Unified Ideal Stability Limits for Advanced Tokamak and Spherical Torus Plasmas Page: 4 of 8
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2
Inverse aspect ratio E = A-1
0.0 0.2 0.4 0.6 0.8100
(a) t M
10
4.0 (b)
3.5 x
3.0
9 (C) NN
8 - -
7 -O ).
6 - - -- - - -- - - .10 5 3.3 2.5 1.81.6 1.41.25
Aspect ratio1.0
'A
a)
0
a.
0U1
FIG. 1: (a) Toroidal beta, (b) plasma elongation, and (c)
normalized beta values versus aspect ratio for wall-stabilized
fully self-sustained equilibria.
9% to 84% as the aspect ratio A is decreased from 5 to
1.25. Figure lb shows that the elongation K nearly dou-
bles from 2.2 to 3.9 for the same range of aspect ratios.
For elongations above those shown in Figure lb, 3t can-
not be further increased without destabilizing low-n kink
modes with a conformal conducting wall at ra/a= 1.1.
Figure lc shows that the normalized toroidal beta ON -
Qt(%)aBto/Ip(MA) increases from 5.7 to 9.0. It can be
shown [13] that 3~ A-1/2(l + n2)9N2 fBS, implying
that the explicit dependence of the beta limit on aspect
ratio is relatively weak when the bootstrap current frac-
tion fBs is held fixed. Thus, the strong dependence of
the elongation and normalized toroidal beta on aspect
ratio are together responsible for most of the increase in
toroidal beta with decreasing aspect ratio. Figure lc also
shows that the normalized volume-averaged beta (ON) -
(3)(%)aBtg/Ip(MA) where (0) - 2p0(p)/(B2) [11] ex-
hibits much smaller variation with aspect ratio than ON
and is an approximate stability invariant. These re-
sults suggest that the beta limit for wall-stabilized self-
sustaining configurations is (ON) ~ 6 nearly independent
of aspect ratio.
The optimal profiles in the stability calculations dis-
cussed above are found to vary only slightly with aspect
ratio with one notable exception. Figure 2a shows that
the safety factor profile for A=1.6 is monotonically in-
creasing as a function of minor radius (square root of
the normalized poloidal flux), while for A=3.3 the q pro-
file exhibits strongly reversed-shear. The shear changes
sign from positive to negative near A=2.0, so this aspect
ratio represents a possible natural dividing line between
the spherical torus and advanced tokamak. Figure 2b
shows that the optimal pressure profiles are generally
quite broad, and the pressure peaking factors p(0)/(p)
are found to increase from 1.38 to 1.57 between A=1.25
and A=5.0. Figure 2c shows that both current density12
10 (a)
8 A=1.6
6
4
2 A=3.3
0
0.0 0.2 0.4 0.6 0.8 1.0
W1I 2S 1.0 - b)
i 0.8
0
0.6 A=3.3
a)
0.4
u 0.2
a- 0.0 - - - -
0.0 0.2 0.4 0.6 0.8 1.0
W112c 1.0 - -1.4O
10 (c) Total (d)
S0.8 . 1.30(
a 0.6c
25, 1.20
0.4 A=3.3 1 A=3.3
S0.2 y/BS 1.1
0.0 1.00 . . . . . . .
0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 6 7 8 9
4T12 Toroidal Mode Number
FIG. 2: (a) q (safety factor) profiles, (b) normalized pres-
sure profiles, (c) normalized current density profiles, and (d)
kink marginally-stable wall position divided by plasma minor
radius for the A=1.6 and A=3.3 equilibria shown in Figure 1.
profiles are hollow and driven completely by the boot-
strap effect except for a small region in the core. Fi-
nally, as shown in Figure 2d, for low aspect ratio there
is a monotonic decrease in marginal wall position with
increasing toroidal mode number. In contrast, for high
aspect ratio the wall position can exhibit oscillations due
the influence of individual mode rational surfaces associ-
ated with lower edge safety factor and shear.
Partially self-sustained no-wall stability The fully
self-driven regimes with very high beta and elongation
outlined above are theoretically achievable but have not
yet been realized experimentally. Elongation values in
excess of those shown in Figure lb have recently been
achieved for A=3.5 [14], but not yet at high beta. The
physical understanding of external kink stabilization uti-
lizing rotation [15, 16] and active feedback [17] has im-
proved significantly recently, but normalized beta values
significantly above those attainable with optimized pro-
files without conducting wall stabilization are not easily
achieved. Further, the pressure profile control techniques
required to realize the highest beta values in fully boot-
strapped regimes are only beginning to be developed.
These factors motivate an investigation of the aspect
ratio dependence of the ideal beta limit for parameters
more typical of present-day experiments.
In the following analysis, ideal beta limits are deter-
mined for equilibria with a fixed self-driven current frac-
tion of 50% (with no local bootstrap current overdrive)
and which are marginally stable to ballooning modes and
n=1-3 kink modes without wall stabilization. For most
aspect ratios treated, the optimization of the pressure
and current profiles results in the equilibrium being si-
multaneously marginally stable to ballooning and n=1
kink modes. With this set of constraints and with a fixed
plasma boundary shape with elongation r 2.0 and tri-
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Menard, J. E.; Bell, M. G.; Bell, R. E.; Gates, D. A.; Kaye, S. M.; LeBlanc, B. P. et al. Unified Ideal Stability Limits for Advanced Tokamak and Spherical Torus Plasmas, report, February 6, 2003; Princeton, New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc740861/m1/4/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.