J. Plasma Fusion Res. SERIES, VoL.2 (1999) 3942

Behavior of lnterchange Mode in Heliotron and Tokamak
WAKATANI Masahiro, ICHIGUCHI Katsujir, TATSUNO Tomoya,
FURUKAWA Masaru, CARRERAS Benjamin A.2and LYNCH Vicky E.2
Graduate School of Energy Science, Kyoto University
INational Institutefor Fusion Science
2oak Ridge National Laboratory(Received: 24 December 1998 / Accepted: 3 June 1999)

Abstract
When the pressure profile becomes flat locally at the resonant surface, the beta limit due to the idealinterchange mode increases, since the highly localized radial mode structure is suppressed. It is also
found that non-resonant ideal modes become unstable near the magnetic axis, when the pressure profile ishighly peaked in Heliotron E. However, the beta limit of non-resonant mode is higher than that ofresonant mode. On the other hand for negative shear tokamaks, resistive interchange modes aredestabilized, when 4s (4 value at magnetic axis) is larger than q.'n (minimum 4 value). Here the nonresonant ideal modes similar to those in Heliotron E also become unstable.

Keywords:
interchange mode, pressure profile, heliotron, negative shear tokamak

1. Properties of Interchange Modein Section 2.

One well-known property of the localizedinterchange mode is that the growth rate becomes

General properties of interchange modes wereknown for tokamaks [1] and stellarators [2]. Particularly
for heliotrons with large rotational transform and highextremely small near the marginal state, which is shownas lr'o< exp [-const/(0(0) lB"Q) - 1)r/2], where B"(0) is
the beta limit of relevant ideal interchange mode [6].
The associated radial mode structure becomes highlylocalized at the resonant surface even for low mode
number interchange instabilities. This fact means thatshear, the pressure-driven interchange modes are
dangerous [3]. In Heliotron E the origin of the observed

sawtooth oscillations is considered as the resistive or
ideal interchange mode [4]. For explaining the crash
phase of soft-X ray fluctuation, the ideal interchange
mode seems relevant [5].

the stability condition called Mercier-criterion fortoroidal plasmas or Suydam criterion for cylindrical

For the theoretical study of interchange modes
smooth pressure profiles decreasing toward to edge areusually assumed. In Heliotron E with a wide magnetic
hill region, resistive interchange modes are unstableeven for extremely low beta plasmas and these modes
may affect the pressure profile to flatten at the resonantsurface.Thus we consider a stair-like pressure profile orplasmas is applicable to the lowest mode number casewith m = lln = 1, where m(n) is a poloidal (toroidal)
mode number.

For studying the stability of interchange mode in asheared magnetic field, usually resonant modes are
considered dangerous. However, this is not valid when aa profile with a locally flat region for the stability study

Corre sponding author's e-mail : wakatani@ ene rgy. kyoto-u.ac.j p39@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchWakatani M. et al., Behavior of Interchange Mode in Heliotron and Tokamak

here. Even for the case of W = 0.01 the beta limit
increases and the radially localized mode structure
disappears for the nearly marginal state. Also it is notedthat this tendency has been seen for the toroidal model

pressure profile is highly peaked or a low shear region

exists. Several examples are shown in Section 3.

2. Interchange Modes for Locally Flat
Pressure Profile

171.

When the pressure gradient vanishes at the mode
resonant surface, the properties of interchange modechange and the beta limit increases. For the locally flatpressure profile given by PlPo- 7 - rz + 2 (r- r,) exp

In a realistic situation, the flattering of pressure
profile may occur at many resonant surfaces, when theresistive interchange modes are unstable in the wide
region. For a cylindrical model of Heliotron E, a stairlike pressure profile has been obtained to improve thel- (r - r")2/2W21 (see Fig. 1), the growth rate of m = 2ln= I mode is plotted as a function of P(0) = Po l(Bzl2ltn)
in Fig. 2, where L= _-dPldrl,=r. and r, is the resonant

beta limit several times by considering resonant surfaces

forn<4[8].

surface. Here both pressure and rotational transformprofiles are fixed, when B(0) is increased. The current
effect is negligible, since B(0) is low. The
eigenfunctions for W = 0 and lV = 0.01 are also shown3. Non-resonant Modes in Heliotron andTokamak
First we found that non-resonant resistive modes
become unstable in the central region for a highly
peaked pressure profile case in Heliotron E [9]. Recently
we also found that the non-resonant ideal modes becomeooo-unstable in the similar situation [5]. For toroidal
currentless plasmas of Heliotron E, the rotationaltransform at the magnetic axis increases from 0.47 at

No Step''\0.8

w=0.01 ---... w=0.1 .---

0(O) = O to the values larger than 0.5, when F(0) > | 7ofor a highly peaked pressure profile [5]. The property ofthese non-resonant modes is similar to that of theinfernal mode in low shear tokamaks [10].

