J. Plasma Fusion Res. SERIES, Vol. 5 (2002) 487490

The Marginally Stable Pressure Profile and a Possibility ofHigh Beta Plasma Confinement in LHD
WATANABE Tsuguhiro and HOJO Hitoshit
National Institute for Fusion Science, Toki 509-5292, Japant Plasma Research Center, University of Tsukuba, Tsukuba 305-8577,

Japan(Received: ll December 2001 / Accepted:24May 2OO2)

Abstract
It is shown theoretically that LHD has potentiality of the high beta plasma confinement. The chaotic
field line-layer, which surrounds the outside of the outermost magnetic surface, plays a key role for anefficient plasma confinement in LHD.
The plasma in the chaotic field line region prevents an immediate cooling of the core plasma causedby neutral atoms in the vacuum vessel. Furthermore, plasma pressure in the chaotic field line region can
increase the core plasma pressure, even in the magnetic hill condition. The equilibrium and stability of
LHD plasma are analyzed under the ideal MHD model, and a marginally stable pressure profile isderived analytically and also numerically.

Keywords:
LHD, high beta plasma, equilibrium, MHD stability, chaotic field line layer, fusion reactor

1. Introduction
After the first discharge on March 31, 1998, theLarge Helical Device (LHD) of National Institute forfollowing merit for plasma confinement: (l) Unstable
resonant modes are localized on the rational surface sothat these instabilities do not relate directly to the decayof the entire plasma column. (2) The magnetic shear hasFusion Science has extended the parameter range of theplasma confinement, successfully. The electron temperature on the magnetic axis exceeds l0 keV alongwith the increase of the heating input, and, meanwhile,
the observation such as ELM or plasma disruptions,which synchronizes with an increase in the heatinga strong stabilizing effect for the convective instabilityof entire plasma column. (3) The magnetic field line in
the chaotic region has extremely long connection length.

Characteristic of chaotic field line region and its
role for the plasma confinement in LHD can besummarized as follows t2l: (1) The connection length ofthe diverter field line which approaches close to theoutermost magnetic surface exceeds l0 km. The cold

power input, has not yet been observed.

Numerical studies of particle orbits show that theloss cone does not exist in LHD, hence there is an
excellent containment performance to the high-energy
particle in LHD [].

diverter plasma does not cool down the core plasma
directly therefore. (2) The lines of force that are
disengaged from the chaotic field line region reach thevacuum vessel wall soon. Then, it is expected in chaotic
field line region that the plasma pressure can besustained stably by the line-tying effect of the field

The characteristics of LHD magnetic field are thehigh magnetic shear configuration, and the existence ofthe chaotic field line layer which surrounds theoutermost magnetic surface.

High shear magnetic field configuration has@2O02by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author's e-mail: wata@nifs.ac.jp487

Watanabe T. et aI., The Marginally Stable Pressure Profile and a Possibility of High Beta Plasma Confinement in LHD

lines, which are disengaged from the chaotic field

2. Analysis of Equilibrium of LHD

region. (3) The chaotic field line region can confine thereflecting particles (particles whose velocity is almost
perpendicular to the magnetic field). ( ) The plasma

plasma will be analysis of equilibrium. For the tokamak

The first step to analyze characteristics of high beta
which is an axisymmetric torus, equilibrium is describedby the Grad-Shafranov equation. This equation cananalyze the configuration of complete F = 1 plasma thatpressure of the chaotic field line region can increase thecore plasma pressure, even in the case of the magnetic
hill configuration. (5) The plasma contained in thechaotic field line region prevents an immediate core

the magnetic field vanishes in plasma.

Let's extend the Grad-Shafranov equation to 3D
configuration in order to solve the high beta equilibriumof a helical plasma. It is assumed that nested magnetic
flux function can exist and that the plasma pressure is afunction of the flux function.

plasma cooling down caused by the neutral atom flittingin the vacuum vessel (role of the plasma blanket).

