5.6 External Kink Mode Stability Calculations for FTU-D.
The linear stability of the n=1 ideal external kink modes of FTU-D plasmas has been investigated
using the stability code MARS [1, 2]. The equilibrium quantities required by MARS have been
supplied by the equilibrium code CHEASE [3]. The considered equilibria have been computed
using as inputs the boundary of the plasma and the functions dP0/d and TdT/d has obtained by
the FIXFREE code [4] (here, P0(), T(), and  being, respectively, the plasma pressure, the
poloidal current flux function and the poloidal magnetic flux function). Following the standard
procedure for the numerical studies of MHD modes for X-point configurations, the plasma has been
considered to extend up to the flux surface which includes an assigned fraction (namely, 95%) of
the total poloidal flux .
The MARS code can compute the linear stability of MHD modes in presence of a vacuum region
between the plasma and a perfectly conducting wall. The metric tensor quantities characterizing the
vacuum region are again supplied by CHEASE, which allows for a shape of the wall which is
conformal to the plasma boundary. This can, in principle, be a limitation with respect to the FTU-D
layout, which has a circular vacuum chamber and a D-shaped plasma boundary. Nevertheless, at
least for top-bottom symmetric configurations, numerical calculations performed by dr. J.
Manickam (PPPL) [5] have shown that the portion of the vacuum chamber which contributes to the
dynamics of the external mode can be estimated to be between ±60º with respect to the equatorial
plane. Thus, it is sufficient that the conformal wall matches the vacuum chamber shape in the
external region of the torus to have reliable results.
In the preset study the P0(), T() profiles and the plasma boundary shape have been kept constant,
whereas the plasma  and the distance between the plasma boundary and the wall (and in some
cases, the total plasma current) has been varied. Thus, the stability study has been performed on a
restricted set of equilibria. Indeed, the stability of the external kink may depend on several other
parameters, as, e.g., the value of the current density and its derivative at the edge, the pressure and
current profiles, the shaping, etc.. Thus the results presented in this report have to be considered as
preliminary and indicative of a realistic experimental configuration.
The main results of this preliminary study on the stability of the n=1 external kink mode in presence
of a perfectly conducting conformal wall are summarized in table 1. The expression for the
conformal wall coordinates (Rwall, Zwall) are defined by:
Rwall = Rvc + s (Rv - Rvc),
Zwall = Zvc + s (Zv - Zvc),
where s =[1,Rext], Raxis is the magnetic axis major radius, Rvc and Zvc are the cartesian coordinates of
the vacuum chamber center, Rv and Zv are the cartesian coordinates of the plasma vacuum interface
(in the limit of circular, concentric conformal wall Rext coincides with b/a, where b and a are,
respectively, the minor radius of the wall and of the plasma). The conformal wall for case 1 to case
3 has been centered on the plasma geometric center (Rvc / Raxis = 1, Zvc / Raxis = 0.), whereas Rext  b
/ a = 1.22 for case 1 and Rext = 1.24 for case 2 and case 3. The (conformal) wall for the case 4,
Single Null plasma configuration, has been modelled assuming Rext = 1.45, Rvc / Raxis = 1.075, Zvc /
Raxis = 0.07. The (conformal) wall for the case 5, Single Null plasma configuration, has been
modelled assuming Rext = 1.6, Rvc / Raxis = 1.1, Zvc / Raxis = 0.04. All the above mentioned wall
configurations result in a distance between the (conformal) wall and the plasma radius d ≈ 0.04 m.
Note that the values of the toroidal magnetic field and the plasma current shown in table 1 are only
a particular choice: indeed, at fixed geometrical factors, the relevant quantity which enters into the
equilibrium is the ratio B / Ip,95% between the toroidal magnetic field and the plasma current
including inside the 95% flux surface.
1
It has to be noted that the stability of the n=1 external kink mode is strongly dependent on the
considered value of the safety factor at the edge, more precisely, it is very sensitive to qa if it is close
to an integer value. In fact, the interpretation of the results for the equilibrium case 2 is ambiguous,
likely because of the closeness (from below) of the boundary safety factor to the integer value qa,95%
= 3.
case,
toroidal field /
plasma conf.
plasma current
1, Double Null B = 5 (T) /
Ip,95% = 350 (kA)
2, Double Null B = 2.5 (T) /
Ip,95% = 300 (kA)
3, Double Null B = 2.5 (T) /
Ip,95% = 250 (kA)
q0
qa,95%
95%
95%
N,crit,, comments
1.1
4.5
1.4
0.49
N,crit,≈3.6
1.1
2.8
1.5
0.58
unclear result
N,crit,≈4.69
(b/a|crit≈1.3,
dmin,p-w 95% = 0.051)
4, Single Null B = 2.5 (T) /
1.25
2.7
1.5
0.5
stable at N ≈3.6
Ip,95% = 280 (kA)
(Rext, crit≈1.65,
dmin,p-w 95% = 0.052)
5, Single Null B = 5 (T) /
1.16
4.16
1.4
0.4
stable at N ≈3.4
Ip,95% = 360 (kA)
(stable with the wall at
infinity at N ≈2.4)
Table 1. Plasma configurations, toroidal magnetic field, plasma current, safety factor at the center
and at the edge (95% of total poloidal flux) of the plasma column, elongation and triangularity of
the last considered magnetic surface, threshold value of N and minimum distance between plasma
and wall on the external equatorial plane.
1.175
3.2
1.5
0.58
References:
[1] A. Bondeson, G. Vlad, and H. Lütjens, Phys. Fluids B, 4, (1992) 1899.
[2] A. Bondeson, G. Vlad, and H. Lütjens, Proceedings of the International Atomic Energy Agency
Technical Committee Meeting on Advances in Simulations and Modeling of Thermonuclear
Plasmas, Montréal, Canada, 15-18 June 1992, p. 306, Vienna, 1993, International Atomic Energy
Agency.
[3] H. Lütjens, A. Bondeson, and A. Roy, Comput. Phys. Commun., 69, (1992) 287.
[4] F. Alladio, F. Crisanti, M. Marinucci, P. Micozzi, and A. Tanga, Nuclear Fusion 31, 739 (1991).
[5] J. Manickam, Princeton Plasma Physics Laboratory, private comunication.
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