Ideal interchange instabilities instellarators with a magnetic hill
T. Tatsuno∗ , M. Wakatani
Graduate School of Energy Science, Kyoto University,
Kyoto, JapanK. Ichiguchi
National Institute for Fusion Science,
Oroshi-cho, Toki, Japan
Abstract. The prediction of the Mercier (Suydam) criterion for localized interchange modes in stellarators shows stability in cases in which global modes are unstable. One case is non-resonant pressure
driven instabilities with low mode numbers which become unstable even if the mode resonant surface
does not exist inside the plasma column. The other case is interchange instabilities when the pressure gradient vanishes at the mode resonant surface. If the pressure becomes flat in a narrow region
around the mode resonant surface, high mode number instabilities are suppressed and the beta limit
at the particular resonant surface increases. The radial mode structure at nearly marginal beta alsochanges significantly. The properties of the non-resonant mode and the transition from a resonant toa non-resonant mode are clarified with a cylindrical plasma model for a low shear stellarator with amagnetic hill.

1.

Introduction

Although the Mercier criterion is useful for investigating pressure driven instabilities in tokamaks[1] and stellarators [2], it does not predict thelimiting conditions in some cases within the ideal
MHD model. For deriving the Mercier criterion itis assumed that the unstable mode is radially localized near the mode resonant surface. There is a tendency for the radial mode structure to become narrow in the vicinity of the mode resonant surface withan increase of mode number. Even the interchange
mode with (m = 1, n = 1) has the property that theradial mode structure becomes highly localized near
the marginal regime [3], where m(n) is the poloidal(toroidal) mode number. This result explains why
the Mercier limit correlates with the beta limit due
to the interchange instabilities with low mode numbers [4, 5]. However, this situation changes substantially when the pressure gradient becomes locally flatat the mode resonant surface [6]. Details of the effectof pressure profile on the interchange modes will begiven in this article with the use of a cylindricalplasma model for a low shear stellarator with a magnetic hill.
∗

Present affiliation: Graduate School of Frontier Sciences, University of Tokyo, Tokyo, Japan. E-mail:tatsuno@plasma.q.t.u-tokyo.ac.jp

Nuclear Fusion, Vol. 39, No. 10

In order to destabilize the interchange mode, aresonant surface is not always necessary. It is reasonable that in a low shear region, with a steep pressure
gradient, non-resonant modes approximately satisfying the resonant condition are destabilized. First,
unstable non-resonant resistive modes were foundfor a Heliotron-E plasma with a highly peaked pressure profile [7]. Recently, ideal non-resonant modes
were shown to be unstable in the central regionof Heliotron-E [8], which seems consistent with the(m = 2, n = 1) mode triggering a sawtooth [9]. It isnoted that non-resonant modes usually have global
mode structures, which requires numerical analysis
to clarify their properties. For studying the detailsof ideal non-resonant instabilities we use a cylindrical plasma model which saves computational time
greatly. Since it is hard to excite a non-resonantmode in a high shear region, our interest is in a low
shear stellarator with a magnetic hill.
It is noted that Fu et al. [5] studied the relation
between Mercier modes and low-n modes with a full3-D stability code for l = 2 stellarators. They foundthat unstable localized low-n modes are correlatedwith the Mercier criterion. However, the stability of
global type low-n modes was found to be decorrelated from that of Mercier modes for the case with afairly large outward magnetic axis shift. It seems thatstrong poloidal coupling in the toroidal geometry isessential for this type of unstable mode, which may

c 1999, IAEA, Vienna

1391

T. Tatsuno et al.
be a tokamak type ballooning mode. In this article
our interest is in the decorrelation between the low-npressure driven modes and the Suydam modes in thecylindrical model. Thus, both the rotational transform and the pressure profiles are important here.
In Section 2, we derive an eigenmode equation
for studying linear interchange modes in stellarators, which is derived from the reduced MHD equations [10]. In Section 3, we first solve the eigenmode
equation analytically in the low shear limit, and discuss the non-resonant mode. Next, we solve the same
eigenvalue equation numerically for the finite shear
case in Section 4. Here we give examples to highlight various properties for both the resonant andnon-resonant modes. Finally, in Section 5, we summarize the results obtained and give some physicalinterpretations for the behaviour of the non-resonant
mode.

2.

Eigenmode equations

For analysing pressure driven instabilities in stellarators, we use the ideal reduced MHD equations[10], which are written as∂ψ= B · ∇u∂t(1)

d ∇2⊥ u= −B · ∇Jz + ∇Ω × ∇p · ez
dt

(2)

dp
=0
dt

(3)

ρ

∂
+ ∇ψ × ez · ∇
∂z

d
∂
=
+ ∇u × ez · ∇dt∂tι(r) ≡

R0 dψ0rB0 dr(10)

where the equilibrium poloidal flux function is givenby ψ0 (r). Since the correction due to the diamagnetic
current becomes higher order in this formulation, therotational transform in this approximation includes
only the vacuum helical field contribution.
For the stability analysis, we use the following normalization for the variables:ψ → aB0 ψ, u →aR0u, t → τPA t
τPA

p → p0 (r = 0)p, r → ar, Jz →

B0
Jz
µ0 a

(4)

