75m; #68996
1.0

0.4

0.6

Fig. 10 Contour plots of temporal/spatio evolution of temperature profiles from Thomson scattering (fitted with a polynomial) after pellet injection for configurations with (a)Rax = 3.65m and (b) Rax = 3.75m.

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