J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)362–365
000–000

MHD Equilibrium and Stability of Spherical Tokamak Plasmawith Current HoleMIZUGUCHI Naoki and HAYASHI Takaya
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 2003 / Accepted: 24 February 2004)

Abstract
The potential characteristics of spherical tokamak configurations with current hole are investigated from thepoint of view of magnetohydrodynamic (MHD) equilibrium and stability. The effect of the toroidal shear flows is also
considered by using a modified Grad-Shafranov equation. Linear and nonlinear stability for low-n kink modes andintermediate-n ballooning mode is analyzed by means of numerical simulations.
Keywords:
spherical tokamak, current hole, ballooning mode, magnetohydrodynamics

1. Introduction

current, respectively. Two arbitrary functions p and F are
expressed that p = p(ψ) and F = F(ψ) = rBθ , where p and B
are the plasma pressure and the magnetic field, respectively.
By specifying p and F, the Grad-Shafranov equation can besolved under the appropriate boundary conditions. The
inversion of the elliptic operator ∆* is carried out by meansof an iterative method with keeping the value of the total
current fixed.
The ST configurations with a current hole can becalculated by choosing the functions p and F as suitable
forms. In the current hole, both p and ψ are flat and jθ = 0.
Therefore, by setting p′(ψ) and F ′(ψ) to go zero at both themagnetic axis and the edge, the current hole configurations
can be modeled. We use a power of trigonometric functions
for them to insure the continuity and smoothness of thequantities at the boundaries.
An example of numerical solution of (1) for parametersof ST with current hole is shown in Fig. 1. The radial profileof the pressure, the toroidal current, and the safety factor q
are plotted in Fig. 1(a), and the contours of the poloidal flux
are drawn in Fig. 1(b). The parameters of the equilibrium isthat the aspect ratio A = 1.5, the plasma beta at the current
hole β0 = 20%, and the minimum of the safety factor q0 =2.76.
To investigate the effect of the plasma flow on themagnetohydrodynamic (MHD) stability, we consider theequilibria, or steady state, problem of an axisymmetrictokamak with stationary shear flow. The flow considered here
does not change in time, but varies in space. In this paper, weinvestigate a specific case that only toroidal shear flow exists.
More general treatment, including poloidal component ofIn conventional large tokamak experiments such as JT60U [1] and JET [2], a very good confinement, which isnamed the internal transport barrier (ITB), is observed. In theITB tokamak the pressure gradient is mainly sustained in theedge region, and becomes almost flat in the core region, on
the other hand. As an extreme of ITBs, an uniqueconfiguration has been observed recently. The plasma current
goes to zero around the magnetic axis, and the plasma is
confined at very high beta. This is named ‘current hole’. Such
configurations attract a great deal of attention as a candidateof fusion core plasma, because high beta plasma confinement
can be obtained with large bootstrap fraction.
In the past several years, the low-aspect-ratio, orspherical tokamak (ST) concept [3] has been studiedintensively, since good confinement and stability at high beta
can be obtained experimentally. Therefore, the capability of
ST configurations with current hole should be examined inorder to explore the reactor-sized STs in future.
In this paper, we investigate the properties of theequilibrium and the stability of STs with current hole.
Furthermore, the effect of plasma shear flow due to the ITB
is discussed.

2. Equilibria
An axisymmetric equilibrium is obtained analytically byusing the Grad-Shafranov equation,

11jθ = – ∆ * ψ = rp' + FF' ,rr(1)

where the elliptic operator ∆* is given by ∆* ≡ r ∂∂r r1 ∂∂r + ∂ 2 .
∂zψ and jθ are the magnetic flux function and the toroidal
2©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: mizu@nifs.ac.jp1362

Mizuguchi N. et al., MHD Equilibrium and Stability of Spherical Tokamak Plasma with Current HoleFig. 1

Numerical equilibrium of ST with current hole.
(a) radial profile of p, jθ and q, (b) poloidal flux.
Fig. 2

flow, is our future interest.
Now we start from the force balance equation of MHD
fluid,– ∇p + j × B – rˆvθ2=0rflow to exist only in the region where the current exists. Themaximum Mach number is about 0.03 for this case. All theother parameters used here are the same as the stationary case.
The field variables do not differ so much from the stationarycase.