R0
∇2 u, Az → aB0 Az(11)aτPA ⊥√
where τPA = R0 µ0 ρ/B0 denotes the poloidal
Alfvén time and a the minor radius of the plasma
column. Then the linearized reduced MHD equations
can be written as(5)γ(∇2⊥ u) = −∇2⊥ u →

whereB · ∇ = B0

where R0 denotes the major radius of the torus, r theminor radius, and θ and ζ = z/R0 the poloidal andtoroidal angle, respectively. Here a perfectly conducting wall is placed at the plasma boundary, and theboundary conditions are given by Br = ∂ψ/∂θ = 0,vr = ∂φ/∂θ = 0 and p = 0 at r = a.
In the following study, we neglect the toroidal
effect in the reduced MHD equations. We also assume
that the equilibrium quantities do not depend on thepoloidal angle θ. This assumption means that theflux surfaces have a circular cross-section in the large
aspect ratio limit. Then the rotational transform is
written asn − mι 2
Ds m2∇⊥ [(n − mι)u] −
u
γ
γ r2
(12)

2r cos θ (∇φ)2
Ω=
+
R0
B02

(6)

where Ds and the averaged helical curvature Ω areexpressed as

Jz = −∇2⊥ Az(7)

1∇hφi × ∇φ · ez .
Az = ψ +2B0(8)

β0 0 0
pΩ
22


Z
2
2
Ω =  N r ι + 2 rι dr .

Here ψ, u and Ω denote the poloidal flux function,
the stream function and the average curvature of themagnetic field, respectively. The quasi-toroidal coordinates are introduced and the metric is writtenasdl2 = dr2 + r2 dθ2 + (R0 + r cos θ)2 dζ 2
1392

(9)

Ds = −(13)(14)

Here  ≡ a/R0 denotes the inverse aspect ratio,
β0 ≡ 2µ0 p0 (r = 0)/B02 the central plasma beta
value and N the toroidal period number of thehelical field. In order to derive Eq. (12), all perturbed quantities are assumed to be proportional to
Nuclear Fusion, Vol. 39, No. 10 (1999)

Article: Ideal interchange instabilities in stellarators
exp[γt − i(mθ + nζ)]. In Eq. (13), the prime denotes
the derivative with respect to the normalized minor
radius r. The perpendicular Laplacian operator in
Eq. (12) is expressed as


1 d
d
m2
2
∇⊥ =
r
− 2.
(15)
r dr
dr
r
Then the ordinary differential equation (12) for the
stream function u with mode numbers (m, n) is written as
2mι0 (n − mι) du
d2 u
1
−
+
dr2
r γ 2 + (n − mι)2 dr
 2
m
1
−
+ 2
r2
γ + (n − mι)2


 0
mι
Ds m2
+ mι00 (n − mι) −
u=0
×
r
r2
(16)
which is an eigenmode equation with eigenvalue γ 2 .
For solving Eq. (16) the boundary condition at theplasma surface r = 1 is u = 0 under a fixed boundary
condition. We also impose regularity of the solution
at r = 0. With these boundary conditions, we canset up an eigenvalue problem for the eigenvalue or
growth rate γ 2 and the corresponding eigenfunctionu.

3.

Analytic solution of eigenmode

In this section we assume ι0 = 0 for obtaining ananalytic solution, then Eq. (16) is written as
d2 u 1 du m2
Ds+−1
u = 0.
+dr2r dr
r2 γ 2 + (n − mι)2

unstable mode has a global structure without localization in the radial direction unlike the resonant
mode. Further we notice that, when there is no magnetic shear, the radial mode structure, Jm (Z(m, k)r),
is not affected by the beta value. We notice from
Eq. (18) that the more unstable mode has the smaller
node number, and the eigenvalue is discrete withrespect to k for the specified (m, n). This property is
general and can be also shown even for the case withthe method given by Goedbloed and Sakanaka [11] by
writing the eigenmode equation (16) in the Sturmian
form (Appendix). Since the LHS of Eq. (18) is proportional to γ 2 and the RHS is linear with respect tothe plasma beta, the relation (18) gives a parabolicline in the (β, γ) plane. Thus a small variation in
β0 from marginal equilibrium may cause an abrupt
increase of growth.
The beta limit for stability is obtained by substituting γ 2 = 0 into Eq. (18), which yields
β0c =

γ2 =

D̃s m2− (n − mι)2Z 2 (m, k)(18)

where Z(m, k) is the kth zero point of the mth order
Bessel function of the first kind Jm (r̃).
Although the resonant surface does not exist
inside the plasma column, it is seen that the modesatisfying n ' mι is most unstable and that the
Nuclear Fusion, Vol. 39, No. 10 (1999)(19)

In order to examine the beta limit of the higher harmonic modes with the same helicity, we use the transformation of the variables (m, n) 7→ l(m, n), whichyieldslβ0c=

Z 2 (lm, k)(n − mι).
4N ιm2(20)

Since Z(lm, k) > Z(m, k) for l ≥ 2, the beta limit oflthe higher harmonic mode, β0c, is higher than thatfor the l = 1 case, β0c . This is different from theresonant modes with the same helicity, which give thesame beta limit given by the Suydam criterion [3].

