PHYSICS OF PLASMASVOLUME 8, NUMBER 3

MARCH 2001

On the applicability of local asymptotic stability criteria
to stellarator stability
B. A. Carreras and V. E. Lynch
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

K. Ichiguchi
National Institute for Fusion Science, Toki, Japan

M. Wakatani
Graduate School of Energy Science, Kyoto University, Kyoto, Japan

T. Tatsuno
Tokyo University, Tokyo, Japan

共Received 22 November 2000; accepted 12 December 2000兲
An intrinsic consequence of the three-dimensional nature of the stellarator equilibrium may be theexistence of local flattening of the pressure profile at the resonant surfaces. This local flattening of
the pressure profile significantly changes the stability properties. The localized interchange modes
are stabilized, and a new instability branch controls the critical beta. This instability is strongly
stabilized by shear at high poloidal mode numbers. As a consequence, the plasma stability properties
change, and the asymptotically derived local stability criteria often used in stellarator design are nolonger applicable. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1349877兴

I. INTRODUCTION

singular surfaces. In contrast, in dynamical calculations ofequilibria unstable to resistive interchanges, we have seen
the formation of flat spots at the resonant surfaces even forvery low values of beta. In those calculations, we have observed a delicate interplay between resistive and ideal interchange modes. The first causes the local flattening of thepressure profile, which causes a modification of the stability
threshold for the ideal modes. Finally, in experiment, highresolution electron temperature and density measurements in
TJ-II 共Ref. 8兲 show the existence of multiple structures that
may be related to the resonant surfaces. Therefore, it is reasonable to assume that the pressure profiles in stellarators
have a complex structure with zero-gradient at each rationalsurface. The size of these flat spots can be very small, buteven in such cases their presence has important consequences for stellarator stability.
In this paper, we pursue this idea further and investigate
the linear stability of ideal interchange modes in the cylindrical geometry for plasmas with zero pressure gradient at
the resonant surfaces. An analytical expression of the linear
growth rate is derived and compared to numerical results.
From these analytical results, we show that the ␤ limits may
be substantially increased over the estimates made withasymptotic local criteria evaluated with smooth pressure profiles.
The rest of the paper is organized as follows: In Sec. II,
we introduce the equations used in the linear stability studies. The linear stability theory for pressure profiles with local
flattening at the resonance surfaces is discussed in Sec. III. In
Sec. IV, we discuss the effect of magnetic shear on these
instabilities. The implications of these results for theasymptotic stability criteria are presented in Sec. V. Finally,
the conclusions of this paper are given in Sec. VI.

Theoretically, ideal interchange modes should play arole in limiting the accessible beta 共␤兲 values in stellarators.
This instability is particularly important for stellarators witha magnetic hill in the outer plasma region, such as the torsatron and heliotron configurations. In these configurations,
low-n interchange instabilities may be defining the operational range of the device.1 The Mercier criterion2 gives agood estimate of the stability boundaries of both high-n andlow-n ideal interchange modes.3 However, it has been foundthat the limits given by the Mercier criterion are violated insome experiments.4 To explain this disagreement, it may beargued that finite Larmor radius and/or kinetic effects probably stabilize the high-n modes. However, the issue of thestability of the low-n interchange modes remains.
To understand the cause of the discrepancy between
theory and experiment, we turn our attention to the pressure
profiles used in the estimates of stellarator stability. There
are many reasons to suspect that pressure profiles with zero
gradient at the rational surfaces may be the relevant profiles
for stellarators. The existence of a three-dimensional 共3D兲
toroidal equilibrium is still an unresolved mathematicalproblem.5 Of course, numerical solutions of the equilibrium
magnetohydrodynamic 共MHD兲 equations are commonly calculated. However, these solutions may just be weak solutionsof these equations.6 If the 3D equilibrium exists, it may havemagnetic structures 共magnetic islands and/or stochastic regions兲 around the rational surfaces that increase transport in
those regions; this increased transport will naturally lead to a
local decrease of the pressure gradient. Other arguments canbe made for such a pressure profile. From the perspective of
having smooth particle fluxes in a 3D equilibrium, Boozer7
suggested that the pressure-gradient should be zero at the1070-664X/2001/8(3)/990/7/$18.00

990© 2001 American Institute of Physics

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Phys. Plasmas, Vol. 8, No. 3, March 2001

On the applicability of local asymptotic stability criteria . . .

