Abstract In this paper we present some preliminary results of a coupling between the MHD stability code MARS-F and the three-dimensional eddy currents code CARIDDI, aimed at the analysis of Resistive Wall Modes with 3D conducting structures

F Villone

G Rubinacci

Y Gribov

Y Q Liu

CiteSeerX

Analysis of RWM with a 3D Model of Conducting Structures  
 
F. Villone
1
, G. Rubinacci
2
, Y.Q. Liu
3
,
  
Y. Gribov
4
  
 
1
 Ass. EURATOM/ENEA/CREATE, DAEIMI, Univ. di Cassino, Cassino (FR), ITALY 
2
 Ass. EURATOM/ENEA/CREATE, DIEL, Univ. Federico II di Napoli, ITALY 
3
 Ass. EURATOM/VR, Chalmers Univ. Technology, Gothenburg, SWEDEN 
4
 Physics Unit, ITER Naka Joint Work Site, Naka, Ibaraki, JAPAN 
 
Abstract 
In this paper we present some preliminary results of a coupling between the MHD stability  
code MARS-F and the three-dimensional eddy currents code CARIDDI, aimed at the analysis 
of Resistive Wall Modes with 3D conducting structures. 
 
1. Introduction 
Ideal external kink instabilities of low n”0 toroidal mode numbers limit the achievable 
normalized dN. The presence of a stabilizing ideal conducting wall can mitigate this limit, 
thanks to eddy currents induced by plasma perturbations. The non-vanishing resistivity of any 
real wall causes a decay of eddy currents and hence a loss of stability. The resulting resistive 
wall modes (RWM) grow at the time scale of the wall time, thus allowing a possible active 
stabilization of such modes with magnetic coils. A critical issue is a correct three-dimensional 
model of the conducting structures (walls) surrounding the plasma, and of the feedback coils. 
Aim of this paper is to describe the coupling between the toroidal stability code MARS-F [1] 
and the three-dimensional magneto-quasi-static code CARIDDI [2], with the final goal of 
analysing Resistive Wall Modes in the presence of 3D conducting structures. 
The paper is organized as follows. Section 2 describes the formulations used by MARS-F and 
CARIDDI, and the basic ideas of the coupling scheme. In Section 3 some preliminary results 
are presented, while Section 4 draws the conclusions and illustrates future work. 
 
2. Formulation 
MARS-F is an extension of the stability code MARS [3], that solves the single fluid MHD 
equations. This code has been modified to study stabilization of the RWM by plasma rotation 
[4, 5], using various damping models to approximate the ion Landau damping. A recent  
modification is to introduce in the code a semi-kinetic damping model, based on a large 
aspect ratio calculation [6]. With these modifications, the code has been used to predict the 
32nd EPS Conference on Plasma Phys. Tarragona, 27 June - 1 July 2005 ECA Vol.29C, P-5.049  (2005)
critical rotation speed required to stabilize the RWM, as well as to model the resonant field 
amplification experiments on JET and DIII-D. Another extension is to add feedback coils 
directly in the code [1], thus allowing feedback control of the RWM to be studied by solving 
both MHD equations and the feedback equation in a single code. MARS-F has been used also 
for numerical simulation of RWM in ITER [7]. 
One limitation of the code is the 2D representation of the conducting structures, including the 
vacuum vessels and the feedback coils, since the code assumes exp(jnl) dependence for the 
n-th harmonic along the toroidal angle l. Also, a thin wall approximation is assumed for the 
wall along the radial coordinate. On the other hand, the vacuum vessels often have 3D 
features such as slits and ports. The feedback coils have finite toroidal width, and may not 
cover the whole toroidal circumference.  Accurate modeling of these coils also require a 3D 
representation. The other limitation comes from the Fourier representation of the feedback 
coils along the poloidal angle.  
CARIDDI is a 3D eddy currents finite elements code based on an integral formulation [2]. It 
requires a discretization only of the conducting structures, which can be also topologically 
complex, and allows an easy coupling with external circuitry [8]. The introduction of a two-
component electric vector potential and the use of edge elements give rise to a very accurate 
(imposing the right continuity conditions) and effective (requiring a minimal number of 
unknowns) code, able to deal with arbitrary 3D geometries of conducting structures, hence 
overcoming the limitations mentioned above. CARIDDI has been extensively used for fusion 
applications, also coupled with the CREATE-L 2D linearized plasma response model [9].  
The magnetic field calculated by the CARIDDI code is used as the boundary condition for the 
MARS-F code. The boundary surface (i.e. the coupling surface) is chosen as a surface 
between the plasma boundary and the first conducting structure, which is usually the resistive 
wall.  We propose two types of boundary conditions for the coupling. The first one uses only 
the normal component of the magnetic field computed by CARIDDI, together with the 
estimation on the eigenvalue from the previous iteration.  The other type uses the ratio of the 
normal to the poloidal field as the boundary condition for the MARS-F code. 
The current density perturbation predicted by MARS-F is used in CARIDDI to compute the 
induced voltage in the conducting structure. This is the forcing term that allows the 
computation of the eddy currents induced in conducting structures, from which the magnetic 
field fed to MARS-F is calculated. Since CARIDDI works in the spatial domain, Fourier 
transforms must be numerically performed to provide the correct input to MARS-F. 
 