4.

Numerical solutionof eigenmode equation

4.1.

Resonant and non-resonant modes
for standard pressure profiles(17)
For the parabolic pressure profile, p = p0 (1 − r2 ), theanalytic solution is readily obtained with the transformation r̃ ≡ {D̃s m2 /[γ 2 + (n − mι)2 ]}1/2 r, whereD̃s = 4β0 N ι. From the solution u ∝ Jm (r̃) for the(m, n) mode and the boundary condition u = 0 at
r = 1, the growth rate is written as

Z 2 (m, k)(n − mι).
4N ιm2

We have solved Eq. (16) numerically by the ‘shooting’ method using the fourth order Runge–Kutta formula. At first we tried the same eigenvalue problem
as that shown in Section 3 in order to validate thenumerical code. The growth rates obtained for the(m, n) = (2, 1) mode coincide well with the analytic
solution, Eq. (18), and the radial mode structuresdescribed by the Bessel function J2 (r̃) seem to beunchanged by variation of β0 .
Next we have investigated the effect of magnetic shear on non-resonant modes for the standard
1393

T. Tatsuno et al.
0.025

σ=0.05
σ=0.5
σ=2.0

(a)

β limit

0.02
0.015
0.01
0.005

(b)
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Shear Intensity σ

0

0.2

0.4
0.6
0.8
Normalized Radius

1

Figure 1. (a) Dependence of the beta limit on the magnetic shear parameter σ for the non-resonant(2, 1) mode. (b) Radial mode structures in the cases of σ = 0.05, 0.5 and 2.0 for a parabolic pressure
profile with β0 = 0.03.
6e-02

Growth Rate

5e-02

Resonant
Non-resonant

Resonant
Non-resonant

4e-02
3e-02
2e-02
1e-02

(a)

0e+00
0e+00 2e-03 4e-03 6e-03 8e-03 1e-02
β value

(b)
0

0.2

0.4
0.6
0.8
Normalized Radius

1

Figure 2. (a) Dependence of the growth rate of the (2, 1) mode on the central beta value β0 for
p = p0 (1 − r4 ). Diamonds denote numerical results. The white diamonds correspond to the resonantcase and the black diamonds to the non-resonant case. (b) Radial mode structures corresponding
to the resonant (β0 = 1.35 × 10−3 ) and non-resonant (β0 = 5.97 × 10−3 ) cases. Here the rotationaltransform profile is ι(r) = 0.499 + 0.2r 2 for the resonant case and ι(r) = 0.501 + 0.2r 2 for thenon-resonant case.

parabolic pressure profile. For the assumed rotationaltransform profile, ι = 0.51 + σr2 , σ is changed from0.05 to 2.0. The rotational transform profile in thecase of σ = 1.69 is approximately coincident withthat in Heliotron-E [8]. When the beta value is fixed,
the growth rate of the non-resonant (2, 1) mode isdecreased with the increase of the magnetic shear
intensity σ or the beta limit is increased almost linearly with the increase of σ, as shown in Fig. 1. The
radial mode structure is shifted to the inner region
when σ is increased (Fig. 1). This result can be interpreted in the following way. As σ is increased, there
are two effects. First, the magnetic shear becomeslarger in the outer region compared with the inner1394

region. Second, the outer region is removed further
away from the resonance than the inner region. These
effects may account for the mode structure becoming more localized towards the magnetic axis. Also
when β0 is decreased, since the destabilizing effectdue to the plasma pressure gradient becomes weak,
the non-resonant mode can be excited only in the
inner region. However, since there is no resonant surface, the radial mode structure is not highly localized and still has a global structure. The behaviourof the growth rate near the marginal beta value forthe non-resonant mode is different from that for the
resonant mode, as shown in Fig. 2. The growth rateof the non-resonant mode decreases to zero without
Nuclear Fusion, Vol. 39, No. 10 (1999)

