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High-beta stability of a toroidal plasma
Volume 84A, number 3
PHYSICS LETTERS
20 July 1981
HIGH-BETA STABILITY OF A TOROIDAL PLASMA L. SUGIYAMA and J. W.-K. MARK1 Massachusetts Institute of Technology, Cambridge, MA
02139, USA
Received 10 December 1980 Revised manuscript received 18 April 1981
The high toroidal mode number ideal MHD ballooning mode is shown to stabilize again at large beta, in several sequences of numerical toroidal equilibria. Raising the q profile improves high-beta stability for equilibria with the same p0-
loidal beta and flux surface geometry. A smaller aspect ratio enhances the stabilizing effect of raising the q profile and allows second stability to occur at smaller poloidal beta, though larger total beta.
1. Introduction. In an earlier work [1] it was shown that ideal MHD ballooning modes exhibit a second critical value of 13 (the ratio of the volume-averaged plasma pressure to the vacuum magnetic pressure
~,(x,t)
= ~(i/i, ~
(1) [
X exp[iwt
/ +
S~(
in
0~
—
f q2(iji,
s~)ds~
at the axis) beyond which the plasma is again stable. The existence of this second region of stability was
where
originally discussed using a model equilibrium config-
magnetic surfaces are represented by i/i
uration where the flux surfaces were represented by shifted circles but the main physical effects on the mode were included the increase of the poloidal magnetic field toward the outer edge of the plasma colunm and the dependence of the rate of magnetic shear on poloidal angle. Subsequent numerical studies [2,3] have confirmed the existence of the second stable region for numerical models of toroidal equilibria and experimental studies have been interpreted as evidence for a second stable region [4]. Here we investigate the effects of varying the q profile and the aspect ratio on the second stable
is the poloidal arclength along a flux contour, and ~ is the toroidal angle of symmetry. These coordinates are characterized by the line elements h1~dli = RB~~ ds~,and h~d~ = R d~, where R is the distance from the axis of symmetry and B~is the poloidal magnetic field. The local pitch of the field lines is defined to be qQ = d~/th~, where the derivative is taken along the field line. To first order in an expansion in 1/n0, the equations governing MHD ballooning equations can be written as [5,6] 2IVSI2 T — B~ ~ IVSI2B- VT + K T —pw B2 4irB2 B2 ~
—
2. The normal mode equations. The perturbed plas ma displacement is represented in normal mode form
(~4’,s,~, ~)
w
~ =
are curvilinear coordinates in which =
constant, 5~
(2)
T
B2 as
_ypB.VB~ç_(1+13)W=13K~T,
(3)
pwB 1
Present address: University of California, Lawrence Livermore Laboratories, Livermore, CA 94550, USA.
where KT = (B- K X
VS)/21T,
ture of the magnetic field B,
K being the normal curvai~,= (B X VS Vp)/B2,
123
Volume 84A, number 3
PHYSICS LETTERS
and f3 = 4ir~/B2is a local beta. Here T/B is related to the normal component of the displacement ~, and the quantity W is proportional to ‘)p(V~and measures the compressibility. At marginal stability the most unstable mode is incompressible and W = W(14~)= 0 to lowest order in 1/n 0. The set of equations then reduces to the well-known ballooning equation, which we write in the form R~r2B~
a 1 + ~2 ~
~2
(d~i/dr)2
~XR2B~ as~ (4) GR
÷(_r21 + ~2) + d~’/~r B~(_Ric~+RKg~))T=0. Here r(~J~) is a generalized minor radius, defined j~j
terms of the volume V(~)contained within a flux surface, r(~L’)= [V(~i)/2ir2R o]~2,where R 0 is the location of the magnetic axis. The equilibrium is characterized by two flux surface quantities — the shear =
~i)
din q/d in r,
(5)
and the pressure gradient parameter 2~ —8irR0r G(i~’) d~L’/dr d~i
20 July 1981
