We examine the effect of an external vertical magnetic field on the shape of the boundary magnetic surface of an arbitrary vacuum stellarator configuration. A numerical solution of the fixed-boundary problem makes it possible to determine the cylindrical coordinates R, φ, Z as functions of flux coordinates (e. g. , Boozer coordinates). We derive an equation for the deformation of magnetic surfaces under the influence of vertical field in linear approximation. The equation is a magnetic differential equation, whose right-hand side is expressed in terms of R, , Z, written in flux coordinates, and derivatives. The described approach can be used to estimate the response of divertor islands to an applied vertical field. We show that the effect of the vertical field on the island structure is described (along with the rotational transform and magnetic shear) by the corresponding resonant harmonic in R2 (here, R is the major radius of the cylindrical coordinate system expressed in flux coordinates a, θ, φ with straightened magnetic field lines). The proposed method also makes it possible, without calculating the external conductors that build up the initial configuration, to determine the new vacuum boundary magnetic surface obtained when a vertical field is imposed, and thus to evaluate—using standard equilibrium codes—the properties of the displaced configuration.
Khanaeva et al. (2026) studied this question.