Improved stability due to local pressure
flattening in stellarators
K. Ichiguchi
National Institute for Fusion Science,Oroshi, Toki, Japan

M. Wakatani, T. Unemura
Graduate School of Energy Science, Kyoto University,Gokasho, Uji, Japan

T. Tatsuno
Graduate School of Frontier Science, University of Tokyo,Tokyo, Japan

B.A. Carreras
Oak Ridge National Laboratory,
Oak Ridge, Tennessee, United States of America
Abstract. It is demonstrated that the stability of low n pressure gradient driven modes in stellarators
is improved by local flattening of the pressure at low order rational surfaces. Here n is the toroidalmode number. Results are presented for the Large Helical Device with an inward magnetic axis shiftof 25 cm.

1.

Introduction

The largest stellarator/heliotron device, the Large
Helical Device (LHD), has successfully started
physics experiments [1]. The electron and ion temperatures Te ≈ 3.8 keV and Ti ≈ 2.8 keV wereobtained in the low density range n̄e ≈ 1.5×1019 m−3[2]. The energy confinement time obtained was about
50% better than the International Stellarator Scaling of energy confinement [3]. The maximum averagebeta value, β̄ ' 2.4%, exceeded 2%, which was thehighest beta obtained in stellarator/heliotron devices[4]. Because β̄ ' 2.4% is not limited by MHD instabilities and the target beta value of LHD is 5%, ahigher power heating is expected in an optimized
magnetic configuration.
In this paper we will discuss the effects of local
pressure profile flattening on interchange modes
which may affect the stability and confinement
properties of LHD. For studying MHD stability instellarator/heliotron devices, the Mercier criterion [5]is valuable. For three dimensional MHD equilibriaunder the assumption of the existence of flux surfaces, the Mercier criterion is usable for evaluating
the beta limit [6]. Another important ingredient of
three dimensional MHD equilibrium and stability is
the formation of magnetic islands [7]. This is related
to the existence of three dimensional nested flux
Nuclear Fusion, Vol. 41, No. 2

surfaces [8]. The magnetic islands may be produced
by resonant perturbed magnetic fields which aregenerated by internal resistive MHD instabilities orexternal error fields. If the magnetic islands appear
at low order rational surfaces, it is expected thatthe pressure profile will become flat in the island
regions. It is shown that the MHD stability based on
the Mercier criterion changes significantly, although
the pressure flattening is highly localized in theneighbourhood of the relevant rational surfaces [9–11]. For this situation the stability limit to the lowmode number interchange modes is the importantcriterion.
In Section 2, reduced MHD equations and a modelfor describing a pressure profile with locally flat
regions at rational surfaces are given. In Section 3,
improvement of the stability limit is discussed briefly
for cylindrical model stellarators. In Section 4, therelation between the width of local pressure flattening and the growth rates of interchange modes
is shown for the LHD case in the toroidal model.
Examples of pressure profiles to suppress the low nmode are also shown. In Section 5, we summarize ourresults and discuss how they allow reinterpretation ofthe analysis of the high β CHS and LHD experimentsthat seem to violate the Mercier criterion when evaluated using smooth pressure profiles.

c 2001, IAEA, Vienna181

K. Ichiguchi et al.

2.

Reduced MHD equations and amodel pressure profile with locallyflat regions at rational surfaces

and Pres (ρ) plays a role in flattening the pressure atrational surfaces

Pres (ρ) =For analysing pressure driven instabilities instellarator/heliotron devices, we use the ideal
reduced MHD equations [6, 10, 12], which are writtenas∂ψ= −(R/R0 )2 B · ∇u
∂tρ

(4)

∇⊥ = ∇ − ∇ζ(∂/∂ζ).

(5)

Here ψ, u and P denote the poloidal flux function,
the stream function and the plasma pressure, respectively. The axisymmetric component of the magnetic
field is given by B, and R and ζ denote the major
radius and the toroidal angle, respectively. The magnetic axis is R = R0 , and the toroidal field at R = R0
is B0 . Because the free boundary effect is not significant for the stability of currentless plasmas, a perfectly conducting wall is usually placed at the plasmaboundary [10]. Note that the equilibrium state of ψis consistent with the rotational profile due to stellarator fields.
For describing the locally flat pressure profile,P (ρ) = C[P0 (ρ) + Pax (ρ) + Pres (ρ) − A](8)

is assumed, where P0 (ρ) denotes a standard smoothpressure profile, Pax (ρ) corresponds to a flatteningof pressure profile near the magnetic axis given by"

1Pax (ρ) = [P0 (0) − P0 (ρ)] exp −2182



ρwa



ρ−ρmwm

4 #.