where the angle 0 is between the normal to a flux contour and the cylindrical radius R. The nontrigonometric terms are important only for shifted or noncircular flux surfaces and although apparently O((B~/ BT)2(RO/a)) they have a large destabilizing effect on the ballooning mode. 2The operator the Grad— V(R2 V) of~ eq.is(11). Shafranov operator R 3. Equilibrium. The numerical equilibria are obtained using a program developed by Stevens and modified by Englade [7]. The relevant equation to be solved is the Grad— Shafranov equilibrium equation, which, in the usual cylindrical coordinates, is written as a~L~1 a~i~ + = — 4irR2 c3~~ d~ R ~R az2 d’,ji — ~ (11) —
‘
where F(i/i) = RBT is the poloidal current stream function. The boundary is assumed to be a circular conducting shell. To generate a sequence of flux-conserving equilib-
(6)
6
I
R
I
0/o’ 2 5
— -
The squared growth rate [‘2 has been normalized to =
~
(7)
in terms of the Alfvén speed VA = B0/~/4~, and the effects of the gradient operator on the pitch of the field lines is represented by the quantity RB
~Ji, s~) ~
IV~’l~
a
f
q (4’) 4) d4,
qQ(~,1i,
(8)
x0
where the derivative ata ~ is taken at constant orthogonal poloidal angle x- The magnetic curvature terms are decomposed into normal and geodesic components cos0 =
—
BT Kg
R
—
Bp 2 RB BTB~ a
BR sin 0 +
—
lnIVgiI,
3 as~ B
124
___
B2 a~p
(9) (10)
MAGNETIC AXIS
________________________ ‘I’ BDY
Fig. 1. q(~i)for the Ro/a = 2.5 sequences. The flux-conserving sequencehasq and 2.0. 0 = 1.0,andthescaledsequenceshaveq0 = 1.4
Volume 84A. number
20 July 1981
PhYSICS LETTERS
3
a
b
d Fig.2. Sample equilibrium flux contours for the heating sequences with Ro/a
= 2.5.
(a) ~
0.1 .(h) iJ,~, 1.6. Ic) ~p
2.9.(d)
125
Volume 84A, number 3
PHYSICS LETTERS
20 July 1981
na, we choose an initial low-beta configuration, with a certain q(i~i)profile that remains invariant during
1
-~------~-~-
the pressure raising process. Specifically, we have q(i,li)
=
0.459/(0.062
653,
—
iji)O
L
(12)
with the flux variable ~1inormalized to —0.237 at the magnetic axis and zero at the boundary. The values of q range from 1.01 to 2.82, and are shown in fig. 1.
d ~(‘I’I
-
A heating function has also to be specified. The assumption introduced in the method of producing a sequence of equilibrium states is that the heating can
b
be chosen to satisfy the condition
p(~li,t) = ii(Ii, t)/(dV/d~i)7,
(13)
where ‘y is the adiabatic exponent, taken to be 2. The function r~(i~i,t) defines the evolution of the energy input. For the initial configuration, ij is found to be
o~
pono(~)7.594~2—7.6l8~, (14) and the subsequent states are generated by increasing the value of r~,according to some given function of time N(t), ~(i,1i, t) = N(t)i~ 0(1LI).Here ~o is the value of the permittivity of free space, in mks units. The initial low-beta configuration, which specifies the functions q(i1Li) and fl0(1~L/),was calculated by means of a conventional code, and corresponds to pressure and poloidal2 and current profiles F2 =stream+function c, respectively. of the formsdetails p = as,1i Further about the method of solution and other uses of the code are given in ref. [7]. In figs. 2a—d we depict vertical cross sections of
b
the magnetic surfaces for a flux-conserving sequence.
~
1
‘I’ _________________________________
a -
C F) )
d
0
0
I
_______
1 tiFig. 3. Pressure P(I~)and F(~i)= RBT profiles for members
These equilibria have major and minor radii R 0 = 50 cm, a = 20 cm, R0/a = 2.5, and vacuum toroidal field of 17.3 T at the center of the chamber. The low-13 equilibrium was taken to have a toroidal current of 3 MA. Sample pressure and poloidal current stream function profiles for this sequence are shown in fig. 3. A second flux-conserving sequence with R0 = 1 m, a = 20cm, BTO = 18.OT, and the same form of q and p profiles, is also calculated. The scaling of Bateman and Peng [8] has been applied to the two flux-conserving sequences, in the form — ~ / 2~-~,-~l/2 ~ where C is a constant chosen to fix the value of q at the magnetic axis. Representative q profiles are shown in fig. 1, for R0/a = 2.5, ~ = 3.4. The profiles for the
126
of the Ro/a = 2.5 flux-conserving sequence, plotted against poloidal flux ~ in arbitrary units. ~i = 0 is the magnetic axis and ~ = 1 the plasma boundary. (a) 3,,~ 0.1, (b) i3,,, = 1.6, (c) ~ = 2.9, (d) ‘~~‘ = 3.9.