991

II. STABILITY MODEL

Interchange modes, resistive and ideal, extend uniformly
along the magnetic field lines. They are flutelike instabilities.
Therefore, for these instabilities it is possible to average over
the toroidal magnetic field modulation induced by the helicalwindings. Using the Greene and Johnson formalism9 and assuming a straight helical system, the averaged equilibrium
magnetic field geometry has cylindrical symmetry. In thissystem, the magnetic field line curvature is given by the averaged magnetic field line curvature,

␬⬅

r 2
d⍀
⫽
B V ⬙,
dr R 0 0

共1兲

where prime indicates the derivative with respect to the toroidal flux, and V ⬘ ⫽ 兰 dl/B is the specific volume enclosed by
a flux surface. In Eq. 共1兲, R 0 is the major radius of the stellarator, r the averaged minor radius of a flux surface, and B 0
the toroidal magnetic field at the magnetic axis.
We use a reduced set of MHD equations to describe theideal interchange stability properties. The geometry is cylindrical with minor radius a and length L 0 ⫽2 ␲ R 0 , and thecylindrical coordinates are r, ␪, and z. The reduced set of
MHD equations consists of the poloidal magnetic flux evolution equation,

⳵␺⫽⫺R 0 ⵜ 储 ⌽;⳵t共2兲

the perpendicular momentum balance equation,

␳m

⳵ Ũ
⫽⫺ ␳ m V⬜ •ⵜŨ⫺ⵜ 储 J 储 ⫹z• 共 “⍀⫻“ P 兲 ;⳵t

共3兲and the equation of state,

⳵ p̃
dp eq
⫽⫺Ṽ⬜ •ⵜp̃⫺Ṽ ⫹D⬜ ⵜ⬜2 p̃.
⳵t
dr r

共4兲

Here, p is the pressure, V⬜ is the perpendicular flow velocity,
U is the z-component of the vorticity, ␺ is the poloidal magnetic flux function, and ␳ m is the mass density. The totalmagnetic field can be expressed in terms of the poloidal flux
function asB⫽⫺ 共 ⵜ ␺ ⫻ẑ 兲 /R 0 ⫹B 0 ẑ;共5兲and the perpendicular flow velocity can be expressed interms of a stream function ⌽/B 0 asV⬜ ⫽ 共 ⵜ⌽⫻ẑ 兲 /B 0 .

共6兲Here, ẑ is the unit vector in the toroidal direction. The
z-component of the vorticity can be expressed in terms of thevelocity stream function by Ũ⫽ⵜ⬜2 ⌽̃. The parallel derivative to the magnetic field, ⵜ 储 , is defined as ⵜ 储 f ⫽B•“ f .
In Eqs. 共2兲–共4兲, a tilde identifies perturbed quantities,
and the subindex eq identifies equilibrium quantities. For the
linear stability calculations presented here, each equation has
been linearized in the perturbed quantities.
The driving term of the interchange instability is thepressure gradient in the bad curvature region ( ␬ ⬎0). Thatis, these modes are driven by ⫺ ␬ (dp eq /dr)⬎0. The second

FIG. 1. Linear growth rate of the 共m⫽6; n⫽3兲 mode for an equilibriumwith parabolic pressure profile and ␫ ⫽0.32(1⫹2.2r 2 ⫺0.46r 4 ⫹2.5r 6 ).

