J. Plasma Fusion Res. SERIES, Vol. 5 (2002) 491494

Tearing Mode in a Heliotron Plasma
ICHIGUCHI Katsuji
National Institute for Fusion Science, Toki 509-5292, Japan(Received: ll December 2001 / Accepted: 1 May 2002)

Abstract
Tearing mode is studied in a heliotron plasma by varying the net toroidal current. Ideal internal kinkmode can be unstable when two resonant magnetic surfaces are generated. The tearing mode is found at
the current below the marginal stability value for the ideal internal kink mode. The mode structure islocalized around the inner resonant surface like the internal kink mode. The growth rate and the mode
structure can be affected by the unstable internal kink mode. Tearing and resistive interchange hybrid
mode is also obtained at a finite beta.

Keywords:
tearing mode, heliotron, intemal kink mode, resistive interchange mode, reduced MHD equations

1. Introduction

straight stellarator configurations [3,9]. However, it isdifficult to apply it to the 2D or 3D configuration. In thestudy of the tearing mode in the toroidal geometry, theglobal mode analysis is useful. Thus, in the present

Heliotron configurations have an advantage that nonet toroidal current is necessary for generating theconfinement magnetic configuration. However,substantial net toroidal currents are observed in theLarge Helical Device (LHD) experiments [1]. Suchcurrents have a potential to drive current driven modes.
Several theoretical analysis for the ideal internal andexternal kink modes in the stellarator plasmas have been
carried out [2-6]. On the other hand, in the helias
configuration, the m = 2 and n = | tearing mode wasidentified in the W-7 AS experiment recently [7]. Herework, the method based on the reduced MHD equations
is employed. The mode structure of the tearing mode,
the relation with the ideal mode and the finite beta
effects are discussed.

2. Numerical Method
In order to examine the global structure of thetearing mode, we utilize the RESORM code [10], whichm and n are the poloidal and the toroidal mode numbers,

is a linear stability code for ideal and resistive

respectively. Thus, it is interesting to know how thetearing mode behaves in the heliotron plasma. In thepresent paper, we study the feature of the tearing modein one of the LHD configurations numerically.
The stability of the tearing mode is often studied
based on the so-called A' analysis [8]. Here A' is thejump of the logarithmic derivative of the outer solution
for the poloictal flux at the resonant surface. Thismethod is uset'ul in the determination of the stability
boundary in a parameter scan and was used in theinstabilities in the heliotron plasmas. This code solves
the linearized equations of the following reduced MHD
equations for the poloidal flux Y, the stream function @and the plasma pressure P as an initial value problem;#='(*)'" 'YeD+41,'o^ff=-n .Yl, +n'oYdlxYr.Y( ,(l)A)@2002 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author's e-mail: ichiguch@ nifs.acjp
491

Ichiguchi K., Tearing Mode in a Heliotron PlasmadPdt=0,(3)

where ( R, p. and 4 are the toroidal angle, the major
radius of the torus, the plasma density and the resistivity, respectively. Here O is the averaged magnetic curvature, of which the explicit expression is given in ref.
[10]. The subscript 0 means the value at the magneticaxis. The magnetic differential operator and the convective time derivative are given by B . V = (RoBolR\(ala0-VY xY(.Y anddldt = @laD + (R/R0)2VY x
V 6 . V, respectively. The toroidal current density isIdH---- I=rr0kA---"-- - I=r20kA

aI=130kA
I=150kA

rL-r

0.5

-0

0.5

given by J€ = R2V.(Vr/R2)Y.

1

pp

The Fourier transform is employed in the angle
directions in the code, and the toroidally averaged 3D
equilibrium which is calculated by the VMEC code [11]is examined. Therefore, the toroidal mode number is
specified as a parameter, while the poloidal mode
coupling appears through the convolution with theaveraged equilibrium quantities.

Fig. 1 Profiles of the rotational transform for the various

net toroidal currents. The radius po denotes thesquare root of the normalized equilibrium poloidal
flux, which is also used in Figs. 3, 5 and 6.

sinoleres6nonce

doublgresononce

noresononce

0.010

3. Numerical Results
We examine the tearing mode in one of the LHD

configurations. The helical coils in the LHD arecomposed of three layers for providing the flexibility inthe minor radius of the helical coils. Here, we choose
the configuration with the coil current flowing only in0.005
sloblethe outerJayer. For the net toroidal current in the plasma
column, highly peaked current density, J = Jo (1 - p)",

100

120

140

160

r (kA)

is assumed, where p denotes the square root of thenormalized toroidal flux. Figure 1 shows the profile ofFig. 2 Growth rate of the n = 1 ideal internal kink mode.

the rotational transform t at Bo = 0 Vo. There are twosurfaces with r = ll2 in the plasma column for the total
toroidal current in the range of 110 kA < 1 < 150 kA,
where the toroidal field at the coil center is assumed tobe Bs - 3 T. Only one r = l/2 surface exists for 1 < 100(o)kA and no t = 1/2 surface for 1 > 160 kA. We
concentrate on the n = I mode which is resonant at t =7-2m=2ell2 in the present paper.
At first. the ideal mode is examined with theRESORM code. Unstable modes are obtained for the---- m=3-4range of 1 corresponding to the existence of the two t =1/2 surfaces as shown in Fig. 2, where the growth rate iso.5

0.5

PpPpFig. 3 Mode structures of (a) stream function and (b)poloidal flux of the n = 1 ideal internal kink mode
for / = 130 kA. The arrows indicate the positions ofplotted as a function of 1. Here 7is normalized by thepoloidal Alfv6n time. Figure 3 shows the mode structureof the eigenfunctions for 1 = 130 kA as an example. Them = 2 andn = I component is dominant and the modestructure is localized in the region between the magnetic

r = 112 surfaces, which are also used in Figs. 5and 6.

