23P131-B
Analysis of plasma rotation eﬀects on ballooning stability
in magnetospheric plasma conﬁnement
磁気圏型プラズマ閉じ込めのバルーニング安定性に対する流れの効果の分析
Masaru Furukawa
古川 勝
Grad. Sch. Frontier Sci., Univ. Tokyo, Kashiwa-shi, Chiba 277-8561, Japan東京大学 大学院新領域創成科学研究科 〒 277-8561 千葉県柏市柏の葉 5 − 1 − 5Magnetohydrodynamics ballooning mode stability of magnetospheric plasma conﬁguration is studied via time-dependent eikonal formulation. Plasma rotation eﬀects on the ballooning mode stability can be categorized into four: (i) change of equilibrium which enters the ballooning equation
through the metric elements, (ii) new self-adjoint terms in the ballooning equation originatingfrom plasma rotation, (iii) time varying wave number due to rotation shear and (iv) non-selfadjoint terms including ﬁrst-order time derivative. (i) and (ii) modiﬁes the instability growth ratein the order square of Mach number, rotation speed divided by thermal speed. (iii) leads to stabilization after long time similar to slab geometry, not in a sense of time average as tokamak case.
(iv) partly works like friction.

1. Introduction
Plasma conﬁnement by magnetospheric conﬁguration was proposed to achieve advanced fuelnuclear fusion[1], and has been studied in laboratories toward the goal[2,3]. Since these experimental devices have only poloidal magnetic ﬁeld,
pressure-driven magnetohydrodynamics (MHD)
instabilities might occur. Thus the MHD stability was studied for limiter [4] and separatrix conﬁgurations[5]. The separatrix was shownto have stabilizing eﬀect because of its bigﬂux expansion[5]. These studies assumed static
plasma equilibria. If a plasma is rotating, thepressure-driven instability is aﬀected considerably. For example, rotation shear stabilizes ballooning mode[6,7] in tokamaks on time average[811]. The stabilization occurs due to energy transfer from unstable to stable modes[11].
These studies mainly focused on plasma rotation shear. However, the magnitude of the rotation can also play a role. In the present paper, westudy how to categorize the plasma rotation effects on ballooning instabilities by examining thegoverning equation. We also point out an important diﬀerence between the magnetospheric conﬁguration and tokamaks. The conclusions given
below have been veriﬁed by numerical simulations, which will be presented elsewhere.
2. Examination of governing equation
We consider a magnetospheric, axisymmetric plasma with only poloidal magnetic ﬁeld andtoroidal rotation. The equilibrium is describedby the Grad–Shafranov equation including thetoroidal rotation[12],µR ∇·21∇ψR2¶¯
β0 2 ∂p ¯¯=− R.
2∂ψ ¯R(1)

Note that all quantities are normalized by their
typical values in this paper. The equilibrium
magnetic ﬁeld is expressed by B = ∇ψ × ∇φ,
where φ is the toroidal angle. The equilibrium
toroidal rotation speed is given by RΩ(ψ). The
ratio of plasma pressure to magnetic pressure
is denoted by β0 := 2µ0 p0 /B02 , where p0 and
B0 are typical values of pressure and magnetic
ﬁeld, respectively. The pressure takes the formp = p̄(ψ) exp[M02 Ω2 (ψ)(R2 /Rc2 − 1)] when thetemperature T is constantp on each magnetic surface. Here M0 := Rc Ω0 / 2T (ψ)/mi is the Machnumber based on thermal velocity of ions. The
ion mass is mi , Rc is the major radius of the internal ring current, and Ω0 is the typical valueof toroidal rotation frequency. The plasma rotation changes the source term of Eq. (1) and thus
the solution ψ too. The change of the equilibrium, or the metric elements equivalently, enters
the ballooning equation explained below throughthe change of its coeﬃcients. Assuming M02 ¿ 1,
we may expand the source term and obtain¯∂p ¯¯= p̄0 + O(M02 ),∂ψ ¯R(2)where the prime denotes ψ derivative. Since thesource term changes in O(M02 ), the metric elements also change in O(M02 ).
Next we examine the linearized ideal MHD
equation including equilibrium plasma ﬂow[13]:∂2ξ∂ξ+ 2MA ρv · ∇= Ff (ξ),
2∂t∂t
Ff := (∇ × B̃) × B + (∇ × B) × B̃
β0+ ∇(ξ · ∇p + Γp∇ · ξ)
2+ MA2 ∇ · (ρξv · ∇v − ρvv · ∇ξ),

ρ(3)