corresponding R0/a = S sequences are almost identical. The scaling changes BT but leaves ~ and the flux surface geometry %(I(R, Z) unchanged. 4. Results and interpretation. The ballooning equation (4) was solved as an eigenvalue problem in ~2 along equally spaced flux contours, for the sequences of equihbna described above. As a first appnoxlma-
tion, the disconnected mode of Coppi [9], which decays to zero at the inside edge of the torus [T( ±ir) = 0], was used. These marginal stability curves appear
Volume 84A, number 3
PHYSICS LETTERS
C
3
0 7
3 2
/
I
/
I
I
/
R,/o ‘2 5
.71.
4
G
the disconnected mode approximation and the heavy dashed curves marginal stability in the infinite domain. The ballooning unstable region is above and inside the U formed by the marginal stability curves. Note that (a) has no high-beta sta— everything to the right of the heavy curves is Unstable. The heavy short dashes near the origin represent the
Mercier marginal stability boundary (Mercier unstable below I
2
Fig. 4. Marginal stability curves for the Rota = 2.5 sequences plotted against the shear and pressure gradient parameters i 2/(dip/dr)J(dp/d~p). (a) q~ = dlnq/dlnr and G = —[8irRor = 1.0, (b)qo = 1.4, (c)qo = 2.0. Each thin solid line represents one equilibrium (one value of ~3)from a sequence; the numbers along the top give f3 (%) for a few equilibria. The heavy solid curves represent ballooning marginal stability in
biization
2 0
/1
0
20 July 1981
6
8
in figs. 4 and 5 as solid heavy lines. The stability boundaries in the infinite domain [the exact solution where T(±oo)= 0] appear as the heavy dashed lines. In these figures the light solid curves each represent one equilibrium corresponding to one value of /3 from the heating sequence, parametrized by ~i the s’—G origin corresponds to the magnetic axis and the large~endapproaches the plasma boundary. The numbers at the top refer to values of 13 (%) for representative members of the sequences. The heavy short-dashed line at small s’ represents the Mercier stability boundary the Mercier unstable area is below the line. We see that at low 13 the two ballooning descniptions differ markedly, but when the mode stabffizes at
the curve).
high (3 they give rather similar results, confirming the analytical prediction that away from the axis (at high
ing on the strongly ballooning mode, which is localized towards the outer edge of the torus. Raising the q profile improves stability both by improving the low-shear stability near the magnetic axis and by decreasing the width of the unstable region. The unstable region decreases in width due to the stabilizing effect of the scaling on the local shear ~ and the decrease in the magnitude of the destabilizing higher-order curvature terms. In addition, merely by scaling /3 1 /q~while leaving /3~unchanged, the critical value of (3 for second stability decreases significantly. A comparison of the corresponding sequences of different aspect ratio shows that, while the smaller aspect ratio sequences are more unstable for q 0 1, the stability improves more rapidly as q0 increases above 1. In addition, the lower aspect ratio sequences
shear) the second stability is the result of effects act-
stabilize again for much less diamagnetic equffibria
—
—
127
Volume 84A, number 3
PHYSICS LETTERS
a
20 July 1981
than the higher aspect ratio sequences. Note that the maximum shown in the R0/a = 2.5 sequences is
6
In figs. 4 and 5 a thin region near the plasma boundary has been omitted due to unphysical edge effects introduced by the model and the lack of numerical resolution there. In addition, the ballooning stability
~
was evaluated could =3.9comparedto9.9forR0/a=5. (the origin not be in evaluated the instead. diagram), We directly do but notthe near regard Mercier the the magnetic criterion Mercier axis instability of some of these equilibria as a serious
ia 5
~O
q0’ .0
C0 -U”
2
I
4
0
I
6
drawback because it could be eliminated by an appro-
8
priate shaping of the flux contours, for instance by choosing a D-shaped boundary. Finally, we should point out that the ballooning stability results presented here are pessimistic, since the stabilizing 1/n0 corrections have been neglected. Low-n ideal MHD ballooning modes are even more stable, but the equilibria found here to be ballooning stable may still prove unstable to other internal MHD modes.
