Proceedings of ITC/ISHW2007

Effects of the stochasticity on transport properties in high- LHD
Yasuhiro SUZUKI, Kiyomasa WATANABE, Hisamichi FUNABA, Satoru SAKAKIBARA,
Noriyoshi NAKAJIMA, Nobuyoyshi OHYABU and LHD experiment group
National Institute for Fusion Science, Toki 509-5292, Japan
Effects of the stochasticity of magnetic field lines on transport properties are investigated. In a high- LHD
plasma, the structure of field lines in the edge region becomes stochastic by finite- effects but the finite pressure
gradient exists in the region. The radial diffusion coefficient and the Kolmogorov length of stochastic field lines areestimated. In the edge region, the radial diffusivity of stochastic field lines becomes large and the Kolmogorov
length becomes short due to the increased . In the region, the radial heat diffusivity becomes large due to thestochasticity of field lines.
Keywords: stochasticity, transport, radial heat diffusivity, HINT2

1. Introduction
Generating and keeping clear flux surfaces are an aim
of magnetic confinement researches, because thestochasticity of magnetic field lines leads the degradationof the confinement connecting between core and edge
region. There are some analytical works investigating theimpact of the stochasticity of magnetic field lines on theradial transport property [1-4]. In those works, Rechester
and Rosenbluth pointed out the radial heat diffusivity due
to stochastic field lines relates to both the stochastic
diffusion parallel and perpendicular to the magnetic field[1].
The stochasticity of field lines due to finite- effects
is an intrinsic property in stellarator/heliotron. Since thepressure-induced perturbed field breaks the symmetry of
the field, the structure of magnetic field lines becomesstochastic, especially in the edge. In order to aim
stellarator/heliotron reactors, the study of the transport due
to stochastic field lines is critical and urgent issue.
The LHD is an L=2 heliotron device. A numericalcode to calculate 3D MHD equilibrium without theassumption of nested flux surfaces predicts the fieldstructure becomes stochastic due to the increased .
In addition, in numerical simulations, the finite pressure
gradient ∇p can exist in the stochastic region [5]. In theedge region of LHD plasmas, the connection length ofstochastic field lines is still long compared to the parallel
electron mean free path. That is, there is a possibility tokeep the finite pressure on stochastic file lines. LHD
experiments suggest the plasma pressure spread over theregion expected stochastically [7]. This supports above
speculation. However, that speculation does not include theeffect of stochastic diffusion perpendicular to the field.
author’s e-mail:ysuzuki@LHD.nifs.ac.jp

In this study, the radial heat diffusivity due tostochastic field lines is investigated in a high- LHD
equilibrium. In next section, the degradation of flux surface
quality due to finite- effects is studied in a LHD
configuration. Then, the diffusive property of stochastic
field lines is studied. Lastly, results are briefly summarized
and shown future subjects.

2. Degradation of flux surface due to plasma Figure 1 shows Puncture maps of magnetic field linesfor (a) the vacuum field and (b) a finite- equilibrium on
the horizontal cross section. The configuration is an inward
shifted configuration (Rax=3.6m, =1.254, BQ=100%). The
profile of the normalized plasma pressure p/p0 is alsoplotted as the function of R on the equatorial plane. Thefinite- field is calculated by HINT2, which is a 3D MHD
equilibrium calculation code without the assumption ofnested flux surfaces [5]. Since HINT2 uses the real
coordinate system, it can treat the magnetic island andstochastic field. The diamagnetic beta β dia is about 3%.
For the finite- the region with closed flux surfaces
decreases and field lines in the edge region becomesstochastic. Chains of small magnetic islands appear. Thefinite plasma pressure exists in spite of field lines
becoming stochastic in the edge (see fig. 2(b)). Two arrows
indicate the position of the vacuum last closed flux surface(LCFS) on the equatorial plane. The finite pressure spreads
over the vacuum LCFS.
In fig.2, profiles of the electron temperature Te(#46465, t=1.625), the distance along the magnetic field
LC started along R on the equatorial plane, contour lines
with p=const. (Z<0) and puncture map of field lines(Z>0) for comparison are shown, respectively. The lengthof the calculation tracing field lines is limited to 2000m.