Article: Ideal interchange instabilities in stellarators
the tail at β0 ' β0c , where β0c is the beta limit for
the non-resonant (2, 1) mode.
Here we study the transition from the resonantmode to the non-resonant one. For currentless plasmas in Heliotron-E, MHD equilibria show that thecentral rotational transform is increased with anincrease of beta value. When the vacuum rotationaltransform at the plasma centre is lower than 0.5,
a resonant surface for the (2, 1) mode exists inside
the plasma column. The resonant mode may not
be excited due to the low beta value at the initial state. Experimental results show that the (2, 1)
mode becomes unstable for β0 & 0.7% in neutralbeam heated plasmas, which leads to the occurrenceof sawtooth [8]. However, when ECRH is applied tothe central region, the pressure profile becomes more
peaked and the (2, 1) mode is stabilized. These data
could be understood with the disappearance of theι = 0.5 surface according to the increase of the central beta value. Linear stability of the ideal (2, 1)
mode in toroidal geometry shows that a resonantmode appears first, which then changes to a nonresonant mode with an increase of β0 . Finally, thenon-resonant mode becomes stable, when ι(0) deviates far from 0.5 [8].
In the cylindrical model we simulate the above
situation by changing the central value of the rotational transform artificially. To clarify the properties of the non-resonant mode, we consider a weak
shear configuration with the resonant surface forthe (2, 1) mode at first. Then we exclude the resonant surface of ι = 0.5 by increasing ι(0). Figure 2 shows numerical results for the pressure profile p = p0 (1 − r4 ). White diamonds correspondto the growth rates for the equilibria with rotational transform profile ι = 0.499 + 0.2r2 , which
has the resonant surface for the (2, 1) mode at thenormalized radius r ' 0.07. Black diamonds correspond to the growth rates for the equilibria with
ι = 0.501 + 0.2r2 , which has no resonant surface forthe (2, 1) mode. The beta limit for the resonantcase seems to be 1.14 × 10−3 or less, while for thenon-resonant mode it is 5.97 × 10−3 . The difference
between these beta limits correlates with the radialmode structure. In the small growth rate regime,
when β0 is decreased, the radial mode structure of
the resonant mode becomes more localized. Thus,a highly localized mode with an extremely small
growth rate is possible, as shown in Fig. 2. Thus,in β–γ space the line for the resonant mode caseextends to the lower beta region with small growth
rates. On the contrary, since the non-resonant mode
Nuclear Fusion, Vol. 39, No. 10 (1999)

cannot be localized at a particular surface, thegrowth rate decreases to zero without a tail with thedecrease of β0 .
We can apply the Suydam criterion to resonantmodes, which can be derived from the indicial equation of Eq. (16) at the singular point, or the resonantsurface. It is written as
Ds
1<
ι02 rs2
4(21)

for the stability case, where Ds and ι0 are evaluatedat the resonant surface, r = rs , for the corresponding mode. In the case of Fig. 2, the resonant surfaceof the (2, 1) mode is rs ' 0.07. Here the beta limit
obtained from the criterion (21) is β0 ' 1.05 × 10−3 .
Generally it is difficult to obtain the beta limit for
the resonant mode numerically. One reason is theextension of the growth rate to the low beta side as
mentioned above, and the other is the localizationof the mode structure in the vicinity of the resonantsurface. In Fig. 2, however, the difference between
the analytic evaluation and the numerical result is
less than 10%, and the growth rate at the numerically obtained beta limit is 4.49 × 10−11 , which isnormalized by the poloidal Alfvén time.
It is noted that global type modes in toroidal
stellarators are discussed in Ref. [5]. However, thesemodes are different from the non-resonant modes discussed here, since in toroidal geometry the poloidal
coupling is essential to destabilize global typemodes.
4.2.

Resonant modes
for locally flattened pressure profiles
at the resonant surface

Here we consider the equilibria with resonant surface at ι = 0.5 for the (2, 1) mode in the plasma
column, but without the pressure gradient on theresonant surface. In the experimental situation of
Heliotron-E, there may exist small magnetic islands
due to resistive interchange instabilities at the loworder resonant surfaces [12, 13], which may be nonlinearly saturated at low fluctuation levels. In sucha case the equilibrium may not be violated by theresistive mode; however, the local plasma profile
will change and the pressure gradient becomes small
near the resonant surface [8, 14]. For this situation
the Suydam criterion (21) predicts stability at theι = 0.5 surface. Here we will show that low-m modes
can be unstable due to the finite negative pressure
gradient at locations other than the resonant surface.
1395

T. Tatsuno et al.
For simplicity the pressure profile is assumed to
be
"

2 #
1 r − rs2
p = 1 − r + λ(r − rs ) exp −
(22)
2
W

1396

0.8
Pressure Profile

where rs is the position of the mode resonant surface,
and the choice λ = 2rs makes p0 vanish at r = rs . The
width of the flat region is controlled with the parameter W . Several pressure profiles given by Eq. (22)are shown in Fig. 3. We assume ι = 0.45 + 0.2r2 andconsider the (2, 1) mode again. The resonant surface
exists at rs = 0.5 where the pressure gradient vanishes.
For the three cases with W = 0, 0.01 and 0.1
shown in Fig. 3, growth rates of the (2, 1) mode areshown as a function of β0 in Fig. 4(a). Although thehighly localized mode structure is observed in thecase of W = 0, it is not localized even in the case of
W = 0.01, and the beta limit is increased by a factor of 2. Furthermore, in Fig. 4(a) the growth rate
decreases to zero without a tail near the beta limit
for W = 0.01, while the growth rate in the higher
beta regime is not affected. The growth rates andthe radial mode structures in the case of W = 0.1
are shown separately in Fig. 5, where both the firstgrowing mode with the maximum growth rate andthe second growing mode with the next growth rate
are shown. In Figs 4(b) and 5(b) we see that theradial mode structures are quite different from thecase with W = 0. They are restricted to one sideof the mode resonant surface and change sharply at
the mode resonant surface in the case of W 6= 0.
In order to understand the role of the second growing mode, it is interesting to study the non-linear
behaviour of the (2, 1) mode for an equilibrium witha flat pressure region in the neighbourhood of theresonant surface. It is considered that, since theaverage magnetic shear is weak on the inner sideof the resonant surface, the first growing mode isrestricted to the region [0, rs ] in the case of W = 0.01,
whereas in the case of W = 0.1, it is restricted tothe outer region since the average pressure gradientseems larger on the outer side. It is noted that thebeta limit of the W = 0.01 case, β0c = 1.0 × 10−3 ,is lower than that of the W = 0.1 case, β0c =2.7 × 10−3 . In both cases the second growing mode
appears in the opposite region to the first growingmode.
To investigate why the steep mode structureappears at the resonant surface, we expand the coefficients in Eq. (16) in the neighbourhood of themode resonant surface r = rs . Since the rotational