term on the right-hand side of Eq. 共2兲 is the field-line bending term, which is responsible for the magnetic shear stabilization effect.
For the numerical calculations, we use the averaged curvature of a helically symmetric system,d⍀⫽␧ 2 M 共 4r ␫ ⫹r 2 ␫ ⬘ 兲 .
dr

共7兲

Here, ␧ is the inverse aspect ratio, M is the number of toroidal field periods, and ␫ is the rotational transform. In calculating the stability properties, we use the linearized form ofEqs. 共2兲–共4兲 and calculate the linear growth rate at different
␤ values. The threshold ␤ is obtained by extrapolating to azero linear growth rate. In all calculations presented here, wehave chosen a configuration with M ⫽20 and ␧⫽0.1.
III. LINEAR STABILITY PROPERTIES OF LOW-N
INTERCHANGE MODES FOR PRESSURE PROFILESWITH ZERO GRADIENT AT THE SINGULAR
SURFACES

In studying the stability properties of the pressure profiles with zero gradient at the rational surfaces, we begin
considering a smooth pressure profile. For a smooth pressure
profile, like p 0 (r)⫽ p̄ 0 (1⫺r 2 ), the linear growth rate of aninterchange mode as a function of ␤ has been plotted in Fig.
1 共continuous line兲. For this instability the eigenfunction is
sharply localized at and symmetric with respect to the singular surfaces 共Fig. 2兲.
To investigate the changes in the stability properties
when we consider a pressure profile, p(r), which is likep 0 (r) but with zero gradient at the resonant surfaces, we
modify the pressure profile in the following way:
p 共 r 兲 ⫽ p 0共 r 兲 ⫺冋兺m 共 d p/dr 兲 兩 r⫽r 关共 r 2 ⫺r m2 兲 / 共 2r m 兲兴

⫻exp ⫺m

共 r⫺r m 兲 2
2
2W m

册

,

共8兲

where W m is a measure of the size of the flat spot. The index
m refers to the different resonant surfaces considered in thecalculation. In studying the linear stability properties, we canstart with one singular surface. For the rotational transform

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992

Carreras et al.

Phys. Plasmas, Vol. 8, No. 3, March 2001

FIG. 2. Eigenfunctions for the m⫽6 localized interchange mode for asmooth parabolic pressure profile.

profile considered, we will carry out most of the calculations
at the ␫ ⫽0.5 surface located at the radius r s ⫽0.497.
Even for very small values of W m , there is a qualitativechange of the stability properties. For instance, Fig. 1 shows
the linear growth of the instability 共broken line兲 after modifying the pressure profile as described by Eq. 共8兲 withW m /a⫽0.004. The result is that the instability threshold has
increased by more than 60%, and the form of the eigenfunction has changed. The localized interchange instability
branch 共Fig. 2兲 is stable, and two other types of modes are
now the fastest growing modes 共Fig. 3兲. Their eigenfunctions
are mirror-symmetric with respect the resonant surface,which is the one with the largest growth rate depends on
details of the profiles and the exact location of the flattening.
Some of these changes in the stability properties of cylindrical plasmas have already been discussed elsewhere.10–13
Here we will derive an analytic form for the dispersion relation and test the analytical results with numerical calculations. In Fig. 1, the linear stability results are for the 共m⫽6; n⫽3兲 mode. The lower-m (m⬍3) radially symmetricinterchange modes may require a larger flat spot for fullstabilization.10 Note that the modification of the pressure
profile by local flat spots with a width of W m /a⫽0.004 is
hardly noticeable. For instance, Fig. 4 shows a parabolicpressure profile modified in the way indicated by Eq. 共8兲 at

FIG. 3. Eigenfunctions for the m⫽6 mode for the most likely unstablemodes once the localized interchange mode has been stabilized.