0.6o,U'oo0.4(L0.2

It is shown that the resistive interchange modes
I I,l2]. It

become unstable in negative shear tokamaks

oo.2 0.4 0.6 0.8

is noted that the localized ideal interchange modes arestable in tokamaks when 4 > I in the whole plasma
region []. In order to destabilize the localized resistive
interchange modes the condition, eo) e^in, is necessary;

1Normalized RadiusFig. 1 Locally flat pressure profiles for W = 0.01 and W= 0.1. As a reference W= 0 case is also plotted.

9e-02
8e-02
7e-O2g(!

6e-02

arE

5e-02

E4e-02o3e-02eNo Step+No Step

W=0.01 -e-W=0.1 -o--W=0.01 ----"-2e-O2
1e-02

0e+00
Oe+OO

1e-03 2e-03 3e-03 4e-03 5e-03
pvalueFig.2Growthrateofm=2ln=lmodeisshownasafunctionofB(0)o

0.2 0.4 0.6 0.8

1Normalized Radius

foracylindrical plasmawithrotational profileof{rl = O.4 + O.2 12 in the left hand side figure. Eigenfunctions lor W = 0 and W= 0.01 at nearly marginal state areplotted in the right hand side figure.

40

Wakatani M. et al., Behavior of Interchange Mode in Heliotron and Tokamak

n=1'-'- m=l- -' m=2
m=3cl"""" m4--"- m=5gn
Fig. 3 Growth rates for interchange modes with 1 < n <8 in a negative shear tokamak. For n = 3 and n =5. only the resistive interchange mode becomes
unstable and growth rate for S= 107 is shown. ForFig. 4 Radial mode structure of n = 1 non-resonant idealmode in a negative shear tokamak. The growthrate is shown in Fig.3. Here Ydenotes a poloidal
flux function.

n = 1,2,7 and 8, the non-resonant ideal mode
becomes unstable.
however, by deforming the outermost flux surface toelliptic or dee-shape, the stability is improved and BN 2 is possible for fairly broad pressure profiles, where B1q
is a normalized beta given by B^ = B(a(m)B(T)/le(MA)). Here a is a radius, B is a toroidal field and /o is a
plasma current. By applying the RESORM code to anegative shear tokamak with a circular cross-section,
low-n resistive modes are shown unstable. Figure 3
shows the growth rate of unstable mode for different
toroidal mode number with 1 < n < 8 [12]. lt is notedthat n = 3 and n = 5 cases for S = 107 shown withsquares belong to the resistive interchange modes;
however, other cases with n = 1,2 and n = 7,8 are nonresonant ideal modes. Here S is a magnetic Reynolds
number. For the cases in Fig. 3, Mercier modes arestable and the unstable non-resonant modes have
similarity to the infernal modes. The radial mode
structure of the n = I non-resonant mode is shown in
Fig. 4. Here 4-profile is almost flat with 4o = 2.8 near
the magnetic axis and the (m, n) = (3, l) mode gives theminimum of l4o - mlnl. lt is quite similar to thatobtained for the Heliotron E [2].

pressure profile may be obtained at the saturated state

4. Nonlinear Interchange Mode
When we include both ideal and resistive

interchange mode may explain the sawtooth observed in[8]. For this profile the beta-limit due to the linear idealinterchange will be higher than the Mercier or Suydam

limit for a standard smooth pressure profile [7]. On theother hands, when the source term or heating term isincluded in the pressure evolution equation, a relaxation
oscillation similar to the sawtooth appears for thedominantmode with m=lln = I [4] or m=2/n= ll5l
in the case of Heliotron E. When the magnetic axis isshifted inward and the pressure profile is highly peaked,
the m = 2ln = | resonant interchange mode is easily
destabilized. Nonlinear evolution of this mode seems
consistent with the sawtooth observed by the soft-X ray
tomography.

Non-linear evolution was also studied for the nonresonant mode with m = 2/n = I [5]. When the mode
amplitude becomes finite, the rotational transformchanges and the mode resonant surface of I = 0.5 for them = 2ln = I mode appears and the reconnection ofmagnetic field line occurs in the nonlinear stage. Thus
the global nonlinear behavior of the non-resonant mode
looks similar to that of the resonant ideal interchange
mode. Thus the m = 2ln = | resonant or non-resonant
the Heliotron E.

interchange modes in the nonlinear evolution ofHeliotron E plasma, the flattening of pressure profile
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41

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