Furthermore, LHD are showing the excellent
experimental results as follows: 1) A smooth highenergy proton spectrum of 300 keV or more is observedin the ICRF experiment. 2) Excellent plasma is

Several surface quantities, which are integral
constants of equilibrium equations, exist in an
equilibrium of plasma. It is necessary to set up

sustained even when magnetic field strength isdecreased to the order of 1/6 (B* = 0.5 T) of the normal

appropriately surface quantities beforehand to determine

values (B* = 2.75 - 3 T). 3) Plasma exists in the chaotic

the plasma equilibrium.

Plasma current is not driven actively for thefield line region surrounding the outermost magnetic

confinement, in LHD. Hence, we adopt a distribution ofrotational transform t(yr) and a pressure profile p(t4), as
surface quantitie which provide for equilibrium.
Let's introduce a coordinate system (V, X, Q) as
shown in Fig. 2. The coordinates I and Q are thepoloidal and toroidal angles ($d7 = $Op = 2n), which aresurface.

According to the fact, that a large plasma current is

not needed for the plasma confinement, and the above
mentioned theoretical and experimental results, we haveguessed that LHD can sustain high beta plasma stably.
In Sec. 2, we have summarized briefly an algorithm
for the numerical treatment of high beta equilibrium ofLHD. In Sec. 3, we have analyzed the stability of LHD
plasma under the ideal MHD model, and have obtained
numerical results for the marginally stable pressure

introduced, appropriately and arbitrarily, to describe a
toroidal system. Magnetic flux function yr is determinedby the condition of the equilibrium, numerically.
Let's express the magnetic field B, which has aflux function ur. as follows:

profile. Section 4 is devoted to summary anddiscussions.

VyDIVERTOR FIELD LINES IN LHD

8^w=2.7t T

1=LtFig. 1 Chaotic field lines of LHD. The lines ol' force areclassified by the connection length.

Fig. 2 Coordinate system

488

Watanat'e T. et al., The Marginally Stable Pressure Profile and a Possibility of High Beta Plasma Confinement in LHD

fi=YryxYg.

number (m, n) into the (X, Q) direction. Furthermore, weneglect the coupling effects with the inhomogeneity of(1)The function 5, can be written by the use of a periodic

the equilibrium state.

function lof ()C, Q):€ = €rY l( x V@ + €rY Q xY ry + (.Y ryxY S

8 =P'(vttx+Q'WlQ+4da(t€,tr\ (t1rt"nt^x + nil\*(P+A )YV.YX+(e+Ar)YUt.V0(Yv)'

IJ€rs, | = | 4(v/)cos(mlg + nQ) |\t€rs rJ \((rr)cos{, X

where ' denotes the derivative with respect to yand thesuffix 7 (@) represents the partial derivative with respect

Stability against the helical type perturbation withfinite wave number (m,n) is analyzed, here. The mostdangerous mode is the one that resonance condition m +nllzn = 0 is held. In this case, the stability condition ist2n--Q'gt)
P' (v)'

given by the following form, which can be reduced tothe Suydam condition in the case of straight cylinderQ)

The functions ry and A are determined by the following
set of equations, which can be reduced to the Grad-plasma.

I Il ,.2(B,xBp)2
4 pot'T

Shafranov equation in the case of axially symmetricplasma.

(,,o=:-J

3.1 Stability condition for resonant modes

to X @).Function P(ry) and Q(V) are arbitrary functions of ty and specify the distribution of rotationaltransform il2n of the line of force.

p' (v) =#+ nQ)\B- p'>0. (3)B' *B" o'))'+ u 1y, - u-o=ru -.

tor V x (V@ x Vs))\"

where gn is permeability of vacuum. Toroidal (poloidal)
component of magnetic field is expressed by B, (80),and, p and U represent the shear parameter and the specific volume, respectively:{r.5t"tl. d l-a,rgt+qzz8o\-a@\TI

B,= grYryxY26

where all, ep, d22 are given by the following equations.

