Multimode Analysis of the Resistive Wall Mode Instability

We have developed a code [D. A. Maslovsky and A. H. Boozer, \textit{Phys. Plasmas}, \textbf{12}, 42108 (2005)] that uses a full spectrum of ideal MHD plasma modes and energies calculated by Alan Glasser's DCON, to compute the plasma effective inductance matrix $\stackrel{\leftrightarrow}{\Lambda}$ and the plasma stability matrix $\stackrel{\leftrightarrow}{S}$. The $\stackrel{\leftrightarrow}{\Lambda}$ matrix provides the normal magnetic field on the plasma surface produced by a surface current in the presence of plasma, thus describing plasma response properties to an applied external magnetic perturbation. Combined with the plasma surface inductance matrix $\stackrel{\leftrightarrow}{L}_{p}$, a purely geometric quantity, the plasma stability matrix can be obtained $\stackrel{\leftrightarrow}{S} \equiv \stackrel{\leftrightarrow}{L}_{p}^{1/2} \cdot \stackrel{\leftrightarrow} {\Lambda}^{-1} \cdot \stackrel{\leftrightarrow}{L}_{p}^{1/2}$, with the stability coefficients $-s_{i}$ as its eigenvalues in a non-rotating ideal plasma. By including the complete multimode plasma representation into VALEN, which models the surrounding conducting structures, the plasma response properties for arbitrary values of plasma stability coefficients $s_{i}$, and mode coupling effects, can be accurately assessed. In particular, cases when the value of the stability parameter of the least stable mode is of the order of unity, $s_{u} \sim 1$, and when the value of the stability parameter $s_{2}$ of the second mode is closer to zero than that of the least stable mode, $|s_{2}| < |s_{u}|$ can be analyzed. We discuss ITER-relevant cases of plasma stability calculation and feedback systems optimization.