Plasma and Fusion Research: Rapid Communications

Volume 9, 1203020 (2014)

Kinetic Ballooning Mode Turbulence Simulation based on
Electromagnetic Gyrokinetics∗)
Shinya MAEYAMA, Akihiro ISHIZAWA1) , Tomohiko WATANABE1) , Motoki NAKATA,
Naoaki MIYATO and Yasuhiro IDOMURA
1)Japan Atomic Energy Agency, Rokkasho 039-3212, Japan
National Institute for Fusion Science, Toki 509-5292, Japan(Received 20 January 2014 / Accepted 30 January 2014)

The kinetic ballooning mode (KBM) turbulence in Tokamak plasma is investigated by electromagnetic gyrokinetic simulations. From the entropy balance analysis, it is revealed that the field-particle interactions transfer
a significant fraction of the ion entropy produced by the instability to electrons. Then, the produced ion entropy
balances to the sum of the ion and electron dissipations at the saturation of the KBM instability growth, in contrast
to ITG turbulence where ion entropy production mostly balances to ion dissipation.
c 2014 The Japan Society of Plasma Science and Nuclear Fusion Research

Keywords: gyrokinetics, Vlasov simulation, kinetic ballooning mode
DOI: 10.1585/pfr.9.1203020
In the next generation fusion devices, such as JT-60SAand DEMO, high-β operations are planned to make the design of a nuclear fusion reactor more realistic and attractive (where β ≡ μ0 nT/B2 is the normalized plasma pressure). A finite β eﬀect introduces coupling of drift andAlfvén waves in magnetized plasma, and changes the nature of plasma turbulence from electrostatic to electromagnetic, where stabilization of the ion temperature gradientmodes (ITG) and destabilization of the kinetic ballooning
modes (KBM) are observed as β increases [1]. Since the finite β eﬀect may have large impacts on turbulent transportlevels, understanding of the electromagnetic turbulence iscritically important for high-β operations.
In this paper, we have carried out KBM turbulencesimulations by means of the flux-tube gyrokinetic code
GKV [2, 3]. Plasma parameters are set to be the so-called
Cyclone base case parameters without the electron temperature gradient (R/Ln = 2.2, R/LTi = 6.82, R/LTe = 0,
T i /T e = 1, q = 1.4, ŝ = 0.78, r/R = 0.18 and β = 2%). We
have analyzed the saturation mechanism of KBM instability growth from the viewpoints of the entropy balance
equation [4]. The equation describes the time evolution ofthe perturbed gyrocenter entropy S sk for ions and electrons(s = i, e) and electromagnetic field energy W k in the perpendicular wave number space k,dS sk= Θsk + Dsk + Isk + Esk + Rsk ,(1)dt
dW k=−Rsk ,(2)dts where, Θsk , Dsk and Isk = Re[ dv3 k k δ k +k +k=0 b ·author’s e-mail: maeyama.shinya@jaea.go.jp∗) This article is based on the invited talk at the 30th JSPF Annual Meeting(2013, Tokyo).

Fig. 1 Snapshot of electrostatic potentials of KBM in (a) physical space and (b) field-aligned coordinates of −π ≤ θ ≤ π.

k × k (χ∗sk g∗sk − χ∗sk g∗sk )g∗sk T s /(2BFsM )] denote theentropy production due to the particle and heat transport, the collisional dissipation and the nonlinear transfer, respectively (where gsk = fsk + es FsM J0sk φ k /T sand χsk = J0sk (φ k − v A k ) with the symbols sameas Ref.
The parallel streaming term Esk = [3]).
−∇ dv3 v |gsk |2 T s /(2FsM ) represents the entropy transfer between diﬀerent radial wave number components dueto the periodic boundary condition for twisting mode structures in the sheared magnetic geometry [5]. As shownin Fig. 1, the electrostatic potential perturbations havelarge amplitudes in the bad curvature region (near at the1203020-1

c 2014 The Japan Society of Plasma
Science and Nuclear Fusion ResearchPlasma and Fusion Research: Rapid Communications

Volume 9, 1203020 (2014)

Fig. 3 Schematic picture of the entropy transfer at the saturation
mechanism of KBM instability growth.

