Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)589–592
000–000
J. Plasma Fusion

Effects of Self-Consistent Flow on Island Generation inInterchange Mode
ICHIGUCHI Katsuji and CARRERAS Benjamin A.1
National Institute for Fusion Science, Toki 509-5292, Japan
1Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA(Received: 9 December 2003 / Accepted: 23 April 2004)

Abstract
The time evolution of magnetic island in the nonlinear phase of the resistive interchange mode is examined in thecylindrical geometry. The effects of self-consistent uniform poloidal flow is taken into account. In this case, the reconnection of the magnetic field line occurs both at the saturation of the dominant unstable mode and at the saturation of
the n = 0 mode, where n is the toroidal mode number, through the curvature change of perturbed poloidal flux surface.
Keywords:
magnetohydrodynamics (MHD), resistive interchange mode, magnetic island, driven reconnection

1. Introduction

expressed as

The interchange mode is a dangerous magnetohydrodynamic (MHD) instability in zero-current stellarators. In thelinear analysis, the eigenfunction for the poloidal magnetic
flux corresponding to the maximum growth rate is an oddfunction in the radial coordinate. Therefore, the flux is close
to zero at the resonant surface even with finite resistivity.
This means that magnetic islands are hardly generated spontaneously in the linear phase, not like the tearing mode. On
the other hand, the numerical study for the Large HelicalDevice (LHD) plasma in the toroidal geometry showed thatthe magnetic islands can be generated in the nonlinear saturation phase of the interchange mode evolution [1]. The number of the island on a poloidal cross section is twice of thepoloidal mode number of the driving interchange mode.
Similar islands were obtained in the nonlinear calculationsfor Heliotron E plasmas [2,3]. Thus, we consider the mechanism of the island generation in the nonlinear saturation of
the interchange mode. In the present study, we also take theeffect of the self-consistent poloidal uniform flow into
account by including both parity of sine and cosine in theFourier expansion for each perturbation.

∂ψβ= −B·∇Φ + S1 Jz, dU= −B·∇Jz + 2ε02 ∇Ω ×dt∂t2∇P·ez + ν∇⊥2U, and dPdt = κ⊥∇⊥P. The convective time deriva-d∂ + v · ∇ where v = ∇Φ × e . The
tive is given by dt= ∂t⊥⊥zmagnetic field B is given by B = B0 ez + ez × ∇ψ, where ezmeans the unit vector in z direction. The subscripts of ’0’ and’⊥’ mean the value at the magnetic axis and the vector component perpendicular to z direction, respectively. The factorof ε ≡ a/R0 is the ratio of the minor and the major radii of thecorresponding torus, and β0 denotes the beta value at theaxis. The time is normalized by the poloidal Alfvén time, τA.
The vorticity U and the toroidal current density Jz are given
by U = ∇⊥2Φ and Jz = ∇⊥2ψ, respectively. The vector of ∇Ωdenotes the average curvature of the field line which drives
the interchange mode with the pressure. The perturbation∼X (r, θ, z) are expanded in the Fourier series in the way of∼c∼s∼
X = ∑mnX mn (r) cos (mθ − nζ ) + ∑mnX mn (r) sin (mθ − nζ ),where X denotes ψ, Φ or P and ζ = z/R0. The component of∼cΦ 00 corresponds to the poloidal uniform flow.
In the present study, we employ a straight equilibrium
corresponding to the LHD configuration with the vacuum
magnetic axis located at Rax = 3.6m [1]. The equilibrium iscalculated with no net current and the pressure profile of P =P0(1 − r4)2 with β0 = 2%. In this equilibrium, the rational surface with ι = 1/2 is included in the plasma column. Thus, weconsider the single helicity perturbation with m/n = 2/1. The
modes of 0 ≤ n ≤ 7 and 768 radial grids are used in thenumerical calculation. In the nonlinear calculation, weemploy a fairly large magnetic Reynolds number, S = 104, so

