J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)210–213
000–000

A Model Equation for Ballooning Modesin Toroidally Rotating Tokamaks
FURUKAWA Masaru* and TOKUDA Shinji
Naka Fusion Research Establishment, Japan Atomic Energy Research Institute, Ibaraki, 311-0193, Japan(Received: 9 December 2003 / Accepted: 11 May 2004)

Abstract
A model equation for ballooning modes in toroidally rotating tokamaks is derived. It is confirmed that the model
equation is appropriate for analyzing the stabilization mechanism of the ballooning modes by comparing the numericalsolutions of the model equation with those of the original ballooning equations.
Keywords:
toroidal rotation, ballooning mode, tokamak, magnetohydrodynamics

1. Introduction

solutions of the model equation are compared with those of
the original ballooning equations. Summary is given inSec. 4.

In the H-mode [1] pedestal region, the reachable pressure
gradient is often limited by magnetohydrodynamic (MHD)
activities, called edge localized modes (ELMs) [2]. One of
the causes of the ELMs is considered to be ideal (nondissipative) MHD ballooning instabilities [2]. Thus the linear
stability analysis of the ballooning mode [3,4] has been a
crucial issue in tokamak fusion research.
Plasmas in the pedestal region of tokamaks often rotate.
The plasma rotation affects linear stability of MHD
ballooning modes. It was numerically found that toroidal
rotation shear stabilizes ideal MHD high-n ballooning modes(n: toroidal mode number) [5-8]. We found that the toroidal
rotation shear damps the perturbation energy of the ballooning
modes, which results in the stabilization of the mode [8].
However, it has not been clarified how the toroidal rotation
shear damps the perturbation energy.
An analytical study may clearly show the damping
mechanism of perturbation energy, however, the ballooning
equations [6] solved numerically in ref. [8] are toocomplicated to analyze; they are coupled wave equations fortwo components of the displacement vector. Here, we will
derive a model equation which is a wave equation for onevariable. We will also solve the model equation numerically,and compare the solutions with those of the original
ballooning equations. It will be shown that the model equation
is appropriate for analyzing the damping mechanism of theperturbation energy of ballooning modes.
In Sec. 2, the model equation is derived by simplifying
the original ballooning equations. In Sec. 3, numerical

2. Derivation
In this Section, we derive a model equation appropriate
for analyzing the damping mechanism of the perturbationenergy of ballooning modes.
It was numerically shown that toroidal rotation shear
stabilizes ballooning modes even in the incompressible limit[5,7]. Thus, we adopt the incompressible model. The
incompressible ballooning equations are obtained by taking
the variation of the action with a constraint: the action isI≡dt

d x (T – V + λ W ) ,3wheredξ,dt
V ≡ – ξ * ⋅ F (ξ ) ,2T ≡ρ2W ≡ ∇⋅ ξ ,F (ξ) ≡ µ1

0

∇× Q × B+ ∇× B × Q

ξ v ⋅∇v ,+ ∇ γ p∇⋅ ξ + ξ ⋅∇p + ∇⋅ ρξQ ≡ ∇× ξ × B .
The incompressible condition is imposed by λW, where λ isa Lagrange multiplier. The displacement vector is ξ , ρ and p

Corresponding author’s e-mail: furukawm@fusion.naka.jaeri.go.jp*Present address: Graduate School of Frontier Sciences, The University of Tokyo,Kashiwa, 277-8561, Japane-mail: furukawa@ppl.k.u-tokyo.ac.jp1210©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchFurukawa M. et al., A Model Equation for Ballooning Modes in Toroidally Rotating Tokamaks(0)
2Here, ξˆ (0) = ξ̂ (0)⊥ B × k̂/B + ξ̂ || B, R̂ and Ẑ are unit vectors in
the R and Z direction of a cylindrical coordinate system,
respectively, and κ ≡ (B/B) · ∇(B/B) is a magnetic curvature.
Equations (1) and (2) are coupled wave equations for ξ̂ (0)⊥ andξ̂ ||(0) along the extended poloidal angle. Thus, theincompressibility does not simplify the ballooning equations
significantly.
Next, we set Ω = 0 in eqs. (1) and (2), since it was shownthat the perturbation energy damps even for Ω = 0 if Ω′ ≠ 0[7,8]. The resulting equations areare equilibrium mass density and pressure, respectively, γ is aspecific heat ratio, and µ0 is a vacuum permeability. The
equilibriu magnetic field and velocity are B = χ ′( χ̂ ) (∇ζ ×∇χ̂ – q∇ χ̂ × ∇ θ ) and v = Ω ( χ̂ )e ζ , respectively, whereχ̂ ≡ ( χ – χaxis)/( χedge – χaxis) is a normalized poloidal flux, χis a poloidal flux function, the prime denotes the derivativewith respect to χ̂, θ and ζ are poloidal and toroidal angle,
respectively, q is a safety factor, Ω is a toroidal rotation
frequency, eζ is a covariant basis vector in the toroidal
direction. The operator F is self-adjoint [9].
The displacement vector is expressed in the eikonal formand expanded by 1/n << 1 as(0)∂ ξ⊥
2µ0 ρξ = ξ (0) χ , ϑ , t + 1 ξ (1) χ , ϑ , t + ⋅ ⋅ ⋅ e in S χ , α , t ,n2k(0)– 2 k⋅∇ Ω∂ t2∂ξ ⊥∂t2