FIG. 4. Parabolic pressure profile modified in the way indicated by Eq. 共8兲
at the 15 lowest rational surfaces.

the 15 lowest rational surfaces. Such a modification of the
profile would require very high-resolution diagnostics to be
detected in an experiment. Of course, in reality W m may belarger than 0.004, and such detection may be possible. Low
aspect ratio devices may be the most suitable ones for theseexperiments. The linear stability calculations also requirevery high resolution, here, we have used a radial grid of⌬r/a⫽4⫻10⫺5 .
To understand the changes on stability properties, wewill derive an analytic form for the dispersion relation of thisnew instability branch under some simplifying assumptions.
From the set of reduced MHD equations, one can derive theeigenfunction equation for a mode (m;n),

冋
冋冉

册 再

1
1 2m ␫ ⬘ 共 n⫺m ␫ 兲 d⌽
m2
d 2⌽
⫺ 2
⫺ 2⫹ 22 ⫹2drr ␥ ⫹ 共 n⫺m ␫ 兲 drr
␥ ⫹ 共 n⫺m ␫ 兲 2
⫻

冊

册

m␫⬘
D sm 2⫹m ␫ ⬙ 共 n⫺m ␫ 兲 ⫺ 2 ⌽⫽0.
r
r

共9兲

Here, ␥ is the linear growth rate of the (m;n) mode and theeigenvalue for this problem, and D s isD s ⫽⫺

␤ 0 d p d⍀,2␧ 2 dr drwith ␤ 0 ⫽2 ␮ 0 p(0)/B 2 .
An analytical solution for the linear stability problem fora pressure profile with zero gradient at the resonant surface
can be found by dividing the minor radius in three regions asshown in Fig. 5. First we consider the case in which the
eigenfunction is zero in the outer region of the radius 共Fig. 3,
broken line兲. One region is between the magnetic axis andr s ⫺W/2, where r s is the radial position of the singular surface. In this region, we assume a constant rotational transform, ␫ 0 , and parabolic pressure profile. A second region ofwidth W is centered at the singular surface, and we assume
that the profiles have zero pressure gradient and small magnetic shear. In the outer region, the values of the parameters
do not matter because the eigenfunction is taken to be zero,⌽ III(r)⫽0.
In region I, the eigenfunction equation, Eq. 共9兲, simplifies to

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Phys. Plasmas, Vol. 8, No. 3, March 2001

On the applicability of local asymptotic stability criteria . . .

FIG. 5. Pressure and rotational transform profile used in the analytical determination of the dispersion relation Eq. 共15兲.

FIG. 6. Square of the linear growth rate of the 共m⫽6; n⫽3兲 mode vs ␤ fordifferent values of W. Numerical results are compared to the analytical
calculation. The equilibrium parameters are given in Fig. 5.

冋册

d 2 ⌽ 1 d⌽ m 2
Ds⫹ 2 2⫺1 ⌽⫽0.
2 ⫹drr drr ␥ ⫹ 共 n⫺m ␫ 兲 2

共10兲

For the smooth pressure profile p 0 (r)⫽ p̄ 0 关 1⫺(r/r 0 ) 2 兴 , we have D s ⫽(r/r 0 ) 2 D̂ s with D̂ s ⫽4 ␤ 0 M ␫ (0).
In this case, Eq. 共10兲 can be transformed into a Bessel equation, and the solution in region I verifying the origin boundary condition is⌽ I共 r 兲 ⫽␭ 1 J m 共 ␮ r 兲 ,993

we still have Eq. 共15兲 for the linear growth rate but substitutez ms for ẑ m , where ẑ m is a weak function of beta.
If the deviation from the z m1 value, ⌬z⬇ẑ m ⫺z m1 , issmall, we can estimate the correction by linearizing the transcendental equation in ⌬z. We obtained
⌬z⫽

共11兲

2
Az m1 冑B⫺z m1

2
2
2
A 共 4B⫺5z m1⫺ 共 1⫹z m1⫺m 2 兲 /z m1
兲 / 冑B⫺z m1

,

with
D̂ s
m2
␮ 2⫽ 2 2.
r 0 ␥ ⫹ 共 n⫺m ␫ 0 兲 2

共12兲

In region II, the eigenvalue problem near the marginal
point reduces to the Rosenbluth, Dagazian, and Rutherford
solution14 for the internal kink mode:⌽ II共 r 兲 ⫽␭ 2

冋冏 冏

再

2
r
m␫⬘
1⫺ tan⫺1
共 r⫺r s 兲
2
␲
␥

册冎

.