(4)

where ρ and v are equilibrium mass density andﬂow velocity, respectively. The speciﬁc heat ratiois Γ. A displacement of the plasma element fromthe equilibrium trajectory is ξ, deﬁned via a perturbed ﬂow velocity ṽ =: ∂ξ/∂t + v · ∇ξ + ξ · ∇v.
A perturbed magnetic ﬁeld is given by B̃ :=∇×(ξ ×B). The Alfvén Mach number is deﬁned√
by MA := Ω0 τA with τA := L/(B0 / µ0 ρ) and L
a typical length. While the second term of thel.h.s. of Eq. (3) introduces anti-self-adjointness,the r.h.s. of Eq. (3), a generalization of the MHD
force operator[14], is still self-adjoint even including the new terms of rotation v. The ratio of therotation term to the pressure term is roughly M02
by the relation MA2 = β0 M02 . Thus we expect anO(M02 ) modiﬁcation of growth rate for pressuredriven instabilities by the new terms, which isthe same order as the equilibrium change.
For ballooning modes, we adopt the eikonal
formulation based on a large toroidal mode number n[6,7]. Using n À 1, we express ξ =P ³ −j (j) ´ i nS(j)e , where ξ̂ represents an enj n ξ̂velope and S is an eikonal. We assume B·∇S = 0and ∂S/∂t + v · ∇S = 0 for the eikonal[8-11], toobtain S = −φ + MA Ω(ψ)t + S0 (ψ). The wavevector then becomes k̂ := ∇S = −∇φ + k̃ψ ∇ψ,and the radial wave number is given byk̃ψ = MA Ω0 t + kψ0 ,kψ0 := S00 .

(5)

The t dependence of k̃ψ expresses the stretchof wave in time by rotation shear. The rotation shear determines the time scale of k̃ψ variation as (MA Ω0 )−1 . Note that this k̂ does nothave dynamical lattice symmetry[9] as in tokamak case, leading to similar behavior as in slab
geometry[15]. Collecting terms at each order in(0)n, we obtain ξ̂ = ξk B + ξ⊥ B × k̂/B 2 and thecoupled wave equations for x = (ξk , ξ⊥ )T , called
the ballooning equation. Although the explicit
representation of the equation will be presented

elsewhere, the abstract form is as follows:∂2x∂x+ M1+ M0 x = Lx,
(6)
∂t2
∂t
where Mj s and L are matrices of operators. Thel.h.s. (r.h.s.) of Eq. (6) comes from the l.h.s.
(r.h.s.) of Eq. (3). The M1 term yields an eﬀect
analogous to friction[16]. The operator L has
O(M02 ) terms as similar to Ff . Most important
point is that these operators include t throughk̂; Eq. (6) is non-autonomous. However, if we
consider t just as a parameter, we ﬁnd that L isstill self-adjoint. Thus we may be able to utilize
the spectral decomposition of L at each instanceto analyze the solution of Eq. (6). If the timescale of the dynamics is much faster than that ofk̃ψ variation, the wave evolves as an eigenmodeof L at each instance.
M2Acknowledgments
The author is grateful to Mr. T. Sugiura for
useful discussion. This work was supported by
KAKENHI 19760595 and KAKENHI 23760805.
References[1] A. Hasegawa, Comments Plasma Phys. Control.
Fusion 1 (1987) 147.
[2] Z. Yoshida, Y. Ogawa, J. Morikawa et al.: J.
Plasma Fusion Res. 1 (2006) 008.
[3] J. Kesner, L. Bromberg, D. T. Garnier, and M. E.
Mauel: Fusion Energy 3 (1999) 1165.
[4] D. T. Garnier, J. Kesner, and M. E. Mauel: Phys.
Plasmas 6 (1999) 3431.
[5] M. Furukawa, H. Hayashi, and Z. Yoshida: Phys.
Plasmas 17 (2010) 022503.
[6] J. W. Connor, R. J. Hastie and J. B. Taylor: Phys.
Rev. Lett. 40 (1078) 396.
[7] R. L. Dewar and A. H. Glasser: Phys. Fluids 26(1083) 3038.
[8] W. A. Cooper: Plasma Phys. Control. Fusion 30(1988) 1805.
[9] F. Pegoraro: Phys. Lett. A 142 (1989) 384.
[10] F. L. Waelbroeck and L. Chen: Phys. Fluids B
3 (1991) 601.
[11] M. Furukawa and S. Tokuda: Phys. Rev. Lett.
94 (2005) 175001.
[12] H. P. Zehrfeld and B. J. Green: Nucl. Fusion 12(1972) 569.
[13] E. Frieman and M. Rotenberg: Rev. Mod. Phys.
32 (1960) 898.
[14] I. B. Bernstein, E. A. Frieman, M. D. Kruskal
and R. M. Kulsrud: Proc. R. Soc. London, ser A
244 (1958) 17.
[15] T. Tatsuno, F. Volponi, and Z. Yoshida: Phys.
Plasmas 8 (2001) 399.
[16] T. Sugiura, Master thesis, Grad. Sch. FrontierSci., Univ. Tokyo (March, 2011).