B r= BoYtYxVTg

J =(YV.Vf).VQ,=(*)'&(+)

a,, = (Y ry.Y ,26)2

an=(YQ-YW)',=##*Y=l'*#l#an=(YV.Y7).(QxVry)
3. MHD Stability Analysis of LHD
LHD does not carry a large plasma current and isStrong stabilizing effects is present against off-resonant
perturbations in high magnetic shear configuration.
Therefore, in LHD, the whole plasma column will not
be collapsed immediately, by an excitation of resonant

surrounded by chaotic field line region plasma, which isunstable mode at a rational surface.

Let us study the MHD stability of LHD plasmausing the potential energy I4z1 within the plasma, because

stabilized by line-tying effect.

3.2 Stability condition for a convective mode
Since the mode of m = n = 0 is not related to theresonance condition at any rational surface, the(Y€)' + €. vpvE
' |2J| lypL"

6w, =-(vx(6xr))'
Iro-GxJ).y xG'Bl]av

displacement of this instability grows up traversing theplasma column. Hence, we call this type of instability as.

convective instability.

For simplicity, we assume that the perturbation ( is localized on some rational surface (V = Vd and has wave

Stability condition for convective modes (m = n =0) is reduced to the following form:
489

Watanabe T. et al., The Marginally Stable Pressure Profile and a Possibility of High Beta Plasma Confinement in LHD

Irt\Z,r,R2

simplicity, we assume LHD vacuum magnetic field with1rolt) + p't + P'"rtr' fr {*} ,',.
B^B\'/| ,,n'\-'\rrb+ P'+ P-fr) \ro-"n)R* = 3.6 m and B"* = 2.75'1. Toroidal plasma current isassumed to be d = 0. Numerical examples of marginallystable pressure profile are shown in Fig. 3, together withe)where d(t1r) is the toroidal plasma current.

the distribution of the rotational transform and themagnetic hill profile. The plasma pressrue of the chaotic
field line region is assumed to be 0 or to be some finite

3.3 Numerical example of stable pressurevalue.

profile in LHD

4. Summary and Discussions

We show the numerical result for the marginallystable pressure profile given by eqs. (3)and (4). For

We have shown that the plasma pressure in LHD
can be sustained by the magnetic shear and by theplasma pressure in the chaotic field line region. Plasmapressure in the chaotic field line region is sustained
stably by the line-tying effect of the lines of force which
are disengaged from the chaotic field line region. Thispressure specifies the boundary condition for theequation of marginally stable pressure profile given byeq. (3) or eq. (4). The core plasma pressure can beboosted up by the plasma pressure in the chaotic field
line region as shown in Fig. 3. From these results andva"r"=roo u3, vcxros=2 r #, Rer3.6 M, BAx=2.?5 TclI:-the experimental results we may conclude that LHD has
a potentiality of high-beta plasma confinement.

3'o''t4'oStability analysis with finite beta equilibrium
magnetic field is the next step for high beta plasma

4'5", ^r,study of LHD. Global mode analysis is also important
for the high beta plasma stability.

Fig. 3 Numerical examples of marginally stable pressure
profile of LHD in the oblong cross section. Axis ofabscissas is the major radius F. Marginal pressure
profile for the helical (convective) mode is shownby P".,."* (P"o*u). The plasma pressure of the chaotic field line region (n.crsT.crs) is assumed to be 0or to be 5 x 101s x 5 (m-3.keV). LCFS specifies thepositions of the last closed flux surface. The distri-References

tll

T.Watanabe et al.,Kakuynugou Kenkyuu 68,298(1992).

tzl T. Watanabe et aI., J. Advanced Science ll,2ll(leee).

bution of the rotational transform tl2n and theprofile of magnetic hill U are also plotted. Thechaotic field line region is also shown in the fig-t3l B.R. Suydam, Proc. 2nd U.N. International Conf.
on Peaceful Uses of Atomic Energy, Geneva, 31,
157 (1958).

ure.

490