where S is the eikonal, n is a toroidal mode number, ϑ is anextended poloidal angle in the covering space, α ≡ ζ – qθ isa field line label. The eikonal satisfies= B B⋅∇+ 2 B× k⋅ κB ⋅ ∇S = 0 ,dS = 0 .
dt

2

ρ k

∂2 ξ∂t2= µB B ⋅∇0–

ρR Ωp2∂ξ+ 2ρ k ⋅∇Ω(0)∂t– 2ρΩ B2(0)B ⋅∇ξ ⊥ + 2µB02k⋅ Z∂ξ∂t(0)(0)
B ⋅∇p B × k ⋅ R ξ ||2+2R∂2 ξ (0)||∂tRR2ρΩ dΩ
B ⋅ Z ∇χ2B dχ

24– µ B × k ⋅κ0–

ρ 2 R2Ω 4pB

2B× k⋅ R

∂ξ

(0)
⊥2

ξ (0)
,
⊥2(3)

= B ⋅∇ B B ⋅∇ξ || – 2 B × k ⋅ κ ξ ⊥ . (4)
2(0)

(0)

Equations (3) and (4) are much simpler than eqs. (1) and (2),(0)however, they are still coupled equations for ξ̂ (0)⊥ and ξ̂|| . This
coupling does not come from the rotation shear. For a static
plasma, we commonly examine the ballooning stability bysolving a second-order differential equation for aperpendicular component of a ballooning displacement. Thisequation is derived from an incompressible part of potential
energy and kinetic energy of only the perpendicularcomponent of the displacement. When we take a variation of
the potential energy and the kinetic energy of both theperpendicular and parallel components with imposing theincompressibility, we obtain coupled equations even for astatic plasma. It is also noted that eqs. (3) and (4) coincidewith the incompressible limit (the specific heat ratio γ is taken
to be infinity) of eqs. (51) and (52) of ref. [4].
Here, we propose to decouple ξ̂ ||(0) from eq. (3) bydropping the last two terms in that equation;(0)||B × k ⋅ κ B ⋅∇ξ ||

∂ ρRΩ
∂p
κ–B× k ⋅ 2∂χ∂χ R+2µ 0 ρ B22k
B(0)
⊥(0)
µ0
B× k⋅∇p– 2B× k⋅ κ ξ ⊥B2+ 2 B2 B× k⋅ κ B⋅∇ξ || ,The wave vector is defined as k̂ ≡ ∇S = ∇ζ – q∇θ – (ϑ – θk⋅⋅+ Ω t)∇q, where θk is a ballooning angle and Ω ≡ dΩ /dq. The
displacement vector in the eikonal form, as well as λ = λ(0) +λ(1)/n + … , are substituted in the action, then we obtain
incompressible ballooning equations in O (n0);(0)⊥2k(0)
B⋅∇ξ ⊥B2

22µ0ρ

(1)

k

∂2 ξ (0)⊥∂t 2– 2 k ⋅∇ Ω∂ ξ (0)
⊥
∂t2

ρB

∂2 ξ (0)2||∂t 2– 2ρΩ k ⋅ Z∂t= B B ⋅∇2(0)+ 2ρΩ ∇Ω ⋅ Z ξ ⊥12(0)(0)= µ B ⋅∇ B B ⋅∇ξ || – 2 B × k ⋅ κ ξ ⊥+B0–B ⋅∇pp

ρR Ω 2(0)(0)
B ⋅∇p ξ || +B× k⋅ R ξ ⊥ .
2B2µ 02kBB ⋅∇ξ ⊥(0)

2B × k ⋅κ

(0)
B × k ⋅∇p ξ ⊥ .