1

0.6
0.4
No Step
W=0.01
W=0.1

0.2
0
0

0.2

0.4

0.6

0.8

1

Normalized RadiusFigure 3. Pressure profiles given by Eq. (22) for λ = 1.
The effect of pressure flattening is very small and visible
only at r ' 0.5 for W = 0.01.

transform is expanded as ι(r) ≈ ι(rs ) + ι0 (rs )(r −rs ) + . . ., the resonant denominator is expressed asn − mι ≈ −mι0 (rs )(r − rs ) + . . . .

(23)

Since the pressure becomes flat at the mode resonantsurface, p0 (rs ) becomes zero, but p0 (r) is still negativeon both sides of the mode resonant surface. Therefore
p00 is also zero at r = rs , thus p0 is expanded in theneighbourhood of the mode resonant surface asp000 (rs )(24)(r − rs )2 + . . .
2where p000 (rs ) < 0. Substituting the leading terms of
Eqs (23) and (24) into Eq. (16) yields
2m2 ι02 (r − rs )
d2 u
1
du
+
+
dr2
r γ 2 + m2 ι02 (r − rs )2 dr
 2
 0

m
mι0 (r − rs )mι00
− 2 − 2+mιr
γ + m2 ι02 (r − rs )2r
m2 β0 N p000 (4rs ι + rs2 ι0 )
2
)
+ 2 2
(r
−
r
u = 0.
s
4rs [γ + m2 ι02 (r − rs )2 ]

p0 ≈

(25)
As seen here, the effect of the pressure near the resonant surface appears in higher order with respect tor − rs . Thus, the pressure is negligible and does notaffect the steep mode structure.
In order to confirm this situation, we have calculated the radial mode structure of the nearlymarginal mode for the following pressure profiles
numerically. One profile is1 (1 − 4r2 )2 + 0.5,
r < 0.52p=(26)
 1 [1 − 4(r − 0.5)2 ]2 , r > 0.5
2
Nuclear Fusion, Vol. 39, No. 10 (1999)

Article: Ideal interchange instabilities in stellarators
9e-02
7e-02
Growth Rate

No Step
W=0.01

No Step
W=0.01
W=0.1

8e-02
6e-02
5e-02
4e-02
3e-02
2e-02

(a)

1e-02

(b)

0e+00
0e+00 1e-03 2e-03 3e-03 4e-03 5e-03
β value

0

0.2

0.4
0.6
0.8
Normalized Radius

1

Figure 4. (a) Dependence of the growth rate of the (2, 1) mode on the central beta value β0 for
W = 0, 0.01 and 0.1. (b) Radial mode structures for W = 0 (β0 = 5.62 × 10−4 ) and W = 0.01
(β0 = 1.05 × 10−3 ). The radial mode structures for W = 0.1 are shown in Fig. 5.
0.45

1st Growing
2nd Growing

0.4

1st
2nd

(b)

Growth Rate

0.35
0.3
0.25
0.2
0.15
0.1

(a)

0.05
0
0

0.01 0.02 0.03 0.04 0.05 0.06
β value

0

0.2

0.4
0.6
0.8
Normalized Radius

1

Figure 5. (a) Dependence of the growth rate of the (2, 1) mode on the central beta value β0 for
W = 0.1. (b) Radial mode structure of the first growing mode (β0 = 2.72 × 10−3 ) and that of thesecond growing mode (β0 = 2.76 × 10−3 ). It is noted that the growth rates of the first growing mode
are the same as those in Fig. 4(a).

and the other is
 
2
1
25 2



1
−
r
+ 0.5,


4
 2
0.5,
p=


2



1
25
2


1−
(r − 0.6) ,2
4

r < 0.4
0.4 < r < 0.6(27)

r > 0.6.

The latter pressure profile contains a completely flat
region whose width is noted as δ in [0.4, 0.6] in order
to eliminate the effect of the pressure gradient. Those
profiles are shown in Fig. 6. By assuming the samerotational transform profile as in the previous case in
Figs 4 and 5, the obtained mode structures are shownin Fig. 7. The reason why the mode structure of thefirst growing mode is restricted to the inner regionis that, since the average pressure gradient is equal
Nuclear Fusion, Vol. 39, No. 10 (1999)

on both sides of r = 0.5, the interchange mode isconsidered to be excited in the weaker shear region.
It is interesting that the mode structure with the
sharp decrease at r = rs is observed even though thepressure is completely flat in a region with a finite
width around the mode resonant surface. This confirms our conjecture that the locally steep profile ofthe mode structure such as that in Figs 5 and 7 iscaused only by the profile of the magnetic shear, not
any more by the pressure profile. We note that a nonresonant feature is seen in the radial mode structure
for the second growing mode in Fig. 5(b) and the firstgrowing mode for the second pressure profile (27) inFig. 7(b), i.e. the peak is shifted from the resonantsurface. This clearly shows that the unstable mode isdriven by the negative pressure gradient at locations
1397

