J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)222–226
000–000

Linear Local Stability of Electrostatic Drift Modes
in Helical Systems
YAMAGISHI Osamu, NAKAJIMA Noriyoshi, SUGAMA Hideo and NAKAMURA Yuji1
1

National Institute for Fusion Science, Toki, 509-5292, Japan
Graduate School of Energy Science, Kyoto University, Uji, 611-0011, Japan(Received: 9 December 2003 / Accepted: 20 April 2004)

Abstract
Linear local gyrokinetic-Poisson equations are solved numerically, to investigate the stability of collisionlesselectrostatic drift modes in a helical system. As a model of helical plasmas, Large Helical Device (LHD) is considered,
whose MHD equilibrium is numerically obtained. A circular tokamak with comparable aspect ratio to that of theLHD is also considered for comparison. As electrostatic drift wave branches, ion temperature gradient modes (ITG),
trapped electron modes (TEM), and electron temperature gradient modes (ETG) can become unstable. For these modes,
the local parameter dependence is investigated and destabilizing mechanisms are discussed in an LHD configuration.
Keywords:linear gyrokinetics, ITG, TEM, ETG, helical system

1. Introduction

toroidal flux, s, as T/T(0) = 1 – s and n/n(0) = (1 – s)0.2,
where ni = ne and Ti = Te are assumed below. Then p = 2nT isused as an input for VMEC code. The central beta (β0 ≡2µ0 p(0)/B02) is 1.0 %, where β0 is averaged magnetic fieldstrength at R = R0 = 3.75[m]. The measure of temperature
gradient, η = d ln T/d ln n = 1/a = 5 holds everywhere, andsufficient instabilities are expected for this value. For
comparison, we also consider a circular tokamak withcomparable aspect ratio R0/a = 3.9/0.57 to the LHD, for whichq profile is assumed to be the same as that in the currentlessLHD.

Drift wave instabilities are considered to cause a large
amount of the transport and it is important to investigate theirstability properties. Many investigations have been done fortokamaks, while there seem to be only numerable studies forhelical plasmas. There exist many helical devices with a wide
variety of geometry and magnetic field structure, so that it isimportant to investigate the stability in each helical systems
and integrate the results, in order to discuss what is common
and what is inherent feature in many configurations including
tokmaks. Thus we investigate the stability of the drift wave
in a helical system, Large Helical Device (LHD) [1], and try
to obtain the driving mechanisms of each linear instability.
For this purpose, we solve the linear local gyrokinetic-Poisson
equations in the electrostatic regime. The equation we applyis rather exact in the framework of linear gyrokinetic theory[2,3], where the ballooning representation is used as aminimal approximation. Here we consider only collisionlesscases. In this formulation, the non-adiabatic dynamics ofcirculating and trapped electrons/ions is formally treated
without any assumptions in all the frequency regime. Then,
as different drift branches, ion temperature gradient modes(ITG), trapped electron modes (TEM), and electron
temperature gradient modes (ETG) can be considered without
missing the correlation between them.
The model MHD equilibria are numerically calculated
by VMEC code with a fixed boundary constraint [4]. The net
current is assumed to be zero for LHD. The temperature anddensity profile are prescribed as a function of normalized

2. ITG cases
In Fig. 1, the k⊥ ρthi dependence of the frequencies isshown for LHD, to investigate the ion finite Larmor radius(FLR) effect. Here ρthi ≡ vthi /Ωi and k⊥ = nk |k/kα | with nk beinga toroidal mode number, and | k/k α | is treated as anequilibrium quantity with a fixed ballooning angle θk [2]. A
case for non-adiabatic electrons and ions is shown by closedcircles. The mode can also be destabilized by only nonadiabatic ions which is shown by open circles. This andnegative real frequency indicate that the modes are the ITG.
From the comparison of the cases with or without the nonadiabatic electrons, it can be seen that the non-adiabatic
electrons are destabilization for the ITG, and this can beexplained such that the fraction of stabilizing adiabatic
electrons is reduced. We also plot a case that the magnetic
drift frequency ωd is ignored in the gyrokinetic equation,©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: yamagisi@nifs.ac.jp1222