(10)

3.

whered∂=+ (R/R0 )2 ∇u × ∇ζ · ∇⊥dt∂t
Z 2π  2 1|B − B|2RΩ=dζ1+
2π 0R0B021× exp −2(7)(3)

R0 B0 ∂− ∇ψ × ∇ζ · ∇⊥R2 ∂ζ"(6)d 2∇ u = −B · ∇(∇2⊥ ψ) + R02 ∇Ω × ∇P · ∇ζ (2)dt ⊥B·∇=mHere ρ denotes the square root of normalized toroidalflux. In Eq. (8), A and C are numerical factors
to fix pressures at both the magnetic axis and theplasma surface. In Eq. (9), wa denotes the width ofa region of flat pressure profile near the magneticaxis. In Eq. (10), ρm denotes the position of them-th rational surface and wm denotes the width of aregion of flat pressure profile at the rational surfaceρ = ρm .
To calculate fixed boundary MHD equilibria for
the LHD configuration with the pressure shown byEq. (8), the VMEC code [13] was applied. In thefollowing calculations the LHD configuration withan inward magnetic axis shift of 25 cm is considered. Ideal MHD stability against pressure driven
modes was studied with the RESORM code [14],which solves the linearized Eqs (1)–(3) as an initialvalue problem.

(1)

dP=0dtX{[P0 (ρm )+Pax (ρm )] − [P0 (ρ)+Pax (ρ)]}

4 #(9)Predictions based on a cylindricalstellarator model for stabilityof interchange modes in locallyflattened pressure profiles
at rational surfaces

For studying the beta limit due to interchange
modes, the linearized reduced MHD equations are
solved for assumed cylindrical stellarator/heliotron
plasmas; the equations are reduced to [9]γ 2 ∇2⊥ u = −(n − mι)∇2⊥ [(n − mι)u] − Dsm2u (11)r2where Ds and the averaged helical curvature Ω areexpressed as
β0 0 0P Ω22ZΩ = 2 N r2 ι + 2 rι dr .
Ds = −(12)(13)

Here  = a/R0 denotes the inverse aspect ratio,
β0 = 2µ0 P0 (r = 0)/B02 the central plasma beta
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Article: Improved stability in stellarators
value and N the toroidal period number of the helical field. The polarity of the helical field is assumed
as L = 2 in Eq. (12). In order to derive Eq. (11), allperturbed quantities are assumed to be proportional
to exp[γt−i(mθ−nζ)]. In Eq. (11), the prime denotes
the derivative with respect to the normalized minor
radius r. The perpendicular Laplacian operator in
Eq. (10) is expressed as
1 ddm2r− 2 .
∇2⊥ =(14)r drdrr
Here results obtained by solving Eq. (11) for interchange modes in cylindrical stellarator plasmas withlocally flattened pressure profiles [9] are briefly summarized: (i) The Suydam condition becomes irrelevant for examining the stability [15]. (ii) The radialmode structure near the marginal stability becomessimilar to that of the m = 1 internal kink mode near
the rational surface and is extended to the inside(r < rs ) or the outside (r > rs ), where rs is theradius of a rational surface. (iii) The marginal beta
value for the low m mode resonant at r = rs increases
in relation to the Suydam limit [15]. (iv) It becomeseasier to stabilize the higher harmonics with a narrower flat pressure region.

4.