(2)with the centrifugal force, which is caused by the existenceof the toroidal flow.
In the same manner as the derivation of the normal GradShafranov equation (1), we obtain a couple of modified GradShafranov equations,∆*ψ = – r 2

prr3 f 2– FF' –,
ψr
ψr(3)

pz– FF' ,ψz(4)∆ *ψ = –r 2

3. Linear stability
To investigate the MHD stability of the equilibria
obtained in the last section, we have executed nonlinear MHD
simulations in a full toroidal geometry. The governing
equation is a standard set of MHD equations,∂ρ / ∂t = – ∇ ⋅ ( ρ v),

ψ rr3 f 2prψ r + pzψ z–'–,FFψ r2 + ψ z2
ψ r2 + ψ z2

ψr( ∆ * ψ + FF' ) – rf 2 ,r2ψpz = – 2z ( ∆ * ψ + FF' ).
rpr = –(8)1∂(ρ v)/ ∂t = –∇⋅ (ρ vv) –∇p + j × B + µ (∇ 2 v + ∇(∇⋅ v)), (9)3where the subscript of p and ψ denotes the partial derivatives.
Here, F = rBθ and f = vθ /r are found to be a function of ψ, or
F = F(ψ) and f = f(ψ), however, p no longer depends on ψalone for vθ ≠ 0. Therefore, (3) and (4) have to be solved
simultaneously for the unknown variables ψ and p, specifying
arbitrary functions F and f.
To obtain more perspective form for numericalcomputation, (3) and (4) are rewritten as,∆ *ψ = –r 2

Birds-eye view of flow profile in poloidal cross section.

∂B / ∂t = – ∇ × E,

(10)

∂p / ∂t = – ∇ ⋅ ( ρ v) – (γ – 1)( p∇ ⋅ v + Φ ),(11)

where the used variables are following usual definitions,j = ∇ × B, and E = –v × B. In numerical solution, thederivatives are expressed by using a finite difference method,
and the time integration is done by the Runge-Kutta scheme.
Before executing the nonlinear simulations, linear
stability is examined by using a linearized version of (8)-(11).
The result shows that the equilibrium is linearly unstable forseveral modes.
The growth rate of the instability modes for each toroidal
Fourier component is shown in Fig. 3, where n is the toroidal
mode number. The growth rate is calculated from theperturbations in the magnetic energy. The higher-ncomponents are suppressed by assuming large dissipation, oraffected much by the number of numerical grids. Therefore,
we focus on only low-n modes, and use 128 × 128 numericalgrids in the poloidal cross section in this paper.
In Fig. 3, the growth rate for the case with flow is plotted
together. The result shows that the existence of the toroidal
flow tends to make each mode more unstable. However, there
are some components that is not affected by flows so much,

(5)
(6)
(7)

The numerical solution proceeds in two steps. First, (5)is solved in the same manner as the normal Grad-Shafranov
equation described above with a given pressure profile. Then,the pressure profile is adjusted by using (6) and (7). For theconvenience of numerical solution, we do not directly
integrate the pressure gradient (6) and (7), but solve an elliptic
partial equation, which is obtained by taking the divergenceof (6) and (7). These two steps are mutually iterated until theconverged solution ψ and p is obtained. An example of thesolution is shown in Fig. 2. Here, we model the toroidal shear2363

Mizuguchi N. et al., MHD Equilibrium and Stability of Spherical Tokamak Plasma with Current HoleFig. 5

Fig. 3Fig. 4

Nonlinear time development of pressure profile. (a) t =0, (b) t = 500, n = 1, (c) t = 620, n = 1, (d) t = 60, n = 6,(e) t = 320, n = 6 (The time t is normalized by the Alfvèn
transit time.)