Proceedings of ITC/ISHW2007
In figs, contour lines p=const. exists in the stochastic
region with keeping the surface structure. HINT2
calculates converged pressure distribution in the finite-field bypi+1 = p =F=L indlℱpi−L inB
L in dl−L in B,(1)

1:for 𝐿𝐶 ≥ 𝐿𝑖𝑛,
0:for 𝐿𝐶 < 𝐿𝑖𝑛

where, i means a step number of iterations, LC is theconnection length of a magnetic field line starting eachgrid point (LC is finite for open magnetic field lines), andLin is prescribed as an input parameter to control thecalculation. Equation 1 calculates the ‘averaged’ plasmapressure on the flux tube. This corresponds to simulate theradial diffusion of field lines. In order to consider thiseffects, profiles of plasma pressure with different Lin(=30m and 300m) are shown in fig. 3. For Lin =300m,contour lines are different, especially in the stochastic
region. If the distance along 𝐵 is shorter than Lin , theaveraged pressure p is set to zero. In fig. 2, since LC is101~102m, the distribution of p is sensitive to Lin . This(a) vacuum(b) <>dia~3%

Fig.1 Puncture maps of magnetic field lines for thevacuum field and a finite- equilibrium (<>dia~3%) areplotted at the horizontal cross section (=/M). A green
line indicates the normalize plasma pressure p/p0 as thefunction of R on the equatorial plane. Two arrows in figs
indicate the position of the vacuum LCFS on theequatorial plane.

Fig.2 Profiles of the electron temperature Te (#46465,
t=1.625) and the distance along the magnetic field LC
on the plane corresponding to fig. 1(b) are plotted as
the function of R. Contour lines with p=const. are also
shown for the comparison.
study has an assumption that the electron temperature is
low because of the consideration of high- experiments(see fig. 2). Thus, we adopt Lin is 30m. As a result,contour lines of p/p0 is consistent to the temperature
profile.
In order to study the degradation of flux surfaces dueto the increased , the change of positions of the LCFS andmagnetic axis is shown in fig. 4 as the function of β dia .
At first, the change of the outward torus is noted. Forlow- equilibria (< 1%), the LCFS slightly expands
compared to the vacuum field. Then, increasing  (>1%),
the LCFS still sustains near the vacuum LCFS. For high-(>2%), the LCFS shrinks sharply. On the other hand, theinward region, the LCFS degrades monotonically due tothe increased . Thus, we guess the degradation of thetransport is significantly important at high- (>2%). The
magnetic axis also monotonically changes due to theincreased . At a high- ( β dia ~3%), the Shafranov shift∆/a is about 0.5. However, the MHD equilibrium does notcollapse and it is sustained.

Fig.3 contour lines with different Lin (Z>0 and Z<0)are shown at the plane corresponding to fig. 1. Greenlines indicate contour lines with Lin =30m and blue lines
indicate Lin =300m. Lines in the edge are different.

Proceedings of ITC/ISHW2007

Fig.4 The change of positions of inward (red) andoutward (green) LCFS on the plane corresponding tofig. 1is plotted as the function of β dia . The shift ofthe axis is also plotted for the reference (blue).