(5)

The reason why we drop not only 2B2(B × k̂ · κ )B · ∇ξ̂ (0)|| butalso –4(B × k̂ · κ )2ξ̂ (0)⊥ is as follows. From eq. (4), the last(2)2211

Furukawa M. et al., A Model Equation for Ballooning Modes in Toroidally Rotating Tokamaks

two terms of eq. (3) express kinetic energy of ξ̂(0)|| . This meansthat we drop kinetic energy of the parallel component of thedisplacement. As noted above, the ballooning equation for astatic plasma is derived similarly. Actually, when Ω′ = 0, eq.
(5) coincide with the ballooning equation for a static plasma.
Therefore, eq. (5) nicely connects ballooning modes in arotating plasma with those in a static plasma.

growth rate was also found in ref. [7].
Figure 2 shows the time evolution of the perturbationenergy ⏐⏐ξ 2⊥⏐⏐ for (a) S = 0.5, α = 0.7 and Ω′ =0.05, (b) S =0.5, α = 0.7 and Ω′ = 0.2, (c) S = 2, α = 2 and Ω′ = 0.1 and(d) S = 2, α = 2 and Ω′ = 0.8. In Figs. 2(a), 2(c) and 2(d), we
found the periodic time evolution occurs with damping phases
even by the model equation. Figure 2(b) shows the time
evolution of ⏐⏐ξ 2⊥⏐⏐ for a stable case. The perturbation energy

3. Numerical solutions
In this Section, we compare numerical solutions of themodel equation (5) with those of the original ballooning
equations including compressibility. At first, we must confirm
that the toroidal rotation shear can stabilize the ballooning
mode even by the model equation. Figure 1 shows the growthrate γ τA as a function of toroidal rotation shear Ω′τA , whereτA is the Alfvén time. The magnetic shear parameter S andthe pressure gradient parameter α are chosen as follows; (i)low magnetic shear case S = 0.5 and α = 0.7 and (ii) highmagnetic shear case S = 2 and α = 2. The equilibrium is alarge aspect ratio and circular cross-section tokamak. Wefound that the ballooning mode is stabilized by increasing thetoroidal rotation shear in both cases. At several values of Ω′τA ,the growth rate suddenly decreases. Such a behavior of theoriginal, S=0.5, α=0.7model, S=0.5, α=0.7

0.12

γ τA

0.1

original, S=2.0, α=2.0model, S=2.0, α=2.00.08
0.06
0.04
0.02
0-0.02

Fig. 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2ΩτA

Growth rate γ τA as a function of toroidal rotation shearΩ ′τA . The ballooning mode is stabilized by increasing
the toroidal rotation shear even by the model equation.

S=0.5
α=0.7
Ω τA =0.2modelln ξ⊥2modelln ξ⊥2

S=0.5
α=0.7Ω τA =0.05

0.14

original

original
0100200300t τA

400

500

600

0200300

400500600t τA

S=2.0
α=2.0
Ω τA =0.1

model

original

S=2.0
α=2.0
Ω τA =0.8ln ξ⊥2modelln ξ⊥2100

original

0Fig. 25001000t τA

1500

2000

050 100 150 200 250 300 350 400 450 500t τATime evolution of perturbation energy ⏐⏐ξ⊥2⏐⏐. Periodic time evolution with damping phases occurs even by the modelequation.

3212

Furukawa M. et al., A Model Equation for Ballooning Modes in Toroidally Rotating Tokamaks⏐⏐ξ⊥2 ⏐⏐ damps with oscillations even by the model equation.

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4. Summary
We have derived a model equation for analyzing themechanism of the damping of perturbation energy ofballooning modes by toroidal rotation shear. We haveconfirmed that the model equation is appropriate for analyzing
the damping mechanism by comparing the numericalsolutions with those of the original ballooning equations. We
speculate that stabilization of ballooning modes by toroidal
rotation shear originates from both (i) the time dependenceof the coefficients of the model equation and (ii) theconvection term. The mechanism of stabilization will bereported elsewhere.

Acknowledgements
The authors thank Dr. H. Ninomiya, Dr. M. Kikuchi andDr. Y. Kishimoto for a discussion and comments.

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