Yamagishi O. et al., Linear Local Stability of Electrostatic Drift Modes in Helical Systems

which is denoted by triangles in Fig. 1. Since the curvatureof a field line is entered only through ωd, the residual
instabilities are driven by slab-like, parallel compressibility.
The growth rate is reduced by ignoring the ωd, but theinstabilities still remain unstable.
In Fig. 2(a), the radial dependence of the ITG
frequencies is shown as a function of normalized toroidal fluxs, for LHD and a tokamak with a comparable aspect ratio.
Here we consider a fixed toroidal mode number nk , and it ischosen as nk = 86 for LHD and nk = 142 for tokamak, suchthat k⊥ ρthi (θ = 0) value is comparable at a fixed radial label.
The value of k⊥ ρthi is also shown on right Y-axis in Fig. 2(a),which is large at the core and small at the edge depending
mainly on the temperature. For LHD and the tokamak, it canbe seen that the frequency dependence is not so different. We
also plot the α dependence of the frequencies in Fig. 2(b),where change of α affects the local curvature and local shear

in the toroidal direction in non-axisymmetric systems throughthe change of field line label. Here M in the figure denotes anumber of field periods (M = 10 for LHD). The equilibrium
and linear instabilities are 2π/M periodic in α and it isnegligible in the axisymmetric case. It can be seen that the αdependence is found to be very weak in LHD.
The value of flux surface quantities like q, q′ is changed
φ/φ(θk)1 (a)0-2020202θ/π

φ/φ(θk)1 (b)2

ω (x10-5 )incl. nonadiabaticions/electrons00ωd is ignored-2incl. only
nonadiabatic ions

h=B0/B(c)-210Fig. 1

θ/π

0.5

1k⊥ρthi 1.50-2k⊥ρthi dependence of ITG frequencies in LHD. Realfrequency and growth rate are shown by dashed andsolid line respectively. Local parameters are (s,θk ,Mα) =(0.7,0,π). Closed circles are for a case of non-adiabaticions and electrons, and open circles are for a case ofeliminating non-adiabatic elecrons. Triangles are for acase of ignoring ωd.

2Fig. 3

ω (x10-5 )k ⊥ρthi (θ=θ )(a)LHD2kθ/π

ITG eigenfunction φ for (a) LHD and (b) comparable εttokamak, at s = 0.7 and k ⊥ρ thi ( θ = 0) = 0.6. Real(imaginary) part is shown by solid (dashed) line.
(c) Modulation of magnetic field strength h = B0/B for
the tokamak (dashed line) and LHD (solid line).

ω (x10-5 )(b)2compara.
tokamak00

1-2-20Fig. 20.5S100.5Mα/π 1(a) Radial dependence of ITG frequencies in LHD (circles) and tokamak (triangles) with θk = 0 and Mα = π. (b) Field line
dependence of ITG frequency in L HD with s = 0.7 and θk = 0. Real frequency and growth rate are shown by dashed and solidline respectively. In (a), k⊥ρthi (θ = 0) value is also plotted by dotted line for right coordinate.

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Yamagishi O. et al., Linear Local Stability of Electrostatic Drift Modes in Helical Systems

with radial label, and local quantities like local shear andcurvature are changed with the field line label, so that thechange of these parameters is expected to affect the stability.
Nevertheless, the above results show that these effects areweak, and the radial growth rate curve seems to be mainly
determined by k⊥ ρthi value, as can also be seen by comparingwith Fig. 1. Then the core region is stabilized by the ion FLR
effects and the edge is also stabilized by the small k⊥ ρthi .
Typical ITG eigenfunctions in LHD and our tokamak
case are compared in Fig. 3. It can be seen that the mode
width along a field line is not so different in LHD and thetokamak, and the modes are not confined in a helical ripple
but in a toroidal period.
From above, the modes at θk ~ 0 are not so different inLHD and the comparable tokamak. However, the slightdifference can be found when ballooning angle θk is changed.
The results are plotted in Fig. 4 for LHD (left) andcomparable tokamak (right). In this figure, we also show acase that ωd is ignored which is denoted by triangles. Theinstabilities in this case are driven by the parallel
compressibility, as already mentioned in Fig. 1. As isreasonable, the growth rates are same level for LHD and thetokamak when ωd is nonexistent. In the tokamak case, thecomparison with or without ωd shows that ωd is unfavorable
in the outer side of torus with θk ~ 0, while it becomes
favorable in the inner side as θk increases. On the other hand,in the LHD case, ωd is always unfavorable. This is because
the locally unfavorable curvature extends to the inner side of
torus in the LHD, which is reflecting the magnetic hill nature.

