23P127-B
2D extended MHD simulation of R-T and K-H instabilities
レイリー・テイラー．ケルビン･ヘルムホルツ不安定性の二次元拡張MHDシミュレーションRyosuke Goto1), Hideaki Miura1),2), Atsushi Ito1),2) and Masahiko Sato2)
後藤涼輔, 三浦英昭，伊藤 淳，佐藤雅彦
1) The Graduate University for Advanced Studies (SOKENDAI)
322-6 Oroshi, Toki, Gifu 509-5292, JAPAN
1) 総合研究大学院大学 〒509-5292 土岐市下石322-6
2) National Institute for Fusion Science322-6 Oroshi, Toki, Gifu 509-5292, JAPAN
2) 核融合科学研究所 〒509-5292 土岐市下石322-6
Hall and gyroviscosity effects to the Rayleigh-Taylor(R-T) instability are studied by 2D numericalsimulations of extended magnetohydrodynamic(MHD) equations. Numerical simulations over wide rangeof parameters such as the Hall parameter, gyroviscosity, beta value, and wave number of unstable modes
are carried out. Effects on the growth of unstable modes and nonlinear saturation process are reported.

1. General
Dynamical aspects of magnetohydrodynamic(MHD) instability such as ballooning instability in a
torus device have been extensively studied. Thoughnumerical simulations using the single-fluid MHD
model for high wave number ballooning modes
have been carried out[1], not all the mechanism of
the ballooning instability are clarified. The singlefluid model ignores some effects such as Hall effectand Finite Larmor Radius (FLR) effect. Theseeffects come under higher order terms than thepressure gradient, the 𝐉 × 𝐁 force and so on in the
MHD ordering, and have short distinctive length.
Since the single-fluid model can be insufficient forhigh wave number ballooning modes, we consider
studying these effects on the instability by using
an extended MHD model. Furthermore, toroidal
and poloidal flows exist in a realistic plasma. Since
the flow is not uniform, the shear of flow (velocitydifference) can affect the growth of the ballooning
instability through the Kelvin-Helmholtz(K-H)
instability.
Since the ballooning instability is RayleighTaylor(R-T) type instability, here we concentrate ona simple R-T instability. The R-T instability is
instability of an interface between two fluids ofdifferent densities, and occurs when the lighter fluid
is pushed by the heavier fluid. Here we assume thatheavy fluid is sustained both by the magnetic andthermal pressures, as it is assumed in Ref.[2]. The
schematic view is shown in Fig. 1.
In this presentation, we study growth of unstable
R-T modes by the use of the extended MHD modelwith the Hall term and the FLR effect as gyro-viscosity[2]. For simplicity, 2D slab geometry isassumed. A change of the growth rate andsaturation level due to the two effects, depending on
the wave number of the unstable modes isconcerned. Effect of shear flow on the growth isalso going to be studied.

Fig.1. Schematic view of the R-T instability

2. Numerical Simulation
Extended MHD equations in 2D slab geometry
are expressed as follows.
∂u∂ρ∂ρ∂ρ
∂u+ ux + uy
= −ρ � ∂xx + ∂yy � ,
∂x
∂y∂t
∂u
∂u
∂uρ � ∂tx + ux ∂xx + uy ∂yx �∂(π )∂p∂(π )
= − + (Jy Bz − Jz By ) − ∂xi xx − ∂yi xy ,
∂x
∂uy
∂uy
∂uy
ρ � ∂t + ux ∂x + uy ∂y �=−(1)

(2)

∂(π )
∂(π )
∂p− ρg + (Jz Bx − Jx Bz ) − ∂xi yx − ∂yi yy ,
∂y(3)

∂u
∂p
∂p∂p∂u+ ux+ uy = −γp � ∂xx + ∂yy � ,∂x∂y∂t∂bx∂E=− z ,∂t∂y∂by∂Ez= ∂x ,∂t∂E∂bz∂E= − ∂xy + ∂yx .
∂t

(4)
(5)
(6)

Numerical simulations are carried out wide rangeof β, δ, ϵ and m to characterize the influence of theHall term and the gyroviscosity effects. We will
also consider the flow shear, which can cause theK-H instability (Fig.3) in the presentation.

(7)

Here we have set uz = 0 , ∂/ ∂z → 0 ineqs.(1)-(7), and the current density isJx =∂B∂Bz∂B∂B, J = ∂xz , Jz = ∂xy − ∂yx .
∂y y

The electric field and gyro-viscosity are given asϵ∂p
Ex = −uy Bz + �Jy Bz − Jz By − ∂xe� ,(8)ρϵ∂pEy = ux Bz + �Jz Bx − Jx Bz − e� ,∂yρϵEz = −ux By + uy Bx + �Jx By − Jy Bx � ,ρ∂u∂u(πi )xx = −(πi )yy = −δpi � y + x � ,∂x∂y∂uy∂ux(πi )xy = (πi )yx = δpi � −�.
∂x∂y(9)

(10)
(11)
(12)

Equations (1)-(12) have been already normalized bysome representative quantities. The symbol g is thegravitational acceleration, and the symbols ϵ, δ, γ
are the Hall parameter, gyroviscosity, the ratio of
the specific heats, respectively. We also denote thetotal pressure as p = pi + pe = (α + 1)pe , α =pi /pe . Where pi and pe are the ion and theelectron pressure, respectively.
In our numerical simulation, the KawamuraKuwahara scheme[3] for advection terms, 4th order
central difference method for the other terms, and4th order Runge-Kutta-Gill method for time
evolution are used. The system size in the x and y
directions are set 2π and 8π, respectively. The
system is periodic in the x-direction, and theboundary condition ∂/ ∂y = 0 is imposed at theupper and lower boundaries y = ±4π.
Figure 2 shows typical evolutions of R-Tinstability. Figs.2(a)-(b) show the R-T instability inthe single-fluid MHD model and (c)-(d) show theR-T instability in the extended MHD model. Thecontrol parameters are m=2 (the wave number ofperturbation in the x-direction), α = 0.5 andβ = 0.1 in the former, and m = 2, β = 0.1, α =0.5, δ = ϵ = 0.01 in the latter. In Figs.2(a)-(b), we
can observe that typical mushroom-like structures
grow due to the R-T instability. In Figs.2(c)-(d), thegrowth of the unstable mode is under the Hall andthe gyro-viscosity effects. Because of these effects,
the structures of Fig.2(c) is slightly different fromthose in Fig.2(a), and the difference becomes
obvious in a later stage, between Fig.2(b) andFig.2(d).

Fig.2. Time evolution of the R-T instability. Panels(a)-(b) are for the single-fluid MHD model, and (c)-(d)are for the extended MHD model.

Fig.3. An example of the K-H instability

References[1] H. Miura and N. Nakajima: Nucl. Fusion 50 (2010)
054006.
[2] P. Zhu et al: Phys. Rev. Lett. 101 (2008) 085005.
[3] T. Kawamura and K. Kuwahara: AIPP paper (1984)84-0340.