resonant surface. The mode structure of the unstablemodes shows the feature of the ideal internal kink mode.
The finite resistivity influences the stability. Figure

axis and the inner resonant surface. The stream function

is an even function and has a sharp gradient at thesurface. The poloidal flux has a null point at the492

Ichiguchi K., Tearing Mode in a Heliotron Plasma

At Bo = O.32 Vo, a tearing and resistive interchange

4 shows the S-,lependence of the growth rate, where S is
the magnetic F:eynolds number and proportional to llr1.

hybrid mode is obtained. The mode structure is shownin Fig. 6. In both of the stream function and the poloidal
flux, the typical structures of the tearing mode and theAn unstable m,)de appears at 1= 110 kA where the ideal
mode is stabl,:. This mode is identified as a tearing
mode with the following feature. The S-dependence ofthe growth rate calculated with the values for S = 108-1

and S = lOe scales /€ 5-0'54. This dependence is close too- --- -- ---o----- ----o---- ----o--- ------ol=lsOkA

that for the tearing mode derived with the A'theory,
S-vs [8]. Figurr: 5 shows the mode structure for / = 110
kA and S = 107. The m = 2 component is dominant andthe mode structure is localized near the inner t = l/2
surface as is in the ideal internal kink mode. The stream
function is odcl with respect to the resonant surface andthe poloidal llux has a finite value at the resonantsurface which corresponds to the generation of themagnetic island. These are the typical characteristic ofa-.. -o'-" -----A.-....-.-a....-....ao""---'-o'-"-2 o"--:.:- --.-A....-...
-'g-. -'---A----^ ""o\ \-"o!

.9 -3"o"'-"'.\.._

l=l3OkA"f,-....._-

l=l20kA'o-4

l=l lOkA

56789

the tearing mo,le.

The devi;rtion from the S-3/5 dependence in thesmall ,S region seen in Fig. 4 is due to the cylindrical

lo916 S
Fig. 4 S-Dependence of the growth rate of the n = 1 re-geometry effe:t. In the A' theory, the inner region isexpanded around the resonant surface for the asymptotic

sistive mode. Solid line shows the dependence
determined with /(S = 108) and 7(S = 1O'g), whichcorresponds to y* S:054.

matching so that the geometry should be slab. However,if the mode strlcture is localized near the magnetic axis,
the distance from the axis can be comparable with theresistive layer width for small S. In this case, neglecting
the cylindrical geometry is not a good approximation. In
the present carie, as is shown in Fig. 5, the position of
the resonant srrface is close to the axis. Therefore, theo)" "' m=l
m=2deviation from the dependence of S-3/5 becomes>.

---- m=3

prominent as S decreases.

As shown in Fig. 4, the S-dependence is varied
when the cur:ent is increased. This is because thetearing mode iri affected by the ideal internal kink mode.
As the current increases and y of the internal kink modeincreases, the clependence becomes weak. This tendency
is quite similar to that in the case of the ideal and theresistive interr:hange modes [2]. The increase of thecurrent also influences the mode structure. The stream
function for I >-: l2O kA is the even function with respect
to the inner t = l/2 surface like the ideal internal kinkmode. On the other hand, the poloidal flux at the surface

0.5

0.5

Pp

Pp

Fig. 5 Mode structures of (a) stream function and (b)poloidal flux of the n = 1 tearing mode with S =107 tor

l= 110 kA.

(o)""'m=lis finite like the tearing mode. This implies that themode structuro shows the feature of both the internalkink and the tearing modes. Hence, this mode has an_ m=2-'- m=3E

island with the flow like the intemal kink mode.
The behavior of the tearing mode at finite beta isalso examined for 1 = I l0 kA and S = lOi. The pressure
profile of P = /"0(1 - po )t is assumed. The tearing mode

0.5

0.5

Pp

Pp

Fis. 6 Mode structures of (a) stream function and (b)poloidal flux of the n = 1 tearing-interchange hybrid mode with S = 1Oj for I = 110 kA at &= O.32

is dominant for B6 < 0.32 7o, and the resistiveo/interchange m,rde becomes dominant for pe > 0.32 Vo.
493

Ichiguchi K., Tearing Mode in a Heliotron Plasma

tearing mode is obtained at a current value which is justresistive interchange mode are seen around the inner andthe outer t = l/2 surfaces, respectively. The side bands
due to the toroidicity are relatively large in the poloidal

below the marginal value for the internal kink mode.
Hence, the mode structure is localized around the innerresonant surface as is in the internal kink mode.
Because the configuration with single resonant surface
or no resonant surface case is quite stable for the idealmode, such configuration cannot be easily destabilizedwith the finite resistivity. Therefore, the configuration
with two resonant surfaces may be necessary also forthe destabilization of the tearine mode in heliotron

flux structure.

4. Summary and Discussion
An unstable tearing mode with m = 2 and n = 1 isfound in a heliotron plasma with two t = ll2 surfaces
for the toroidal current below the marginal stability
value for the ideal internal kink mode. The mode
structure is localized around the inner resonant surface
as is in the internal kink mode case, not like a double
tearing mode. In the region for the unstable internal kinkmode, the growth rate and the mode structure is affectedby the ideal mode. No unstable tearing mode can be

configurations.

Acknowledgement
This work was partially supported by the Ministryof Education, Culture, Sports, Science and Technologyof Japan, Grant-in-Aid for Scientific Research (C),obtained for the current corresponding to a single
resonant surface or no resonant surface in the present
configuration.

200t.

The properties of the tearing mode structure

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