3. TEM cases
In Fig. 5 (left), we show the frequencies of TEM as afunction of k⊥ ρthi in LHD. A case for eliminating nonadiabatic ions is also plotted by open circles, and it can beω (x10-5 )ω (x10-5 )bad good

LHD
1

1

00-1

-1-2-20Fig. 4

θk /π 1

0.5

comparabletokamak

ωD ωD

00.5

θk /π 1θk dependence of ITG frequencies in LHD (left) and comparable tokamak (right), at s = 0.7. Real frequency and growth rate isshown by dashed and solid line with circles. Triangles also show cases that magnetic drift frequency ωd is ignored.

ω (x10-5 )ω (x10-5 )incl. only
nonadiabatic electrons

quasineutrality
150incl. nonadiabaticelectrons/ions
00Fig. 51incl. Debye shieldingk⊥ρthi000.5

1 k ⊥ρthek⊥ρthi dependence of TEM frequencies in LHD. Local parameters are (s,θk ,Mα) = (0.7,0,π). Closed circles show a case of nonadiabatic ions and electrons, and open circles show a case of elminating non-adiabatic ions. (Right) k⊥ρthe dependence of ETG
frequencies in LHD. Local parameters are (s,θk ,Mα) = (0.7,0,π). Olosed circles show a case for quasi-neutrality condition, andopen circles show a case for including Debye shielding effect. Solid (dashed) line shows the growth rate (real frequency) for
both figures.

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Yamagishi O. et al., Linear Local Stability of Electrostatic Drift Modes in Helical Systemsk ⊥ρthi (θ=θ )ω (x10-5 )k

2 1(a)

ω (x10-5 )ω (x10-5 )

(b)

(c)
11
1

φ/φ(θk)φ/φ(θk)Μα=0

Μα=π00-100Fig. 60.5S0 0100θ/π1-10.50θ/π1Mα/π 1000.5θk /π 1(a) Radial dependence, (b) field line (Mα) dependence, (c) θk dependence of TEM frequencies with nk = 86. Solid (dashed) line
shows the growth rate (real frequency). Two of local parameters (s,θk ,Mα) = (0.7,0,π) are fixed for each dependence plot. In (a),k⊥ρthi (θ = θk ) value is also plotted by dotted line on right coordinate.

seen that the modes can be unstable in the absence of thenon-adiabatic ions. This indicates that the electrons are
essential for the instabilities, and this and positive ωr indicate
the modes are the TEM.
As can be seen, the increase of k⊥ ρthi cannot stabilize
the modes, unlike the case of ITG. This is reasonable because
the ion FLR effect should not work for the electron-dominateddynamics. By comparing the cases with or without the nonadiabatic ions, the non-adiabatic ions are found to bestabilization, contrary to the role of the non-adiabatic
electrons for the ITG. The reason is now under consideration.
In Fig. 6(a), the radial dependence of TEM frequencies
is shown. Here the value of k⊥ ρthi (θ = θk) is also shown onright-Y axis. From the Fig. 5 (left), the mode is not stabilized
by the increase of k⊥ ρ thi unlike the ITG, so that thestabilization in the core cannot be explained by the high valueof k⊥ ρthi . Thus in this case, the reduction of trapped electron
fraction is responsible for this.
The θk and Mα dependences of frequencies are shownin Figs. 6(b) and 6(c). For both cases, the frequencies havevery weak dependence on these parameters. For the TEMcases, this is because the modes are very localized in a helicalripple, and the modes can find another ripple to localize whenthe θk or Mα is changed. This can be seen in the eigenfunction
structure as shown in Fig. 6(b). Here Mα is changed as π and0 which correspond to locally unfavorable and favorable
curvature for θ ~ θ k = 0. For the M α = 0 case, theeigenfunction is odd, and it can localize in the neighborhood
bad curvature region.