2. Basic equations and configurations
We study the island generation by using the NORM
code [1]. This code solves the nonlinear reduced MHD equations for poloidal magnetic flux ψ, stream function Φ andplasma pressure P. In order to examine the generation mechanism of the magnetic island precisely, we study the nonlinear saturation of the interchange mode in the cylindricalgeometry (r, θ, z). In this case, the reduced equations are©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: ichiguch@nifs.ac.jp1589

Ichiguchi K. et al., Effects of Self-Consistent Flow on Island Generation in Interchange Modeas to make the feature of the magnetic island prominent. We
also choose the fluid viscosity of ν = 10-3 and perpendicularheat conductivity κ⊥ = 10-4 so that the (m, n) = (2, 1) component should have the largest linear growth rate.

(a)

3. Time evolution of island structure
Figure 1 shows the time evolution of the kinetic energyof the perturbation, which is defined by Ek = ∑n E kn, E kn = 12 ∫∼c∼s[|∇⊥∑mΦ mn cos(mθ − nζ )|2] + |∇⊥∑mΦ mn sin(mθ − nζ )|2]dV.
The dominant mode is (m, n) = (2, 1) mode in the whole time
evolution. All of the components with n ≥ 1 saturate at t =3,500τA, while the n = 0 mode grows quite slowly and saturates around t = 17,000τA.
Figure 2 shows the flow pattern on the poloidal crosssection at z = 0. We also plot the position of the resonant surface where the total rotational transform ι T equals to 1/2,(b)∼R0,which is defined by ι T (r, θ, z) = 1 ∂ [ψeq(r) + ψ (r, θ, z)] Br ∂r0

as well as the position of ι eq = 1/2. Here the tilde and the subscript of ‘eq’ mean the perturbed and the equilibrium quantities, respectively. In Fig. 2(a), four vortices are seen aroundthe resonant surface. This pattern corresponds to the typicallinear eigenfunction of the interchange mode with m = 2. Thesurface with ι T = 1/2 is deformed mainly by the radial component of the flow at the dominant mode saturation. This
flow pattern is almost kept until the n = 0 components
becomes comparable with the dominant mode. As the n = 0
component grows, the flow pattern varies. At t = 18,000τA,
the uniform poloidal shear flow is seen particularly inside theresonant surface in Fig. 2(b).
Since we treat the single helicity perturbation, the structure of the magnetic island can be observed by plotting thehelical magnetic flux ψ h, which is defined by ψ h(r, θ, z) =Fig. 2 Flow pattern at (a) t = 9,600τA and (b) t = 18,000τA for r≤ 0.8. Dotted and dashed lines show the position ofthe surfaces with ιT = 1/2 and ιeq = 1/2, respectively.

3,000τA in the linear phase, two thin islands are observed, as
shown in Fig. 3(a). The island structure is consistent with thelinear eigenfunction of the resistive interchange mode for thelarge resistivity. Therefore, these islands are considered to begenerated spontaneously. The X-points of the spontaneous
islands are maintained through the whole time evolution. We
call them major X-points here.
At the saturation of the dominant mode, the reconnection of the field line occurs and new X-points are generated
at the positions of the O-points of the spontaneous islands, as
shown in Fig. 3(b). These X-points are called minor Xpoints. The number of the island is increased to four. As
shown in Fig. 3(c), the minor X-points are annihilated when
the n = 0 mode is saturated. The number of the island isreduced to two. The shape of the resultant island is asymmetry with respect to the O-point and the whole island structureis rotated in the poloidal direction.

1 n B∼ψ (r, θ, z) + ψ eq(r) − ψ mn(r), ψ mn(r) = 2 r2 m R0 . Figure 30shows the contour of the helical flux on the poloidal crosssection at t = 3,000 τ A, 9,600 τ A and t = 18,000 τ A. At t =4. Mechanism of reconnection
The generation and the annihilation of the minor Xpoints are related to the radial component of the perturbed
magnetic field, Br. Figure 4 shows the profile of Br along theresonant surface in the poloidal cross section. The zero points
at θ = 0.13π and 1.13π correspond to the O-points at t =3,000τA, while they correspond to the minor X-points at t =Fig. 1 Time Evolution of the kinetic energy.

