Efficient computation of the magnetic field created by a toroidal volumetric current of convex cross section with application to the study of the magnetic confinement in tokamaks

In this paper we present an efficient approach for the numerical computation of the static vector potential and the poloidal magnetic field of a toroidal volumetric current with arbitrary convex cross section. The standard integral expressions for both the vector potential and the magnetic field include singularities that have a deleterious effect in the computation of these integrals. In order to handle these singularities, we first introduce a change of variables to polar coordinates with origin at the observation point that makes it possible to remove the singularities of the integrands thanks to the Jacobian factor. Then, two different numerical integration methods are applied to the resulting integrals: Ma-Rokhlin-Wandzura quadrature rules and the double exponential quadrature rule. Both methods efficiently handle the singularities in the derivative of the integrand for the integrals of the vector potential and the magnetic field, and the advantages and disadvantages of each method are discussed. The results obtained for the vector potential and magnetic field are validated by comparing with closed-form results existing for the vector potential and magnetic field of a circular loop and an infinite cylinder, and good agreement is found. Then, the magnetic field code is used to model the plasma toroidal current in a tokamak nuclear fusion reactor, and it is shown that the combined magnetic field of the plasma current and that of the poloidal and toroidal coils leads to magnetic confinement of the charged particles existing in the plasma.