discussed in Sec. 2, and (ii) spatial scale of ETG can becomecomparable to the Debye length so that the Laplacian term in
the Poisson equation (Debye shielding term) is not negligible.
To investigate the latter, we show the ETG frequencies
as a function of k⊥ ρthe in Fig. 5 (right), in cases with orwithout the Debye shielding term. Here we consider only
electron species because it turned out that the bounce
frequency of ions is too small compared to the dynamics;
ω/ωbi >> 1 and a divergent problem often occurs numerically.
However, it seems not so wrong to ignore the ion
contributions because the ion FLR effect should alreadyeliminate the ion’s significance in this high wave numberregime.
It can be seen that the Debye shielding gives anadditional stabilization as k⊥ ρthe increases. On the other hand,
it is negligible for k⊥ ρthe <~ 0.3 and this indicates that thequasi-neutrality condition is well satisfied for the ITG or TEM
regime. Here we fix the bata value in these calculations,however, the Debye shielding effects also strongly affect thestability at the low beta (or low density), as can be understoodfrom the definition of the Debye length, λD2 = ε0Te /(ee2 ne).
The other properties of ETG are expected to be resemble tothe ITG.

5. Conclusions
As a first step of linear stability analysis of electrostatic
drift modes in LHD, the linear gyrokinetic-Poisson mode
equation is solved in a numerically obtained equilibrium witha fixed temperature and density profile, and the results arecompared to a circular tokamak with the comparable aspectratio and the same magnetic shear (safety factor) profile.
The ITG modes are found to be unstable for k⊥ ρthi >< 1.
The radial dependence of the ITG frequencies is considered
to be determined by k⊥ ρthi so that the modes are stabilized inthe plasma core with high k⊥ ρthi , and they are also stabilized
at the edge with low k⊥ ρthi . The comparison with a tokamakwith comparable aspect ratio shows that the ITG modes in
LHD have similar properties to those in the tokamak. The

4. ETG cases
The ETG modes can also be destabilized by electron
temperature gradient. The dynamics is almost the same as theITG, because two types of instabilities can be treated merely
by scale transformation, when ITG/ETG is considered to be
driven only by ions/electrons.
Possible changes between the ITG and ETG come from:(i) non-adiabatic electrons can contribute to the ITG as4225

Yamagishi O. et al., Linear Local Stability of Electrostatic Drift Modes in Helical Systems

reason is that the local (toroidal or helical) curvaturedestabilization is dominated by the slab-like parallel
compressibility because the negative q′ shear tends to reduce
the toroidal effects for the ballooning-like instabilities, andthe slab-like mechanism is common effect in the torus as wellas slab configurations. In addition, the large temperature
gradient η = 5 also makes the instabilities very strong. Thus
both the negative q′ and large η assumed in this study are
responsible for the similarity of ITG properties in LHD andthe tokamak.
The TEM can also be destabilized for k⊥ ρthi >< 1. Unlike
the ITG, the eigenfunction of the TEM is strongly localized
in a helical ripple 2π /M, and thus the modes should bestrongly affected by the local magnetic structure.
Nevertheless, the local parameter (θk,α) dependence is foundto be weak. This is because the modes can find another helicalripple to localize when these parameters are changed. TheTEMs with a fixed toroidal mode number are stable in thecore because the trapped electron’s fraction becomes small
as in the tokamak. As a result, both the ITG and the TEM
tend to be stable in the core, although the stabilizing
mechanisms are different. It is noted that the TEMs areusually stabilized in the negative q′ tokamaks, so that theTEM results here are considered to be inherent to the LHD.

Its importance in the LHD experiments should be studied indetails.
The ETG can also be destabilized for for k⊥ ρthe >< 1. In
addition to the electron FLR effects, the Debye shielding
effects can also gives stabilization for the ETG. Except forthe differences of the temporal and spatial scales associatedwith the mass ratio, the ETG would be rather resemble to theITG, at least for the linear electrostatic gyrokinetic analysis,
as can be checked by normalizing the gyrokinetic-Poisson
equations appropriately.

Acknowledgements
One of the authers (OY) thanks Dr. Yasuhiro Idomura
for fruitful discussions.

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[3] M. Kotschenreuther, G. Rewoldt and W.M. Tang,
Comput. Phys. Commun. 88, 128 (1995).
[4] S.P. Hirshman, Phys. Fluids 26, 3553 (1983).

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