Toroidal current eigenmodes in the vacuum vessel of an elliptical cross-section tokamak

The currents induced in a tokamak resistive wall during fast transients play a significant role in plasma dynamics, which makes their modelling a mandatory part of the equilibrium problem. In numerical models with axisymmetric walls, the wall is usually divided into a vast number of filaments (sometimes several hundred), in each of which the corresponding circuit equation is solved. It is clear that such a treatment is not suitable for analytical calculations. At the same time, it was established (Portone 2005 Nucl. Fusion 45 926) that only three periodic functions of the poloidal angle, or so-called vessel modes, are sufficient to accurately describe the ITER vacuum vessel response to plasma vertical motion. In our paper, we address the problem of eddy current eigenmode derivation analytically, beginning from Maxwell's equations complemented with Ohm's law for a resistive wall. The result is an explicit form of eddy current eigenmodes in a toroidal shell with an elliptical cross section and their resistive decay times as functions of the wall elongation.