Stabilization of low n interchange
modes with flat pressure regionsat rational surfaces in toroidal
plasmasFor the pressure profile given by Eq. (8) in theLHD model configuration, global pressure driven
modes with n = 1, 2, 3 are examined with theRESORM code [14], where n is the toroidal mode
number.
4.1.

n = 1 mode

The smooth pressure profile P0 (ρ) and the rotational transform profile ι(ρ) are plotted in Fig. 1.
The rational surfaces for the n = 1 mode, ι = 1/1
and ι = 1/2, are shown by the dotted lines in the figure. Here the peak beta value is assumed to be 2%.
The unstable n = 1 mode exists with a growth rate of
4.846 × 10−2; it is destabilized at the ι = 1/2 surface.
Here the growth rate is normalized by the poloidalAlfvén time. When the width w of the flat pressure
region is increased at the ι = 1/2 surface, the growthrate decreases, as shown in Fig. 2. A marginally stable pressure profile against the n = 1 mode with
Nuclear Fusion, Vol. 41, No. 2 (2001)

Figure 1. An assumed pressure profile and a rotational transform profile in the LHD configuration withan inward magnetic axis shift of 25 cm obtained with
the VMEC code [13]. The dotted lines show ι = 1/1 andι = 1/2. The central beta value is β(0) = 2% and theaverage beta value is β̄ = 0.632%. The radius ρP denotes
the square root of the normalized poloidal flux, which isalso used in Figs 3–10.

w = 0.045 is shown in Fig. 3. For the weakly unstable n = 1 mode with γ = 8.583 × 10−3 in the case of
w = 0.032, the radial mode structure of the stream
function is shown in Fig. 4. Here the sharp decreaseof u at the ι = 1/2 surface is as expected from thestability theory for the interchange mode in the cylindrical plasma model, as discussed in Section 2.

4.2.

n = 2 mode

Here the same pressure and rotational transformprofiles as shown in Fig. 1 are used for the stability
analysis of the n = 2 mode. However, the number ofthe relevant rational surfaces increases: ι = 2/2, ι =2/3, ι = 2/4 and ι = 2/5. The RESORM code showsthat the n = 2 mode is destabilized at the rationalsurfaces ι = 2/4 and ι = 2/5. Here the growth rate isγ = 7.187 × 10−2 at β(0) = 2%. Thus it is necessary
to introduce two locally flat pressure regions withdifferent widths at ι = 2/4 and ι = 2/5 for stabilizing
the n = 2 mode. When w = 0.02 at ι = 2/4 andw = 0.04 at ι = 2/5, the instability is suppressed
completely. The pressure profile obtained is shownin Fig. 5.
183

K. Ichiguchi et al.

0.05

γτA

0.04
0.03
0.02
0.01
00.01 0.02 0.03 0.04 0.05

w
Figure 2. Growth rate of the n = 1 pressure driven
mode versus the width w of the flat pressure region at theι = 1/2 surface. Here time is normalized by the poloidal
Alfvén transit time.

Figure 3. A marginally stable pressure profile against
the n = 1 pressure driven mode with the width of the flatpressure region w = 0.045. The rotational transform profile obtained with the VMEC code and rational surfaces
ι = 1/1 and ι = 1/2 are also shown.

4.3.

n = 3 mode

For the LHD configuration with the pressure androtational transform profiles shown in Fig. 1, there
are six rational surfaces: ι = 3/3, 3/4, 3/5, 3/6, 3/7
184

Figure 4. Radial mode structure of the stream function
for w = 0.032 at the ι = 1/2 surface. The growth rate isγ = 8.583 × 10−3 .

Figure 5. A marginally stable pressure profile against
the n = 2 pressure driven mode with the flat pressure
regions given by w = 0.04 at ι = 2/5 and w = 0.02 at
ι = 2/4. The rotational transform profile obtained withthe VMEC code and rational surfaces ι = 2/2, 2/3, 2/4
and 2/5 are shown.

and 3/8. The RESORM code gives the growth rate
γ = 8.233 × 10−2 at β(0) = 2%. The radial mode
structure of the stream function is shown in Fig. 6.
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Article: Improved stability in stellarators

Figure 6. Radial mode structure of the stream function
localized near the magnetic axis. The growth rate is γ =8.233×10−2 and the mode structure has ballooning modecharacteristics.

