22F08

Gyro-kinetic Simulation of Ion Temperature Gradient driven
Drift Wave Instability in the presence of a Magnetic Island
Paul P. Hilscher, Kenji Imadera, Jiquan Li, Kota Takaki and Yasuaki Kishimoto
Graduate School of Energy Science
Department of Fundamental Energy Science
Kyoto University, Gokasho, Uji, Kyoto 611-0011 JapanAbstract
The ion temperature gradient (ITG) driven instability in the presence of a meso-scale magnetic island usinga gyro-kinetic simulation code is investigated. We find that a magnetic island causes radial and poloidal modes
to couple with each other, thus playing a stabilizing role by allowing energy transfer to stable modes. For larger
island sizes, new rational surfaces appear, which again may excite destabilization. Furthermore, we study the effectof kinetic electrons and their influence on the ETG-ITG micro-instability growth in presence of a magnetic island.

1 Introduction

form
∂F1σ
∂t = ky

During a Tokamak discharge, MHD instabilities takeplace, which result in a magnetic island. The saturated
magnetic island can span several ion gyro-radii as discovered by Isayama[3]. In the same time, a temperaturegradient develops and the plasma becomes unstable todrift-wave instabilities, like the ITG mode. The ITG
mode happens on a much faster time scale and thus themagnetic island can be assumed to be stationary during
the time the ITG instability develops.
ITG turbulence on a magnetic island was first studied
by Wang[4], who discovered using a gyro-fluid simulation code, that for a smaller island size, the island
has a stabilizing role due to mode-mode coupling effects, above some threshold, the destabilizing effect due
to the appearance of additional rational surfaces dominates. Here, we will study the inclusion of a magnetic
island within the gyro-kinetic framework.

1 + ησ 22vkk⊥
12 − 2 − 2hφi F0+ vkσ ŝxky (F1σ + hφi F0σ )+ vkσ A1k , F1σ − vkσ A1k , hφi F0σ ,√where vkσ = vk 2miσ is the parallel velocity term,miσ ion-species mass ratio, φ the electric potential
2
and the Maxwellian F0σ = √12π e−vk /vkσ , whileF1σ (x, ky , vk ) is its deviation, σ = i, e, is the ion andelectron species and the brackets hAi denotes the gyroaveraging procedure. The last two terms in the Vlasov
equation appear due to the magnetic island for whom
we assume an analytic approximation,
A1k = −w2 ŝ
2πcos(ζx) cosh ( Li y)y32,

where ζ = 0.1, ŝ = 0.4 and w is the assumed magneticisland width. The Poisson equation reads
Z ∞2
qσ hF1σ i dvk = 1 − Γ0 (k⊥) + Ad. φ ,Σσ2 Calculation Modelv=−∞

For the numerical investigation of micro-turbulence,
we employ a self-developed gyro-kinetic code, seeHilscher[1] and [2]. We perform local, sheared slab
geometry assuming a single rational surface. The normalized, linearized gyro-kinetic equations are a coupledintegro-differential equation system of the Vlasov equation and Poisson equation. The Vlasov equation has the2with k⊥= kx2 + ky2 . The last term Ad. = 1 is theadiabatic response term of the electrons if included;
Ad. = 0 otherwise. A static island (∂t A1k = 0) is
assumed, this is valid, as the evolution of the magnetic island is in general much slower than the microturbulence timescale.

160505040403030202010100−30−20−100102030

00.10

−30

−20−10010

2030

Figure 1: Contour plot of φ potential for an island widthw = 0 and w = 14 and ηi = 4.2. The island has asignificant influence on the ITG mode structure due toinduced mode-mode coupling.
0.090´i =3:9´i =4:4

0.06

0.04

0.02

0.00 -1
10100

Poloidal Wavenumber ky

0.075
0.070
0.0650.060
0.055

0.080
0.075
0.070
0.065
0.060
0.055

51015

Magnetic Island width w

Figure 3: Growthrates for poloidal modes for magneticisland widths of w = 0, 5, 10, 20. Mode-Mode coupling results in equal growth of all poloidal modes anddissipates energy by stable modes.

0.085

0.080

Growtrates °[Ln =w ⋆ ]Growtrates °[Ln =w ⋆ ]

w=10
w=20

0.090
´i =4:2

0.085

0.050
0

w= 0
w= 5

0.08

Growthrate °60200.0500´i =4:25´i =3:9´i =4:41015Magnetic Island width w

20

Figure 2: Growthrates of the systems total field energy,
for first order gyro-kinetics (left) and (full order) gyrokinetics (right) simulation.

3 Results and Summary
A contour plot of the φ-potential with the corresponding magnetic island is shown in Fig.1. The mode-mode
coupling due to the magnetic island results in equal
growth of the poloidal modes as shown in Fig.3. The
coupling transfers energy to stable modes and thus total
instability growth is reduced. On the other hand, thistransfer also excites low-k modes, which may increase
the total heat flux. Fig.2 shows the system total growth
in respect of the magnetic island size. The island has astabilizing effect; destabilization due to the appearanceof new rational surfaces can be observed for an island
width of w = 14, but overall stabilization effects are
dominating with larger island sizes - an effect due tostronger coupling of the modes, which leads to more
efficient dissipation through FLR (finite Larmor radius)
effects. We also compared two p
gyro-kinetic models, full2 )A ; and only
gyro-kinetic effects hAi = J0 ( 2µρtσ k⊥modeled to first order (a common approach for gyro2)A. We note thatfluid models) hAi = exp (− 12 ρ2tσ k⊥
for the first-order model instability is reduced due tothe overestimation of the FLR damping.
As a next step we will present recent simulation results, where we investigate the linear instability growth,

in respect to the island’s size, including a kinetic electron species, as well as the influence of the island onheat transport.
Acknowledgments
The author (P.H.) is grateful for the support of the Ministry of Education, Culture, Sports, Science and Technology of Japan via ”Energy Science in the Age ofGlobal Warming” of Global Center of Excellence (GCOE) program (J-051).

References[1] P. P. Hilscher, K. Imadera, J.Q. Li, and Y. Kishimoto. Towards gyrokinetic simulations of multiscale micro-turbulence in Tokamaks. 16, June
2010.
[2] P. P. Hilscher, K. Imadera, J.Q. Li, and Y. Kishimoto. Gyro-kinetic Simulation of Ion Temperature
Gradient driven Drift Wave Instability in the presence of a Magnetic Island. 16, June 2011.
[3] A. Isayama, Y. Kamada, T. Ozeki, and N. Isei. Measurement of magnetic island width in long-pulse,
high-beta discharges in jt-60u. Plasma Physics andControlled Fusion.
[4] Z. X. Wang, J. Q. Li, Y. Kishimoto, and J. Q Dong.
Magnetic-island-induced ion temperature gradientmode. Physics of Plasmas, 16(6), June 2009.