7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Study on Stability Analysis of Non-axisymmetric Tokamak Plasma Equilibriumby 3-D Multi-layers MethodJ. Wang1, H. Tsutsui2
1
Department of Transdisciplinary Science and Engineering, Tokyo Institute of Technology
2
Laboratory for Zero-Carbon Energy, Tokyo Institute of Technology
e-mail (speaker): wang.jingting@torus.nr.titech.ac.jp

Recently, the importance of non-axisymmetric has beenrecognized in tokamaks. In order to consider the effect ofnon-axisymmetric in tokamaks in three dimensions, it isnecessary to derive a three-dimensional MHD
equilibrium that considers non-axisymmetric as a starting
point. Currently, the most widely used 3D MHD balancecode is the VMEC code,[1, 2] which uses the spectral
method with a Fourier expansion for the angularvariables over the magnetic surface. The VMEC code ispresently used for 3D MHD balance calculations on mosttokamaks. In contrast, VMEC uses magnetic coordinates,
so it cannot be operated in the presence of a separatrix.
VMEC also does not compute stability. The goal of thisstudy is to obtain non-axisymmetric equilibrium withseparatrix and to evaluate MHD stability by 3-D
Multi-layers Method (MLM).
Like VMEC, MLM is an equilibrium analysis method
based on the variational principle for finding extreme
free energy values (𝛿𝑈 = 0). MLM is distinguished from
VMEC by the fact that MLM uses the XYZ coordinate
system to compute the equilibrium with a separatrix, andby finding the minimum value of the free energy it ispossible to find the equilibrium solution and find thestability at the same time.[3] The model consists of theplasma magnetic surface model, the external magnetic
field coil model, and the conducting shell model (Figure1.).
The plasma model is composed of multiple layers ofmagnetic surfaces (Ψ=const.), which coincides with thecurrent surface in equilibrium, and the plasma current isassumed to be several ring-shaped line currents flowing
only on the magnetic surface. The shape of the magnetic
surface is defined by the Fourier series as follows.
𝑅𝑘 = ∑𝑚𝑛

𝑍𝑘 = ∑𝑚𝑛𝑘𝑅𝑚𝑛𝑒 𝑖(𝑚𝜃−𝑛𝜑) ,(1)𝑘𝑍𝑚𝑛
𝑒 𝑖(𝑚𝜃−𝑛𝜑) ,(2)where 𝑅, 𝑍, 𝑘 are the plasma magnetic surface
coordinates and magnetic surface number.
The free energy of a line current system is given by𝑈(𝒙) =12∑
𝛹𝑖 =const.

𝛹𝑖 𝐼𝑖 (𝒙) −12∑𝛹𝑗 (𝒙)𝐼𝑗 ,(3)𝐼𝑗 =const.

where 𝛹𝑖 , 𝐼𝑖 , 𝒙 are the linkage flux and current of the𝑖th line current and the parameters determine the plasma

position and shape. The first term represents the energy
in the plasma, the vacuum vessel, and other parts that are
not connected to the energy source. While the second
term represents the contribution of the coil to which thefeeder is connected. The magnetic flux 𝛹𝑖 and current
𝐼𝑖 are related by
𝛹𝑖 = ∑ 𝑀𝑖𝑗 (𝒙)𝐼𝑗 ,(4)𝑗where 𝑀𝑖𝑗 is inductance matrix. Using 𝒙 as variables,
we search for the minimum value of the free energy ofthe system by finding the mutual inductance whilechanging the shape of the plasma. The solution at theminimum is the stable equilibrium solution.
Using MLM, we analyzed the stability of the equilibria
with longitudinal cross sections with no vertical
positional instability by applying a non-axisymmetricmagnetic field to the tokamak. This work is supported by
the Grant-in-Aid for Scientific Research (21H01061).
References[1] S.P. Hirshman et al., Comput. Phys. Commun. 43,143 (1986).
[2] S.P. Hirshman et al., Phys. Fluids 26, 3553 (1983).
[3] H. Tsutsui et. al, J. Plasma Fusion Research 72, p.
1252-1258 (1996)

Figure 1. Multi-layers Method model. (a)Including
plasma magnetic surface (red), conducting shell likevacuum vessel (black), and magnetic field coil (gray). (b)Plasma current is directed to the toroidal and poloidal
rings on the magnetic surface. The conducting shell istreated as if the magnetic flux is constant, as is theplasma magnetic surface. Therefore, as the position andshape of the plasma change, the mutual inductance of theplasma between conducting shell also changes, as do theeddy currents.