______________________________________ b
a
no/a’s
This work was supported by the US Department of Energy. References
q0’I.4
2
4
6
8
G
[1] B. Coppi, A. Ferreira, J.W.-K. Mark and J.J. Ramos, Nucl.
____________________________________________
C
Fusion 19 (1979) 715. [21 L. Sugiyama and J.W.-K. Mark, Massachusetts Institute
of Technology RLE Report PRR-80/7, to be published in
4
Nucl. Fusion. 2
6
[3] H.R. Strauss et al., Nucl. Fusion 20 (1980) 638. [4]C.K. Chu et al., 8th Intern. Conf. on Plasma physics and controlled nuclear fusion research (Brussels, 1980) (IAEA, Vienna) paper CN-38/L-4-2. [5] A.H. Glasser, in: Proc. Finite beta theory workshop (Varenna, 1977), eds. B. Coppi and W. Sadowski (US Department of Energy, 1978) p. 55. [6] T.M. Antonsen Jr., private communication. 20 2
4
6
8
0
Fig. 5. Ballooning marginal stability curves for the Rota = 5 sequences plotted against the shear and pressure gradient parameters.(a)qo = 1.0,(b) qo 1.4,(c)qo = 2.0.Thecurves are analogous to those of fig. 4.
128
[7] R. conserving equilibria Massachusetts of a circular Tokamak:Englade, MercierFlux stability considerations, Institute of Technology RLE Report PRR-77/23 (1977). [8]G. Bateman and Y.-K.M. Peng, Phys. Rev. Lett. 38 (1977) 829. [9]B. Coppi, Phys. Rev. Lett. 39 (1977) 939.
PHYSICS LETTERS
20 July 1981
HIGH-BETA STABILITY OF A TOROIDAL PLASMA L. SUGIYAMA and J. W.-K. MARK1 Massachusetts Institute of Technology, Cambridge, MA
02139, USA
Received 10 December 1980 Revised manuscript received 18 April 1981
The high toroidal mode number ideal MHD ballooning mode is shown to stabilize again at large beta, in several sequences of numerical toroidal equilibria. Raising the q profile improves high-beta stability for equilibria with the same p0-
loidal beta and flux surface geometry. A smaller aspect ratio enhances the stabilizing effect of raising the q profile and allows second stability to occur at smaller poloidal beta, though larger total beta.
1. Introduction. In an earlier work [1] it was shown that ideal MHD ballooning modes exhibit a second critical value of 13 (the ratio of the volume-averaged plasma pressure to the vacuum magnetic pressure
~,(x,t)
= ~(i/i, ~
(1) [
X exp[iwt
/ +
S~(
in
0~
—
f q2(iji,
s~)ds~
at the axis) beyond which the plasma is again stable. The existence of this second region of stability was
where
originally discussed using a model equilibrium config-
magnetic surfaces are represented by i/i
uration where the flux surfaces were represented by shifted circles but the main physical effects on the mode were included the increase of the poloidal magnetic field toward the outer edge of the plasma colunm and the dependence of the rate of magnetic shear on poloidal angle. Subsequent numerical studies [2,3] have confirmed the existence of the second stable region for numerical models of toroidal equilibria and experimental studies have been interpreted as evidence for a second stable region [4]. Here we investigate the effects of varying the q profile and the aspect ratio on the second stable
is the poloidal arclength along a flux contour, and ~ is the toroidal angle of symmetry. These coordinates are characterized by the line elements h1~dli = RB~~ ds~,and h~d~ = R d~, where R is the distance from the axis of symmetry and B~is the poloidal magnetic field. The local pitch of the field lines is defined to be qQ = d~/th~, where the derivative is taken along the field line. To first order in an expansion in 1/n0, the equations governing MHD ballooning equations can be written as [5,6] 2IVSI2 T — B~ ~ IVSI2B- VT + K T —pw B2 4irB2 B2 ~
—
2. The normal mode equations. The perturbed plas ma displacement is represented in normal mode form
(~4’,s,~, ~)
w
~ =
are curvilinear coordinates in which =
constant, 5~
(2)
T
B2 as
_ypB.VB~ç_(1+13)W=13K~T,
(3)
pwB 1
Present address: University of California, Lawrence Livermore Laboratories, Livermore, CA 94550, USA.