T. Tatsuno et al.
1

Pressure Profile

0.8

δ=0
δ=0.2

(a)

δ=0
δ=0.2

0.6
0.4
0.2
0
0

0.2

0.4
0.6
0.8
Normalized Radius

1

Figure 6. Pressure profiles with locally flat regions
around the mode resonant surface. The solid curve corresponds to Eq. (26) and the broken curve to Eq. (27).

0

0.2

(b)

0.4
0.6
0.8
Normalized Radius

1

δ=0
δ=0.2

elsewhere than the resonant surface. A theoreticalexplanation for the appearance of the sharp decrease
to zero at the resonant surface in the radial mode
structure or u(r) is given in the Appendix.
4.3.

Behaviour of non-resonant type modes

We will show that a non-resonant type mode isalso excited even if the pressure profiles do not haveexactly zero gradient at the resonant surface. For
small and non-zero values of p0 at r = rs , we discuss the transition from the resonant mode to the
non-resonant one. We assume p = p0 (1 − r2 )α , where
α is changed from 4 to 14 (Fig. 8). The profile of
the rotational transform is fixed as ι = 0.4 + 0.2r2 ,where the√ resonant surface for the (2, 1) mode exists
at rs = 2/2. Figure 8 shows that the mode structure gradually changes from a resonant one to a
non-resonant type one. Particularly the α = 14 caseshows that the peak of the radial mode structure
exists at a position different from the resonant surface, which is considered as a non-resonant feature. It
does have a step function structure instead of a peakat the mode resonant surface for the nearly marginal
beta value. In other words, the driving force to theinstability comes from the largest pressure gradientregion different from the resonant surface. From the
sharp decrease of u to zero at the resonant surface inthe α = 14 case, it is considered that the pressure has
almost no effect on the mode structure at the resonant surface, since p0 and p00 are negligibly small.
In the α = 10 case, the mode structure has maximum value at the resonant surface; however, there
exists another broad peak on the inner side of the1398

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Radius

(c)

δ=0
δ=0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized RadiusFigure 7. (a) Radial mode structures of the (2, 1) mode
for the pressure profiles corresponding to Eq. (26) (δ = 0
curves) and to Eq. (27) (δ = 0.2 curves). Both curves
show the radial mode structure of each first growing
mode. Here all the perturbed functions are shown: (a) ũ,(b) p̃, (c) ψ̃.

resonant surface. In addition, the growth rate vanishes without the tail near the beta limit (Fig. 8(b)).
In the α = 8 case, the situation is more ambiguous.
The mode structure has a maximum value at the resonant surface and has no other peak. However, thedependence of γ on β near the beta limit is different
Nuclear Fusion, Vol. 39, No. 10 (1999)

Article: Ideal interchange instabilities in stellarators
1
0.8
Pressure Profile

Table 1. Comparison between the marginal beta values
from the Suydam criterion, βS , and the ones shown in
Fig. 8(b), βn . The equilibrium pressure gradient at theresonant surface is also shown.
(The pressure is normalized by the central value and theradial variable by the minor radius of the plasma column.)

α=4
α=8
α=10
α=14

0.6
0.4
0.2

(a)
0
0
1.4e-01

0.4
0.6
0.8
Normalized Radius

1

α=4
α=8
α=10
α=14

1.2e-01
Growth Rate

0.2

1.0e-01
8.0e-026.0e-024.0e-022.0e-02(b)

0.0e+00
0.0e+00

5.0e-03β value

1.0e-02

α=4
α=8
α=10
α=14

0

0.2

(c)

0.4
0.6
0.8
Normalized Radius

1

Figure 8. (a) Pressure profiles given by p = p0 (1 − r )for α = 4, 8, 10 and 14. (b) Dependence of the growthrate on the central beta value β0 for different α. (c) Radialmode structures near the beta limit for different α. The
height of the mode structure is normalized to its own
maximum value.
2 αfrom the standard resonant mode. Thus, the properties of the cases with α = 8 and 10 seem to be intermediate between the resonant and non-resonant typemodes. For the α = 4 case, a clear feature of the resonant mode is seen, i.e. the small growth rate regime
is extended to the low beta side in β–γ space, and the
Nuclear Fusion, Vol. 39, No. 10 (1999)