共13兲

For rⰇr s , ⌽ II⬇0, and the solution in region II matches
the outer region solution. For WⰇ ␥ /m 兩 ␫ ⬘ 兩 , ⌽ II→␭ 2 r. In thiscase, we can match this asymptotic form to the inner region
solution. From this match, we obtain a simple eigenvalue
condition

␮ 共 r s ⫺W/2兲 ⫽z ms .

共14兲From the eigenvalue condition, we can obtain the dispersion relation,␥ 2 ⫽m 2

冋冉冊 冉 冊册

2r s ⫺W/2 2nD̂ s ⫺ ⫺ ␫ 0 .
r 0 z ms
m

共15兲

⬘ (z)
Here, z ms is the s-zero of the function F m (z)⬅zJ m⬘ , where j ms
⬘ is the s-zero of⫺J m (z). For large m,z ms ⬇ j ms
the derivative of the J m Bessel function of order m. Note that⬘ scales as m⫹0.81m 1/3 for large m. Because the first zero
j ms
gives the highest growth rate, we will only consider this zeroin what follows.
For small values of the flattening region W, we cannot
take ⌽ II⬇␭ 2 r. We have to use the full expression in Eq.
共14兲. In this case, the eigenvalue condition is obtained fromthe solution of a transcendental equation. This implies that

共16兲

where
A⫽

2

共 m ␫ 0 ⫺n 兲 2

冋册
r0

2

W ␲ D̂ s 共 r s ⫺W/2兲 m

共17兲

and
B⫽

冉冊

m 2 共 r s ⫺W/2兲 2
D̂ .
r 0 共 m ␫ 0 ⫺n 兲 2 s

To test the analytical results, we have used a rotational transform profile of the form shown in Fig. 5. The rotationaltransform is ␫ ⫽0.32(1⫹2.2r 2 ⫺0.46r 4 ⫹2.5r 6 ) for r⭓r s⫺W/2 and constant for r⭐r s ⫺W/2. Therefore, this profile iscompletely flat in the inner region of the plasma. The pressure profile used is parabolic with a flat region at the ␫⫽0.5 resonance surface.
When the size of the flat spot is very small, of the orderof 1% of the radius, we have to solve the full transcendental
equation obtained from the matching condition to get a good
agreement between the analytical and numerical results. The
result is illustrated in Fig. 6 for the 共m⫽6; n⫽3兲 mode anddifferent values of the width of the flat spot.
For a width of the order of 10% of the minor radius, we
can use directly the dispersion relation Eq. 共15兲 with thecorrection ⌬z calculated by Eq. 共17兲. In this situation we getgood agreement with the numerical results again. Figure 7
shows the linear growth rate of the 共m⫽6; n⫽3兲 for W⫽0.1a as a function of beta and is compared with the dispersion relation, Eq. 共15兲, with and without the correction⌬z.

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994

Carreras et al.

Phys. Plasmas, Vol. 8, No. 3, March 2001

FIG. 7. Square of the linear growth rate of the 共m⫽6; n⫽3兲 mode vs ␤ forW⫽0.1a. Numerical results are compared to the analytical calculation given
by Eq. 共15兲 with and without correction terms. The equilibrium parameters
are given in Fig. 5.