where KT = (B- K X
VS)/21T,
ture of the magnetic field B,
K being the normal curvai~,= (B X VS Vp)/B2,
123
Volume 84A, number 3
PHYSICS LETTERS
and f3 = 4ir~/B2is a local beta. Here T/B is related to the normal component of the displacement ~, and the quantity W is proportional to ‘)p(V~and measures the compressibility. At marginal stability the most unstable mode is incompressible and W = W(14~)= 0 to lowest order in 1/n 0. The set of equations then reduces to the well-known ballooning equation, which we write in the form R~r2B~
a 1 + ~2 ~
~2
(d~i/dr)2
~XR2B~ as~ (4) GR
÷(_r21 + ~2) + d~’/~r B~(_Ric~+RKg~))T=0. Here r(~J~) is a generalized minor radius, defined j~j
terms of the volume V(~)contained within a flux surface, r(~L’)= [V(~i)/2ir2R o]~2,where R 0 is the location of the magnetic axis. The equilibrium is characterized by two flux surface quantities — the shear =
~i)
din q/d in r,
(5)
and the pressure gradient parameter 2~ —8irR0r G(i~’) d~L’/dr d~i
20 July 1981
where the angle 0 is between the normal to a flux contour and the cylindrical radius R. The nontrigonometric terms are important only for shifted or noncircular flux surfaces and although apparently O((B~/ BT)2(RO/a)) they have a large destabilizing effect on the ballooning mode. 2The operator the Grad— V(R2 V) of~ eq.is(11). Shafranov operator R 3. Equilibrium. The numerical equilibria are obtained using a program developed by Stevens and modified by Englade [7]. The relevant equation to be solved is the Grad— Shafranov equilibrium equation, which, in the usual cylindrical coordinates, is written as a~L~1 a~i~ + = — 4irR2 c3~~ d~ R ~R az2 d’,ji — ~ (11) —
‘
where F(i/i) = RBT is the poloidal current stream function. The boundary is assumed to be a circular conducting shell. To generate a sequence of flux-conserving equilib-
(6)
6
I
R
I
0/o’ 2 5
— -
The squared growth rate [‘2 has been normalized to =
~
(7)
in terms of the Alfvén speed VA = B0/~/4~, and the effects of the gradient operator on the pitch of the field lines is represented by the quantity RB
~Ji, s~) ~
IV~’l~
a
f
q (4’) 4) d4,
qQ(~,1i,
(8)
x0
where the derivative ata ~ is taken at constant orthogonal poloidal angle x- The magnetic curvature terms are decomposed into normal and geodesic components cos0 =
—
BT Kg
R
—
Bp 2 RB BTB~ a
BR sin 0 +
—
lnIVgiI,
3 as~ B
124
___
B2 a~p
(9) (10)
MAGNETIC AXIS
________________________ ‘I’ BDY
Fig. 1. q(~i)for the Ro/a = 2.5 sequences. The flux-conserving sequencehasq and 2.0. 0 = 1.0,andthescaledsequenceshaveq0 = 1.4
Volume 84A. number
20 July 1981
PhYSICS LETTERS
3
a
b
d Fig.2. Sample equilibrium flux contours for the heating sequences with Ro/a
= 2.5.
