Expressions are obtained for the magnetic field created by the currents flowing in the tokamak plasma with elliptic cross section. The analytical approach is based on the Green function method, which ensures the exact separation of the desired solution for given boundary conditions and its correct asymptotic behavior at infinity. The system toroidicity is retained in the linear approximation, as it is done in the standard models of the equilibrium theory, but the magnetic field B at the plasma boundary is described with account for the second harmonic in the poloidal angle. This fact, as well as the noncircular shape of the plasma boundary, on which the inner and outer solutions for the field B must be matched, radically changes the nature of the problem. A detailed description of the techniques for including new elements into the general scheme is given. The result is presented in terms of the poloidal flux as a function of coordinates; its gradient determines the local values of the self field.

In the plasma equilibrium theory, Gajewski's analytical expression [Gajewski, Phys. Fluids 15, 70 (1972)] for the poloidal magnetic flux $\psi$ outside the plasma is known. It was obtained as a solution of the two-dimensional Laplace equation outside an infinite straight cylinder with an elliptical cross section and a uniform current density $j_{\zeta }$ . An example of its use for analysis of static configurations is given in the study by Porcelli and Yolbarsop [Phys. Plasmas 26, 054501 (2019)]. Here, we consider the question of its applicability in dynamic problems including, for example, the current quench (CQ) or vertical displacement event (VDE), when the electromagnetic response of the vacuum vessel to the plasma magnetic field evolution has to be accounted for. It is shown that the mentioned cylindrical model does not provide enough information for calculation of the current induced in the wall. Mathematically, this manifests itself in the fact that Gajewski's expression contains an indefinite constant of integration ${\psi }_b$ (hereinafter it is $\psi$ at the plasma boundary), which, in analytical applications, is replaced either by zero or by a value that makes $\psi =0$ on the magnetic axis. This does not affect the magnitude of the magnetic field $\mathbf{B}$ , but it would incorrectly give the electric field at $\partial \mathbf{B}/\partial t\ne 0$ . To eliminate this shortcoming, an additional block of calculations in the toroidal geometry is needed. Here, the problem is solved analytically. The resulting final expression with ${\psi }_b$ well-defined in the toroidal configuration also includes the effects of the Shafranov's shift and inhomogeneity of $j_{\zeta }$ . The proposed extensions allow generalization of the earlier results to a wider area and cover such events as CQ or VDE.

The dynamic problem of plasma equilibrium in a tokamak is considered taking into account the electromagnetic reaction of the vacuum vessel resistive wall. The currents induced in the wall during transient events contribute to the external magnetic field that determines the plasma shape and position. Accordingly, the plasma geometry must evolve so that the inductive excitation of the wall current would properly compensate for the resistive losses. Simultaneous consideration of these factors presents the main difficulty of the description. It is performed in a general form using the Green’s function method that guarantees the mathematical accuracy of expressions for the magnetic fields from each source. At the same time, it is desirable to minimize the related complications, which is one of the goals here. The starting point is the standard solution of the external equilibrium problem given by integral relating the poloidal magnetic flux to the magnetic field at the plasma boundary. In the evolutionary problem, the additional equations for the plasma-wall electromagnetic coupling are transformed to an equation with a similar integral over the wall, but with either the time derivative of the poloidal magnetic flux or the wall current density in the integrand. The mentioned similarity allows to use the already developed techniques, which makes this formulation compact and convenient. It provides the basis for extension of the existing analytical theory of equilibrium to the case with non-circular plasma and wall.

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.

Analytical evaluation of the vertical displacement event (VDE) growth rate is the main subject here. The attention is attracted to the fact that the model with the plasma and wall boundaries treated as confocal ellipses, introduced by Laval et al. [Phys. Fluids 17, 835 (1974)] and used up until now in analytics, not only restricts the applicability range, but also strongly affects the outcome. In particular, improper utilization of the aforementioned model (i.e., by neglecting the assumption of the wall's confocality) leads to the prediction that the resistive wall with circular cross section cannot slow down the VDE—a conclusion beyond the applicability of the model. Here, we propose a new model and a new approach assuming rigid plasma motion, where the plasma is separated from the wall, and the main element in the task is the plasma–wall electromagnetic interaction. First, we propose a VDE growth time evaluation based on integral equation for the poloidal flux diffusion through the thin resistive shell, which is applicable for the walls of arbitrary shape. Then, the task is simplified by using the large-aspect-ratio expansion of the Green's function. As a result, a slow wall-stabilized motion is predicted in wide ranges of the plasma elongations and the width of the plasma–wall gap. The differences between the existing results and our new predictions are explained.