α

βS

βn

p0

4
8
10
14

1.91 × 10−3
1.53 × 10−2
4.90 × 10−20.560

2.07 × 10−3
5.92 × 10−3
7.02 × 10−3
8.39 × 10−3−0.707−8.84 × 10−2
−2.76 × 10−2
−2.41 × 10−3

nearly marginal mode structure is highly localized at
the resonant surface. For comparison, the beta limit
given by the Suydam criterion is calculated for eachequilibrium in Fig. 8. Table 1 shows both the beta
limit obtained from the Suydam criterion (21) at theresonant surface of the (2, 1) mode, βS , and the one
shown in Fig. 8(b), βn . From Table 1, in the caseswith α ≥ 8, the non-resonant type (2, 1) modes are
unstable even when the central beta value is smaller
than the Suydam limit. The beta limit in the case of
α = 4 almost coincides with the Suydam limit since
the radial mode structure is highly localized aroundthe resonant surface.
Finally, we considered a reversed shear profile,which will be realized in the high beta equilibrium ofa toroidal stellarator. Here we assume a cylindricalplasma with ι0 < 0 in the central region and ι0 > 0 in
the outer region. We also assume the following profileof the rotational transform:"2 #1 r − rc2
ι = ι(0) + σr − λ exp −(28)
2W
where σ is the previously defined shear parameter,rc is the parameter for the minimum point of ι(r)and W denotes the characteristic width of the nonmonotonic region of ι(r). Here σ = 0.2, rc = 0.5,
W = 0.15 and λ = 0.2 are chosen as an example. The pressure profile is again assumed to beparabolic, p = p0 (1 − r2 ). We have calculated twocases which are parameterized as follows. One is the
double resonant case, ι(0) = 0.6, in which the radial
positions of the two resonant surfaces for the (2, 1)
mode are at 0.35 and 0.59, where the beta limits predicted from the Suydam criterion (21) are 6.61×10−3
and 8.25 × 10−3 , respectively. The other is the nonresonant case, ι(0) = 0.66, in which the rotationaltransform has its minimum value 0.508 at r = 0.478.
1399

Rotational Transform

T. Tatsuno et al.
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4

Double resonant
Non-resonant

Double resonant
Non-resonant

(a)
0

(b)
0.2

0.4
0.6
0.8
Normalized Radius

1

0

0.2

0.4
0.6
0.8
Normalized Radius

1

Figure 9. (a) Profiles of the rotational transform for ι(0) = 0.6, σ = 0.2, rc = 0.5, W = 0.15,
λ = 0.2 (double resonance) and ι(0) = 0.66, σ = 0.2, rc = 0.5, W = 0.15, λ = 0.2 (no resonance).
(b) Radial mode structure of the (2, 1) mode near the beta limit; β0 = 7.79 × 10−3 for the double
resonant case, or β0 = 4.87 × 10−4 for the non-resonant case.
0.45

β0=2.5×10-2
β0=2.0×10-2
β0=5.3×10-3

(3,2)

0.4
Growth Rate

0.35

(b)

0.3
0.25
0.2
0.15
0.1

(a)

0.05
0
0

0.01 0.02 0.03 0.04 0.05 0.06
β value

0

0.2

0.4
0.6
0.8
Normalized Radius

1

Figure 10. (a) Growth rate as a function of β0 . (b) Radial mode structures for β0 = 2.5 × 10−2 ,2.0 × 10−2 and 5.3 × 10−3 . The beta limit is estimated as 5.27 × 10−3 , which is lower than the
Suydam limit at the resonant surface of the (3, 2) mode.

The profiles of both the rotational transform and thenearly marginal mode structures are shown in Fig. 9.
In the double resonant case the radial mode structure is localized dominantly at the inner resonantsurface. The reason is that, since the beta limit fromthe Suydam criterion is lower at the inner resonantsurface than that at the outer one, the pressure
driven mode is more unstable at the inner resonantsurface. In the non-resonant case the radial mode
structure is restricted near the minimum point of
the rotational transform and the beta limit is much
lower than that in the double resonant case. It canbe interpreted that, since the pressure driven modeis excited near the minimum point of ι in the nonresonant case, which is fairly close to ι = 0.5, thestabilizing magnetic shear is very weak there. On the1400

other hand, the resonant mode is localized at the resonant surface where the magnetic shear is relatively
strong in the double resonant case, thus the beta
limit becomes higher than that in the non-resonantcase.
We have also calculated the (3, 2) mode in the nonresonant case with the rotational transform shownby the dashed curve in Fig. 9(a). This mode has one
resonant surface at rs = 0.691, where the beta limit
from the Suydam criterion is βS = 8.28 × 10−3 . The
numerical results are shown in Fig. 10. Figure 10(a)shows a transition in the growth rate dependingon β0 , which occurs at β0 ∼ 0.02. This transition is understood from the mode structures shownin Fig. 10(b). The maximum point of the unstablemode structure is located at the resonant surface for
Nuclear Fusion, Vol. 39, No. 10 (1999)

Article: Ideal interchange instabilities in stellarators
β0 & 0.025. However, it moves to the inner weak
shear region for the non-resonant type mode with
β0 . 0.02. This type of mode structure has a small
peak at rs near the beta limit; however, it does notdecrease to zero at the resonant surface as shown
previously, since the pressure gradient is not small
there.

5.