Fig. 2 Poloidal wave number spectra of the terms in the entropy
balance for (a) ions and (b) electrons (summed over theradial wave number and normalized by the total ion entropy production). Circular, square, triangle, and crossdots plot the entropy production due to transport Θsk ,nonlinear transfer Isk , field-particle interaction Rsk , collisional dissipation Dsk , respectively. Plots are limited
up to ky ρti = 0.6 for visibility, while we have employed
0 ≤ ky ρti ≤ 1.55 in numerical simulations.

poloidal angle θ = 0) and elongate along the magnetic
field lines. In the presence of the magnetic shear, thestructures twist and create radially-fine mode structures(see the θ = π plane). The field-particle interaction term
∗J0sk ∂t A k )] corRsk = Re[ dv3 (−es J0sk φ∗k ∂t fsk − es v fsk
responds to the energy exchange among the ion entropy,
electron entropy and field energy. We note that there isno entropy transfer between species via collisions, because
we employ a self-collision model operator. In a steady state(d/dt ∼ 0), the entropy production, transfer and dissipation
are statistically balanced, Θsk + Dsk + Isk + Esk +Rsk = 0, andthe field-particle interactions stand for the entropy transfer
among particle species, s Rsk = 0.
The poloidal wave number (ky ) spectra of the terms
in the entropy balance equation (1) are plotted in Fig. 2,
where the parallel streaming term does not appear because
of the conservation relation kx Esk = 0 for the radial
wave number k x . The ion transport term (Θik > 0) drives
the ky ρti = 0.2 mode, and the nonlinear transfer term Iik
transfers the low-ky entropy to the other modes [where
I (0 ≤ ky ρti < 0.2) = 0.037, Iik (ky ρti = 0.2) = −0.407,
 ikIik (0.2 < ky ρti ≤ 1.55) = 0.370, and thus, the conser
vation relation k Isk = 0 is satisfied]. This produces ion
entropy in the wide wave-number-space range, which aredissipated by the ion collisional dissipation (Dik < 0). At
the same time, there is a significant entropy transfer fromions to electrons through the field-particle interaction terms(−Rik = Rek > 0). This is the dominant source of theelectron entropy, since the entropy production due to theelectron transport is small when R/LTe = 0. The low-ky
electron entropy is transferred to the other modes by thenonlinear transfer term Iek , and is dissipated by the electron collision (Dek < 0). As a result, the statistical balanceof entropy production, transfer and dissipation is satisfied
in the saturated state of the KBM instability.
A more detailed analysis of the nonlinear entropytransfer spectrum Isk reveals that the mode coupling among
the dominant mode and its twisted modes in the sheared
geometry are the dominant nonlinear transfer process. Thissuggests importance of the electron dynamics, since therapid parallel motions of electrons create elongated structures along field lines [3]. Details of the mode coupling in
KBM turbulence will be reported elsewhere.
The saturation mechanism of KBM instability growthis summarized as a schematic picture in Fig. 3. The lowwave-number (ky ρti = 0.2) ion entropy is produced by theparticle and heat transport. They are transferred to thesame wave-number modes of electron fluctuations via thefield-particle interactions. Rapid parallel advection of electrons leads to elongated mode structures along field lines,
which in turn enhances the twisted modes in the shearedmagnetic fields. Then, the nonlinear mode coupling withthe dominant unstable mode and its twisted modes transfers the low-wave-number ion and electron entropy to wide
wave-number-space range. Finally, the total ion entropy
production balances to the total ion and electron dissipations.
The importance of the entropy transfer between ionsand electrons via the field-particle interactions is identified.
This is in contrast to ITG turbulence where the ion entropy
production mostly balances to the ion dissipation [2]. In
addition, it is revealed that the KBM instability growthis saturated through the coupling of the dominant and its
twisted modes, rather than the zonal flow shearing.
This work is partially supported by the HPCI Strategic
Program for Innovative Research. The computations were1203020-2

Plasma and Fusion Research: Rapid Communications

Volume 9, 1203020 (2014)

carried out using FX10 at the University of Tokyo, and HELIOS at Computational Simulation Centre of International
Fusion Energy Research Centre (IFERC-CSC).

[2] T.-H. Watanabe and H. Sugama, Nucl. Fusion 46, 24 (2006).
[3] S. Maeyama et al., Comput. Phys. Commun. 184, 2462
(2013).
[4] H. Sugama et al., Phys. Plasmas 16, 112503 (2009).
[5] S.C. Cowley et al., Phys. Fluids B 3, 2767 (1991).

[1] W.M. Tang et al., Nucl. Fusion 20, 1439 (1980).

1203020-3