3. Radial heat diffusivity due to stochastic fieldlines
The radial heat transport increases as it gains thestochasticity. In the collisionless plasma, where theelectron mean free path e is very long, the radial heatdiffusivity χr due to ‘only’ the stochasticity of magnetic
field lines is given byχr = DFL vth(2)
where vth is the electron thermal velocity and DFL is thediffusion coefficient of magnetic field lines and defined by
DFL = Δr 2 /LC(3)
LC is the correlation length to calculate the diffusion
coefficient. Since χe is the contribution of only thestochasticity of magnetic field lines, the effective radialtransport χeff is given byχeff = χr + χ⊥ .
(4)
On the other hand, in the collisional plasma, Krommes etal. identifies three different subregimes with decreasing
collisionalty [4], which are fluid regime ( τ⊥ <τ∥ <τk ),Kadomtsev-Ppgutse( τ∥ < τ⊥ < τk )andRechester-Rosenbluth ( τ∥ < τk < τ⊥ ) regime. In typicalparameters of LHD experiments, the collisionalty is
Rechester-Rosenbluth (RR) regime in the region expectedstochastically. The radial heat diffusivity due to thestochasticity of field lines is given byχr = DFL χ∥ /Lk(5)in the RR regime, where Lk is the Kolmogorov length.
The Kolmogorov length Lk is a characteristic parameter
to mesure the stochasticity [8]. Thus, equation 5 means theparallel contribution of the stochasticity is very important
as well as the perpendicular contribution, because Lk
plays the role of the correlation length along field lines.
In order to study χr , the diffusion coefficient DFL isFig.5 The radial profile of the diffusion coefficient
DFL is plotted as the function of R. The puncture mapof field lines is also plotted as the reference.
estimated at first. In fig. 5, the radial profile of thediffusion coefficient is plotted as the function of R. The
puncture map of field lines is also plotted as the reference.
The procedure to calculate the mean squared radialdisplacement ∆r 2 of field lines is following; (i) thedistribution of the normalized toroidal flux s = Φ Φedge
is given at first, where Φ is calculated by integrating
inside contour lines at p=const. (ii) then, the normalizedminor radius ρ is calculated and field lines are traced
from distributed points on =cont. plane. (iii) in the last,
the mean squared displacement of Δρ2 is calculatedwith tracing field lines and the distribution coefficient isgiven byDFL = reff 2 Δρ2 /LC ,(6)
where reff is the effective minor radius. In fig. 5, DFL issmall in clear flux surfaces. However, in the stochastic
region, the diffusion coefficient increases rapidly toward
the outside of the torus. In the stochastic region, it isexpected the perpendicular diffusion is very large.
The Kolmogorov lengthis also estimated toconsider the parallel contribution of the stochasticity. In
analyses of edge plasmas, especially the Dynamic Ergodic
Divertor (DED), the Kolmogorov length is given by thequasi-linear form [4]. However, since the stochasticity instellarator/heliotron is caused by pressure-induced
perturbation, the number and amplitude of the mode ofperturbations are unclear and the calculation of thosevalues is difficult. Thus, we estimate the Kolmogorov
length using a following definition,d = d0 exp

lLk(7)

where, d is the circumference of small flux tube and l is
the length of the flux tube. Using this definition, the impact

Proceedings of ITC/ISHW2007
finite- effects on Lk is studied in vacuum configurations
in LHD [7] and finite- equilibria in Wendelstein 7-X [8].
In fig. 6, the profile of the inverse of the Kolmogorov

Fig.6 The radial profile of the inverse of theKolmogorov length Lk is plotted as the function of R.
The puncture map of field lines is also plotted as thereference.
length is plotted as the same plot corresponding to fig. 5.
The inverse of the Kolmogorov length rapidly becomeslong exponentially. This suggests the stochasticity of fieldlines increases with short length. In the outermost position
to calculate Lk (R=4.64m), Lk is about 11m.
Finally, we estimate the radial heat diffusivity χr . At
R=4.64m, DFL and Lk are about 10-4 and 11, respectively.
In fig. 2, the electron temperature is about 20~30eV. If
Te = 30eV and ne =1019m-3, χr ~25m2/s. This is verylarge compared to χeff of the local transport analyses in
the plasma core [9]. The comparison of the experiments
and other estimation, which are obtained from the transport
code for the edge plasma [10], is a future subject.

4. Summary
The stochasticity of magnetic field lines and effects on
the transport properties due to finite- effects areinvestigated. Flux surfaces keeps clear structure until theintermediate- (<2%). However, for high-b, flux surfaces
rapidly degrades due to the increased b. Characteristic
properties, which are the diffusion coefficient of magnetic
field lines and the Kolmogorov length are estimated. In ahigh-b equilibrium, stochastic properties appear in the edge.
Using Rechester-Rosenbluth formulation, the radial heatdiffusivity due to only the stochasticity of magnetic fieldlines. The estimated diffusivity is very large. It is necessary
the comparison to the experiments and other estimation.

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[2] T. H. Stix, Nucl. Fusion 18, 353 (1978)[3] B. B. Kadomtsev and O. P. Pogutse, in Proceedings of the
7th International Conference, Insbruck 1978 (International
Atomic Energy Agency, Vienna, 1979), Vol. 1, p.649[4] J. A. Krommes, C. Oberman, and R. G. Kleva, J. Plasma
Phys. 30, 11 (1983).
[5] Y. Suzuki, N. Nakajima, K. Y. Watanabe, Y. Nakamura,and T. Hayashi, Nucl. Fusion 46, L19 (2006)[6] K. Y. Watanabe, Y. Suzuki et al., Plasma Phys. Control.
Fusion 49, 605 (2007)[7] T. Morisaki, et al., J. Nucl. Matter. 313-316, 548 (2003)[8] E. Strumberger, Contrib. Plasma Phys. 38, 106 (1998)[9] H. Funaba, et al., in this conference[10] M. Kobayashi, et al., in this conference