Concluding remarks

We have clarified the properties of the nonresonant pressure driven instabilities and their relation to the resonant instabilities in the cylindricalplasma model. The behaviour of the non-resonant
type mode depends strongly on the profile of both thepressure and the rotational transform. For some cases
the instability has a mixed character between the resonant and non-resonant modes. In addition, a transition from a resonant mode to a non-resonant typemode occurs when the pressure gradient is increasedin the central region or the pressure profile becomespeaked.
First we have solved the eigenmode equation analytically with respect to the perturbed stream function for an equilibrium with a constant rotationaltransform and a parabolic pressure profile. It is notedthat the non-resonant mode has a global structure,and the dependence of γ on β is parabolic (Eq. (18)).
In this case, it can be shown that the mode with thefewest node numbers has the larger growth rate, andthe higher harmonic mode with the same helicity has
the higher beta limit.
With the numerical calculations, it is shown thatthe growth rate of the non-resonant mode decreases
to zero without the tail near the beta limit, while
the resonant mode has a fairly wide small growth√rate regime expressed as γ ∝ exp[−const/ β0 − βS ][3], where βS denotes the central beta value givenby the Suydam criterion. A physical interpretation
is as follows. Although the resonant mode becomes
localized at the resonant surface with the decreaseof beta value, the non-resonant mode does not havesuch a surface in the plasma column. Therefore,
the free energy necessary to excite the non-resonantmodes is always finite, since the parallel wavenumber
along the magnetic field lines is also finite. Thus, thegrowth rate decreases to zero without the tail near
the beta limit. In the resonant case, since the higher
harmonic modes have larger poloidal and toroidal
wavenumbers than the fundamental mode, they canbe more localized in the radial direction. Thus, thegrowth rates at the same beta value are larger than
Nuclear Fusion, Vol. 39, No. 10 (1999)

that of the fundamental mode. However, all modes
can be highly localized at the resonant surface as thecentral beta value decreases, the beta limit does notdepend on the mode numbers and agrees with the
Suydam limit. On the contrary, in the non-resonantcase, since the parallel wavenumber of the higher harmonic mode becomes larger than that of the fundamental mode, the higher harmonics need moreenergy for excitation in the low beta regime. Thus,
the beta limit of the non-resonant mode with a higher
harmonic mode number is larger than that of the fundamental mode.
When the pressure profile becomes locally flattened with a width of W around the resonant surface, the resonant mode shows a non-resonant feature. The beta limit in this case is increased in thesmall flattening region. The marginal mode structureis quite different from the case with W = 0, i.e. it isrestricted to one side of the resonant surface and thegrowth rate decreases to zero without a tail whenβ0 approaches the marginal value. It is noted that
this non-resonant feature also appears in case of anon-zero but small pressure gradient at the resonantsurface.
In Heliotron-E, when the beta value is increased,
the central rotational transform is increased and the
profile becomes non-monotonic. We have studied thissituation by changing the rotational transform artificially. Even if there is no resonant surface for the(2, 1) mode, when the minimum of the rotationaltransform, ιmin , is close to 0.5, the non-resonant(2, 1) mode becomes unstable, which is independentof the Suydam criterion. When ιmin is less than 0.5,
the double resonant mode becomes unstable. In addition, the non-resonant type (3, 2) mode is unstablebelow the Suydam limit at the ι = 2/3 surface. In
this case a radial mode structure is observed in thecentral region when ι(0) is sufficiently close to 2/3.

Appendix
Behaviour of eigenfunctions
in a resonant layer witha negligibly small pressure gradient
By introducing the variable ξ = u/r, the eigenmode equation (16) is written in the Sturmian form
as


d
dξ
p(r)
− q(r)ξ = 0
(29)
dr
dr
1401

T. Tatsuno et al.

References

where
p(r) = r3 [γ 2 + (n − mι)2 ]
q(r) = r (m2 − 1)[γ 2 + (n − mι)2 ]+ (3mι0 r + mι00 r2 )(n − mι) − Ds m2 .
Here we consider a resonant layer satisfying |r−rs | =|x| ∼ . Since our interest is in the small growth rate
limit, γ ∼  is also assumed. Under these assumptions, if we assume p0 ' 0 in the resonant layer, thesecond term in Eq. (29) becomes negligible and


d
dξ
p(r)
=0
(30)
dr
dr
decides the behaviour of the eigenfunction ξ(r) in theresonant layer. It is noted that Eq. (30) is exactly
the same as that given by Rosenbluth, Dagazian and
Rutherford for the m = 1 internal kink mode in acylindrical tokamak [15]. They gave the solution



2
mι0
1
ξ = ξa 1 − tan−1
x
(31)
2
π
γ
for the boundary conditions ξ → ξa as x → −∞ andξ → 0 as x → ∞. Since mι0 /γ ∼ O(−1 ), the eigenfunction ξ has the largest gradient at r = rs andhas a step function structure near the resonant surface. Further, since u ' rs ξ in the neighbourhood of
the resonant layer, this type of solution may explain
the behaviour of the sharp decrease to zero of theeigenfunction with the largest growth rate near the
resonant surface r = rs for γ → 0.

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(Manuscript received 8 February 1999
Final manuscript accepted 5 August 1999)
E-mail address of M. Wakatani:wakatani@energy.kyoto-u.ac.jp
Subject classification: C0, St

1402

Nuclear Fusion, Vol. 39, No. 10 (1999)