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Ichiguchi K. et al., Effects of Self-Consistent Flow on Island Generation in Interchange Mode

(a)

(b)

Fig. 4 Plots of Br along the resonant surface.

corresponds to θ = 0.13π in Fig. 5(a) and (b) and to θ = 0.5π
∼in Fig. 5(c). The surfaces of ψ = const., r = const. and ι T =1/2 are plotted which cross at θ = 0.2π for t = 3,000τA and t =9,600τA and at θ = 0.66π for t = 18,000τA. At t = 3,000τA, the∼radial curvature of the ψ = const. surface is smaller than 1/r .
This means that Br is negative at θ = 0.2π. At t = 9,600τA, as∼shown in Fig. 5(b), the radial curvature of the ψ = const. surface is larger than 1/r at the same position. This means that Bris positive. The enhancement of the curvature is attributed tothe outward flow of the vortex which is shown in Fig. 2(a).
When the n = 0 mode becomes comparable to the n = 1
mode, the position of the maximum outward velocity deviates from the minor X-point in the poloidal direction. At t =18,000τA, the maximum outward velocity position is locatedat θ = 0.66π as shown in Fig. 2(b), while the position of theoriginal minor X-points corresponds to θ = 0.5π. Therefore,∼the ψ = const. surface around θ = 0.66π is pushed out by theflow. On the other hand, the effect of the outward flow is∼small around θ = 0.33π. As a result, the structure of the ψ =const. surface becomes asymmetry with respect to the minor∼X-point as shown in Fig. 5(c). The curvature of the ψ =const. surface is decreased only for θ ≥ 0.5π to the value lessthan 1/r. Then, the direction of Br reverses and the island disappears at only one side of the minor X-point. The resultant
island has asymmetric structure with respect to the O-point asshown in Fig. 3(c).

(c)Fig. 3 Contour of ψh at (a) t = 3,000τA, (b) t = 9,600τA and (c) t= 18,000τA for r ≤ 0.8. Dotted and dashed lines show
the position of the surfaces with ιT = 1/2 and ιeq = 1/2,respectively.

9,600τA. At θ = 0.2π and 1.2π in the vicinity of the minor Xpoints, Br is negative at t = 3,000τA while it is positive at t =9,600τA. At t = 18,000τA, Br becomes negative again at θ =0.66π and θ = 1.66π in the vicinity of the positions of θ =0.5π and θ = 1.5π where the minor X-points are annihilated.
Therefore, the direction of Br reverses in the generation andthe annihilation of the minor X-points.
The change of the sign of Br can be explained with thegeometrical structure of the perturbed poloidal flux surface∼
because the perturbed magnetic field is tangential to the ψcontour. Figure 5 shows the schematic picture of the contour∼of ψ in the poloidal cross section. The horizontal direction

5. Conclusions
The magnetic islands can be generated and annihilated
in the nonlinear saturation of the interchange mode.
In the saturation of the dominant unstable mode, theradial flow due to the vortex can change the O-point to theminor X-point. Then, the original island is divided into twoislands. In this case, the curvature enhancement of the perturbed flux surface is essential in the minor X-point generation. This mechanism is different from the standard driven
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Ichiguchi K. et al., Effects of Self-Consistent Flow on Island Generation in Interchange Modereconnection because the current concentration at the X-point
does not occur.
The poloidal flow generated self-consistently in the saturation of n = 0 mode has shear structure. This flow annihilates the minor X-points and generates asymmetric islands.

Acknowledgments
This work was partially supported by the Grant-in-Aidfor Scientific Research (C) of the Japan Society for thePromotion of Science.

References~Fig. 5 Schematic pictures of ψ at (a) t = 3,000 τ A, (b) t =9,600τA and (c) t = 18,000τA.

[1] K. Ichiguchi et al., Nucl. Fusion 43, 1101(2003).
[2] M. Wakatani et al., Nucl. Fusion 24, 1407 (1984).
[3] B.A. Carreras et al., Phys. Plasmas 10, 3700 (1998).

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