(a) ~
0.1 .(h) iJ,~, 1.6. Ic) ~p
2.9.(d)
125
Volume 84A, number 3
PHYSICS LETTERS
20 July 1981
na, we choose an initial low-beta configuration, with a certain q(i~i)profile that remains invariant during
1
-~------~-~-
the pressure raising process. Specifically, we have q(i,li)
=
0.459/(0.062
653,
—
iji)O
L
(12)
with the flux variable ~1inormalized to —0.237 at the magnetic axis and zero at the boundary. The values of q range from 1.01 to 2.82, and are shown in fig. 1.
d ~(‘I’I
-
A heating function has also to be specified. The assumption introduced in the method of producing a sequence of equilibrium states is that the heating can
b
be chosen to satisfy the condition
p(~li,t) = ii(Ii, t)/(dV/d~i)7,
(13)
where ‘y is the adiabatic exponent, taken to be 2. The function r~(i~i,t) defines the evolution of the energy input. For the initial configuration, ij is found to be
o~
pono(~)7.594~2—7.6l8~, (14) and the subsequent states are generated by increasing the value of r~,according to some given function of time N(t), ~(i,1i, t) = N(t)i~ 0(1LI).Here ~o is the value of the permittivity of free space, in mks units. The initial low-beta configuration, which specifies the functions q(i1Li) and fl0(1~L/),was calculated by means of a conventional code, and corresponds to pressure and poloidal2 and current profiles F2 =stream+function c, respectively. of the formsdetails p = as,1i Further about the method of solution and other uses of the code are given in ref. [7]. In figs. 2a—d we depict vertical cross sections of
b
the magnetic surfaces for a flux-conserving sequence.
~
1
‘I’ _________________________________
a -
C F) )
d
0
0
I
_______
1 tiFig. 3. Pressure P(I~)and F(~i)= RBT profiles for members
These equilibria have major and minor radii R 0 = 50 cm, a = 20 cm, R0/a = 2.5, and vacuum toroidal field of 17.3 T at the center of the chamber. The low-13 equilibrium was taken to have a toroidal current of 3 MA. Sample pressure and poloidal current stream function profiles for this sequence are shown in fig. 3. A second flux-conserving sequence with R0 = 1 m, a = 20cm, BTO = 18.OT, and the same form of q and p profiles, is also calculated. The scaling of Bateman and Peng [8] has been applied to the two flux-conserving sequences, in the form — ~ / 2~-~,-~l/2 ~ where C is a constant chosen to fix the value of q at the magnetic axis. Representative q profiles are shown in fig. 1, for R0/a = 2.5, ~ = 3.4. The profiles for the
126
of the Ro/a = 2.5 flux-conserving sequence, plotted against poloidal flux ~ in arbitrary units. ~i = 0 is the magnetic axis and ~ = 1 the plasma boundary. (a) 3,,~ 0.1, (b) i3,,, = 1.6, (c) ~ = 2.9, (d) ‘~~‘ = 3.9.
corresponding R0/a = S sequences are almost identical. The scaling changes BT but leaves ~ and the flux surface geometry %(I(R, Z) unchanged. 4. Results and interpretation. The ballooning equation (4) was solved as an eigenvalue problem in ~2 along equally spaced flux contours, for the sequences of equihbna described above. As a first appnoxlma-
tion, the disconnected mode of Coppi [9], which decays to zero at the inside edge of the torus [T( ±ir) = 0], was used. These marginal stability curves appear
Volume 84A, number 3
PHYSICS LETTERS
C
3
0 7
3 2
/
I
/
I
I
/
R,/o ‘2 5
.71.
4
G
the disconnected mode approximation and the heavy dashed curves marginal stability in the infinite domain. The ballooning unstable region is above and inside the U formed by the marginal stability curves. Note that (a) has no high-beta sta— everything to the right of the heavy curves is Unstable. The heavy short dashes near the origin represent the
Mercier marginal stability boundary (Mercier unstable below I
2
Fig. 4. Marginal stability curves for the Rota = 2.5 sequences plotted against the shear and pressure gradient parameters i 2/(dip/dr)J(dp/d~p). (a) q~ = dlnq/dlnr and G = —[8irRor = 1.0, (b)qo = 1.4, (c)qo = 2.0. Each thin solid line represents one equilibrium (one value of ~3)from a sequence; the numbers along the top give f3 (%) for a few equilibria. The heavy solid curves represent ballooning marginal stability in
biization
2 0
/1
0
20 July 1981
6
8
in figs. 4 and 5 as solid heavy lines. The stability boundaries in the infinite domain [the exact solution where T(±oo)= 0] appear as the heavy dashed lines. In these figures the light solid curves each represent one equilibrium corresponding to one value of /3 from the heating sequence, parametrized by ~i the s’—G origin corresponds to the magnetic axis and the large~endapproaches the plasma boundary. The numbers at the top refer to values of 13 (%) for representative members of the sequences. The heavy short-dashed line at small s’ represents the Mercier stability boundary the Mercier unstable area is below the line. We see that at low 13 the two ballooning descniptions differ markedly, but when the mode stabffizes at
the curve).