Equation 共15兲 indicates that at a constant beta, the linear
growth rate is strongly stabilized by flattening the pressure
profile. Relatively small values of W/a can cause full stabilization of these modes,冋W⫽2 r s ⫺r 0 z m1 兩 ␫ 0 ⫺n/m 兩D̂ s1/2

册

.

共18兲

A similar analysis can be followed to find the solution,
which is zero in the inner radial region. However, in this caseusing the constant ␫ approximation in the outer region of theplasma is less credible. Therefore, we will limit our study tothe branch with zero eigenfunction in the outer region of theplasma.
IV. EFFECT OF MAGNETIC SHEAR IN THE STABILITYRESULTS

The analytical calculations show that near the stability
threshold, the linear growth rate goes to zero by cancellation
between the two terms in the right-hand side 共RHS兲 of Eq.
共15兲, the pressure gradient drive and the shear stabilizing
term. Therefore, its value is very sensitive to the value ofeach of these two terms. In general, Eq. 共15兲 can be used as
guidance for the behavior of the growth rate, but it cannot be
used as a quantitative measure of the growth rate in the caseof a rotational transform profile with shear. Even in a case ofa very low shear profile, the interpretation of the linear
growth rate on the bases of the analytical is difficult. It isdifficult to know what is the proper value to use for ␫ 0 when
we compare with the numerical results for a realistic ␫ profile
with shear.
Let us consider the ␫ ⫽0.5 resonant surface for the rotational transform profile ␫ ⫽0.32(1⫹2.2r 2 ⫺0.46r 4 ⫹2.5r 6 ).
This is the same equilibrium considered before but without
the flattening of the rotational transform profile in the inner
region. The change of transform is small because the shear in
the inner region is about 1. However, there are significant
changes in the stability properties. The beta critical increases
due to the shear stabilization effect 关second term in the right-

FIG. 8. Linear growth rate of the 共m⫽2; n⫽1兲, 共m⫽6; n⫽3兲, and 共m⫽10; n⫽5兲 modes as a function of ␤ for an equilibrium with parabolicpressure profile with a flat spot of W⫽0.004a and ␫ ⫽0.32(1⫹2.2r 2⫺0.46r 4 ⫹2.5r 6 ).

hand side of Eq. 共15兲兴. In Fig. 8, the linear growth rate vsbeta for different modes and with W⫽0.004 is plotted. The
linear growth rate squared is no longer a linear function ofbeta. The reason for this change is that the type of instabilities discussed in the previous section is only relevant near
the threshold. As beta increases, the modes broaden and become global modes 共Fig. 9兲, and the dispersion relation Eq.
共15兲 is not applicable to these modes. However, the important issue is that the instability threshold is still controlled by
the same type of instability we just discussed.
Also with magnetic shear, by increasing W, one canachieve total stabilization of the pressure-driven modes. In
Fig. 10, this effect is shown for the 共m⫽6; n⫽3兲 and forseveral beta values. The size of W required increases withbeta as indicated by Eq. 共18兲.
The changes in the stability properties for the pressure
profiles with flat spots at the resonant surfaces are not limited
to cylindrical geometry. The averaging method has been also
used to study the stability properties of a realistic stellarator
configuration with pressure profiles with zero gradient at theresonant surfaces. Once the toroidal couplings are also included in the model, one obtains similar changes in the stability properties.12–15

FIG. 9. Eigenfunction of the 共m⫽6; n⫽3兲 mode for different values of ␤.
These results correspond to the same parameters used in Fig. 8.

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Phys. Plasmas, Vol. 8, No. 3, March 2001

FIG. 10. Linear growth rate of the 共m⫽6; n⫽3兲 mode as a function of W
for different values of ␤. These results correspond to the same parameters
used in Fig. 8.