 
32nd EPS 2005; Y.Q.Liu et al. : Analysis of RWM with a 3D Model of Conducting Structures 2 of 4
3. Preliminary results 
The first tests reported in this paper were made assuming that the vessel is axisymmetric, 
although CARIDDI describes this 2D vessel with a fully 3D mesh; this allows us to use the 
results of MARS-F as reference. The vessel is assumed to have a circular cross section of 
major radius R0 = 2 m, and minor radius rw = 0.52 m; its thickness is 1 cm, while its resistivity 
is 6.53e-7 Y m. The plasma minor radius is a = 0.4 m. 
The first check made was a “consistence” test: given the “exact” n = 1 growth rate (i = 294.9 
s
-1
) and the plasma current density perturbation provided by MARS, we compared the vessel 
current density as estimated by MARS-F and by CARIDDI. Figure 1 shows this comparison 
on the various components, showing a satisfactory agreement. Figure 2 reports a 3D view of 
the current density as estimated by  CARIDDI, in which also the mesh is reported. 
Also the various components of the magnetic field on the “coupling surface” have been 
compared, as reported in Figure 3. Again, the agreement is satisfactory. Using the field as 
predicted by CARIDDI, and using the coupling scheme described above, MARS-F provided a 
growth rate which was less than 0.2 % different from the original one. 
We also tried to perform a “convergence” test, running the coupling scheme starting from an 
arbitrary initial guess. Unfortunately, we experienced an undesired sensitivity of the final 
result on the initial guess itself: starting from a point rather close to the right solution we were 
able to estimate the growth rate within a few percents, while an arbitrary starting point 
provided rather large errors. 
 
4. Conclusions and perspectives 
In this paper we have presented some preliminary results of the coupling of the MARS-F 
stability code with the CARIDDI three-dimensional eddy currents code. This coupling has 
been demonstrated to be consistent, while its convergence still needs further work to increase 
robustness against the starting guess. Once this problem will be sorted out, it will be possible 
to model Resistive Wall Modes, in view of their stabilization, in the presence of general 3D 
conducting structures.  
 
Acknowledgements 
This work, supported by the Italian MIUR and the Euratom Communities under the contract 
of Association between EURATOM/ENEA, was carried out within the framework the 
European Fusion Development Agreement. The views and opinions expressed herein do not 
necessarily reflect those of the European Commission. 
 
32nd EPS 2005; Y.Q.Liu et al. : Analysis of RWM with a 3D Model of Conducting Structures 3 of 4
   
Figure 1. Current densities as predicted by MARS- F (solid) and CARIDDI (dotted) 
 
Figure 2. 3D view of the current density (imaginary part) as predicted by CARIDDI 
 
Figure 3. Magnetic fields as predicted by MARS- F (solid) and CARIDDI (dotted) 
 
[1] Y.Q. Liu, et al., Phys. Plasmas 7, 3681 (2000)  
[2] R. Albanese, G. Rubinacci, Adv. Im. El. Phys.,102, 1-86, Academic Press (1998) 
[3] A. Bondeson, et al., Phys. Fluids B, 4, 1889 (1992) 
[4] Bondeson, A. and Ward, D.J.,  Phys. Rev. Lett. 72, 2709 (1994) 
[5] M.S. Chu, et al., Phys. Plasmas 2, 2236 (1995) 
[6] Bondeson A. and Chu M.S., Phys. Plasmas 3 3013 (1996)  
[7] Y.Q. Liu, et al., Nucl. Fusion 44, 232 (2004) 
[8] G. Rubinacci, A. Tamburrino, F. Villone, IEEE Trans. Mag., 38, 581 (2002)  
[9] R. Albanese, R. Fresa, G. Rubinacci, F. Villone, IEEE Trans. Mag., 36, 1804 (2000) 
32nd EPS 2005; Y.Q.Liu et al. : Analysis of RWM with a 3D Model of Conducting Structures 4 of 4


Analysis of RWM with a 3D Model of Conducting Structures

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=4DB205F5D3C910F7264E6780A10B5565?doi=10.1.1.1074.3787&rep=rep1&type=pdf

Analysis of RWM with a 3D Model of Conducting Structures

Abstract

Similar works

Full text

Available Versions

CiteSeerX