Note that the unstable mode is localized at the central region with a ballooning structure. These are
typical characteristics of the toroidal non-resonantpressure driven mode [16]. To suppress both this nonresonant mode and a resonant mode at ι = 3/8,
the central pressure profile is flattened first withwa = 0.6 in Eq. (9), as shown in Fig. 7. Then thegrowth rate decreases to γ = 6.975 × 10−2 , and theunstable mode has a typical interchange mode structure destabilized at ι = 3/4, 3/5, 3/6 and 3/7, asshown in Fig. 8. For suppressing the pressure driven
interchange mode with n = 3 completely, flat pressure regions are generated at the four rational surfaces with w = 0.03 at ι = 3/7, w = 0.03 at ι = 3/6,w = 0.02 at ι = 3/5 and w = 0.02 at ι = 3/4. The
pressure profile obtained with β(0) = 2% is shownin Fig. 9. The average beta value is changed fromβ̄ = 0.632% (Fig. 1) to β̄ = 1% (Fig. 9).
It was demonstrated that the pressure driven
modes with n = 1, n = 2 and n = 3 can be stabilized by generating locally flat pressure regions at therelevant rational surfaces separately. Furthermore,
it is confirmed that the n = 1, 2, 3 modes become
stable simultaneously when the pressure profile isdescribed with wa = 0.6 and locally flat regions withw = 0.025, 0.03, 0.065, 0.03, 0.025 and 0.02 at ι = 0.4,3/7, 0.5, 0.6, 2/3 and 0.75, respectively (Fig. 10).
Nuclear Fusion, Vol. 41, No. 2 (2001)

Figure 7. The central region of the pressure profile
in Fig. 1 is flattened with the parameter wa = 0.6 in
Eq. (15). The rotational transform profile obtained with
the VMEC code and rational surfaces ι = 3/3, 3/4, 3/5,3/6, 3/7 and 3/8 are shown.

Figure 8. Radial mode structure of the stream function
for the n = 3 pressure driven mode with γ = 6.975×10−2
after suppression of the non-resonant mode.

5.

Concluding remarks

We have shown that pressure profiles with local
flattening at the low order resonant surfaces lead to185

K. Ichiguchi et al.

Figure 9. A marginally stable pressure profile against
the n = 3 pressure driven mode with the flat pressure
regions given by w = 0.03 at ι = 3/7, w = 0.03 at ι = 3/6,w = 0.02 at ι = 3/5 and w = 0.02 at ι = 3/4. The rotational transform profile obtained with the VMEC codeand rational surfaces ι = 3/3, 3/4, 3/5, 3/6, 3/7 and 3/8
are shown.

case, we should note that the size and degree of flattening might depend on dissipative effects such as
resistivity and viscosity. However, there are otherways of creating such locally flat pressure profiles.
One way is by externally applying resonant magnetic
fields. In either situation, we have shown that the low
n pressure driven instabilities are stabilized for theLHD model configurations. If the pressure gradientnear the magnetic axis is large, non-resonant modes
with toroidal mode numbers around 3 or 4 may also
be destabilized. Therefore, to increase the ideal beta
limits, broad pressure profiles with several flat spots
in the low order resonant surfaces may be needed inLHD.
There is some evidence that experiments haveexceeded the beta values given by the Mercier criterion when evaluated with smooth pressure profiles.
For instance, there are observations of this type from
CHS [17, 18] and Heliotron E [19]. The above mentioned existence of local flat spots at the low orderrational surfaces may explain the discrepancy. The
resistive interchange modes may be the mechanism
causing these flat spots in the magnetic hill regionsof those plasmas. Of course, there are other possible explanations for this discrepancy. Finite Larmor
radius or kinetic effects may cause the stabilizationof the large n modes. LHD has achieved plasmas
with β̄ ' 2.4%; this value of beta appears to alreadyexceed the Mercier limit calculated with a smoothpressure profile. More high β LHD experiments areneeded to fully test our conjecture.

Acknowledgement
This work was supported by a grant-in-aid from
the Ministry of Education, Science, Sport and Culture (Monbusho), Japan.

References

Figure 10. A marginally stable pressure profile against
the pressure driven mode with n = 1, 2 and 3 simultaneously. Here w = 0.025, 0.03, 0.065, 0.03, 0.025 and 0.02
at ι = 0.4, 3/7, 0.5, 0.6, 2/3 and 0.75, respectively.

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(Manuscript received 5 October 2000
Final manuscript accepted 8 November 2000)
E-mail address of M. Wakatani:wakatani@energy.kyoto-u.ac.jp
Subject classification: C0, St187