V. ASYMPTOTIC BEHAVIOR AND BETA CRITICAL

In looking for the asymptotic behavior of the linear
growth rate with m, we can see from Eq. 共15兲 that the growthrate dependence with m is quite different from the dependence of the local ideal interchange instability. For the latter,the linear growth rate is very weakly dependent on m. However, in the case of pressure profiles with local flattenings,
the high m modes are strongly suppressed. In Fig. 11 and forthe same equilibrium parameters as the cases in Fig. 8, weshow the linear growth as a function of m. These calculations
have been done with W⫽0.004a. In Fig. 11, we can see thedecrease in the growth rates with increasing m and the totalstabilization of the high-m modes. They are contrasted withthe behavior of the linear growth rate in case of a smoothprofile with W⫽0. Hence, the large m asymptotic stability
criteria cannot have any information on the stability of themodes described by Eq. 共15兲. Furthermore, those criteria became undefined in the case of a zero pressure gradient ineach rational surface. Therefore, for those pressure profiles
the asymptotic local stability criteria cannot be applied.
From Eq. 共15兲 we can calculate the corresponding critical beta for these new instabilities. For fixed W, the criticalbeta value scales as m 2 because z m1 ⬀m. Therefore, the beta
critical at a given surface is given by the beta critical for thelowest m value,

On the applicability of local asymptotic stability criteria . . .

995

FIG. 12. ␤ c as a function of m for different values of W. These resultscorrespond to the same parameters used in Fig. 8.

␤ c0 ⫽

冉冊冉 冊

2
r 0z m01 2 n
1⫺␫0 .
4M ␫ 0 r s ⫺W/2m

共19兲

Here m 0 is the lowest m value at the resonant surface considered. For an equilibrium with magnetic shear, the beta
critical scales qualitatively in a similar way as the zero-shear
case 共Fig. 12兲. That is, increases with m. However, the scaling exponent is no longer 2, but somewhat smaller.
If we now take the smooth profile, p 0 (r), corresponding
to the pressure profile with flat spots, p(r), we can calculate
the critical beta given by the Suydam criterion,
c
␤ 0s
⫽

共 ␫ s⬘ r 0 兲 2

16M ␫ 0,

共20兲

where we have taken ␫ s⬘ ⬅d ␫ /dr 兩 r⫽r s . The form is similar tothe one given by Eq. 共19兲. We can compare the real betaccritical, Eq. 共19兲, with the Suydam beta critical, ␤ 0S
. If wetake ␫ 0 ⬇ ␫ (r s )⫹r s ␫ s⬘ , we obtainc␤ c0 ⬇ 共 2z m 0 1 兲 2 ␤ 0s.

共21兲

We see that the real beta critical can be an order ofmagnitude higher than the critical beta from the Suydamccalculated with a smooth pressure profile.
criterion ␤ 0S
Therefore, local stability criteria like the Suydam criterion calculated with a smooth pressure profile cannot be applied to configurations with a pressure profile with zero gradient at each rational surface. Such criteria may be used as ameasure of stability if a convenient normalization constantcan be found.
VI. CONCLUSIONS

FIG. 11. Linear growth rate as a function of m for different values of ␤.
These results correspond to the same parameters used in Fig. 8.

For stellarator equilibrium with zero-pressure gradient at
the rational surfaces local, asymptotic stability criteria cannot be applied. For stellarators, the Mercier criterion has thesame problems as the Suydam criterion in cylindrical geometry. It is a local stability criterion that cannot be applied tosuch pressure profiles. Calculations using the averaged
method approach indicate that the stability properties for the
low-m modes12–15 are similar to the case of cylindrical geometry. The local interchange-like modes are stabilized, andthe more global type eigenfunction are the residual instabili-

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996

ties. The beta critical also increases over the one obtained forsmooth pressure profiles. These results may explain the apparent violation of this criterion when smooth pressure profiles are used in calculation of the stability for interpretationof the experimental measurements.4
ACKNOWLEDGMENT

This research was sponsored in part by Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the
U.S. Department of Energy under Contract No. DE-AC0500OR22725.
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