high (3 they give rather similar results, confirming the analytical prediction that away from the axis (at high
ing on the strongly ballooning mode, which is localized towards the outer edge of the torus. Raising the q profile improves stability both by improving the low-shear stability near the magnetic axis and by decreasing the width of the unstable region. The unstable region decreases in width due to the stabilizing effect of the scaling on the local shear ~ and the decrease in the magnitude of the destabilizing higher-order curvature terms. In addition, merely by scaling /3 1 /q~while leaving /3~unchanged, the critical value of (3 for second stability decreases significantly. A comparison of the corresponding sequences of different aspect ratio shows that, while the smaller aspect ratio sequences are more unstable for q 0 1, the stability improves more rapidly as q0 increases above 1. In addition, the lower aspect ratio sequences
shear) the second stability is the result of effects act-
stabilize again for much less diamagnetic equffibria
—
—
127
Volume 84A, number 3
PHYSICS LETTERS
a
20 July 1981
than the higher aspect ratio sequences. Note that the maximum shown in the R0/a = 2.5 sequences is
6
In figs. 4 and 5 a thin region near the plasma boundary has been omitted due to unphysical edge effects introduced by the model and the lack of numerical resolution there. In addition, the ballooning stability
~
was evaluated could =3.9comparedto9.9forR0/a=5. (the origin not be in evaluated the instead. diagram), We directly do but notthe near regard Mercier the the magnetic criterion Mercier axis instability of some of these equilibria as a serious
ia 5
~O
q0’ .0
C0 -U”
2
I
4
0
I
6
drawback because it could be eliminated by an appro-
8
priate shaping of the flux contours, for instance by choosing a D-shaped boundary. Finally, we should point out that the ballooning stability results presented here are pessimistic, since the stabilizing 1/n0 corrections have been neglected. Low-n ideal MHD ballooning modes are even more stable, but the equilibria found here to be ballooning stable may still prove unstable to other internal MHD modes.
______________________________________ b
a
no/a’s
This work was supported by the US Department of Energy. References
q0’I.4
2
4
6
8
G
[1] B. Coppi, A. Ferreira, J.W.-K. Mark and J.J. Ramos, Nucl.
____________________________________________
C
Fusion 19 (1979) 715. [21 L. Sugiyama and J.W.-K. Mark, Massachusetts Institute
of Technology RLE Report PRR-80/7, to be published in
4
Nucl. Fusion. 2
6
[3] H.R. Strauss et al., Nucl. Fusion 20 (1980) 638. [4]C.K. Chu et al., 8th Intern. Conf. on Plasma physics and controlled nuclear fusion research (Brussels, 1980) (IAEA, Vienna) paper CN-38/L-4-2. [5] A.H. Glasser, in: Proc. Finite beta theory workshop (Varenna, 1977), eds. B. Coppi and W. Sadowski (US Department of Energy, 1978) p. 55. [6] T.M. Antonsen Jr., private communication. 20 2
4
6
8
0
Fig. 5. Ballooning marginal stability curves for the Rota = 5 sequences plotted against the shear and pressure gradient parameters.(a)qo = 1.0,(b) qo 1.4,(c)qo = 2.0.Thecurves are analogous to those of fig. 4.
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[7] R. conserving equilibria Massachusetts of a circular Tokamak:Englade, MercierFlux stability considerations, Institute of Technology RLE Report PRR-77/23 (1977). [8]G. Bateman and Y.-K.M. Peng, Phys. Rev. Lett. 38 (1977) 829. [9]B. Coppi, Phys. Rev. Lett. 39 (1977) 939.