Linear growth rates for each toroidal Fouriercomponent. Solid and broken lines denote that of thecase with and without flows, respectively.

nonlinear simulation is done only for the case vθ = 0 as theinitial condition.
Figure 5 shows the temporal changes in the poloidalpressure profile. Simulations are executed for two
eigenmodes, the n = 1 (Fig. 5(b)-(c)) and the n = 6 (Fig. 5(d)(e)) modes. One can see that for both cases the shape of thecross section varies reflecting the poloidal mode structures ofthe linear eigenmode in the beginning ((b) and (d)), but a lotof smaller scale components appear as a result of thenonlinear couplings. The configuration goes into turbulent anddestroyed in the end ((c) and (e)). So far as we can see fromthis result, there are no indications that the toroidal
confinement recovers after that, which can be seen in ourprevious simulations for conventional STs without current
hole [4-5]. However, it is necessary to survey simulation
parameters in wider range for better understanding of thenonlinear behavior.

Poloidal mode structures of (a) n = 1, (b) 6, and (c) 12
modes.

5. Summary and discussion
The MHD equilibria of ST configurations with a current
hole have been obtained numerically by solving the modified
Grad-Shafranov equation. We have also obtained both
equilibria with and without steady toroidal shear flows.
Linear and nonlinear MHD simulations have been
executed for such equilibria. The linear results show thegrowth of the 3/1 kink mode and the higher-n ballooning
modes. The steady toroidal flows make the higher-n modes
more unstable. It is considered that the destabilization iscaused by the centrifugal force. In our calculations, theplasma pressure profiles tend to be shifted outward under theexistence of toroidal flows, so that the pressure gradient ismore steepened in the low field side.
The preliminary nonlinear simulation result for the casewith no toroidal flows in the initial state shows that a hard
disruption of the configuration occurs. In our previous
simulation result for the internal n = 1 relaxation of aconventional ST plasma [5], the torus configuration recovers
spontaneously through a closed vortex flow inside the q = 1
surface. Another previous result for the internal reconnection
event [4] also shows a spontaneous recovery of thesuch as the n = 12 and n ≤ 3 modes.
The poloidal mode structures are shown in Fig. 4 for then = 1, 6, and 12 components. It can be seen that the dominantpoloidal component of the n = 1 mode is m = 3, where m is
the poloidal mode number. Analysis of the driving source of
the instabilities by using the energy principle shows that this
3/1 mode is a kind of current-driven (kink) instabilities.
Another feature of Fig. 4 is that the structures of the higher-nmodes are localized poloidally, which implies that thesemodes are categorized into the ballooning mode.

4. Non-linear simulation
To investigate further the overall dynamics of plasma, it
is necessary to execute nonlinear simulations by solving
directly (8)-(11). We use a finer numerical mesh with 256 ×
256 grids in poloidal cross section, and 64 grids in toroidal
direction, to follow the mode couplings among as many
modes as possible. A low-pass filtering technique is applied
to remove high-wavenumber components. Since the initial
flow causes no qualitative change in the linear instabilities,
3364

Mizuguchi N. et al., MHD Equilibrium and Stability of Spherical Tokamak Plasma with Current Holeconfiguration through the magnetic reconnection process
between the internal and external magnetic field. Theseprevious results seems to imply that STs have good‘resilience’. However, the nonlinear result in this paper shows
an opposite example. One of the reasons for the discrepancy
might be the existence of the current hole. Because there areno MHD force balances in the current hole region to sustain
the high beta plasma, the restrictive forces against thedeformations would be small. However, more detailed
nonlinear analysis is our ongoing subject.

The authors would like to thank Dr. Y. Ono, Dr. Y.
Nagayama, Dr. M. Furukawa and Dr. H. Miura for theirfruitful comments.

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