Equations of Plasma Equilibrium in a Magnetic Field with Three-Dimensional Magnetic Surfaces

A system of equations that describe the static equilibrium of plasma in a magnetic field with three-dimensional magnetic surfaces of toroidal topology is obtained. The approach is based on the mixed representation for the magnetic field; the representation is similar to that used in the derivation of the classical Grad–Shafranov equation but modified with allowance for the possible axial asymmetry of magnetic surfaces. The system consists of three differential equations for three scalar functions; two of the equations can be written in the form of magnetic differential equations and the third one (the balance of forces in the direction perpendicular to the surface) serves as an analogue of the Grad–Shafranov equation. The obtained equations allow a simple transition to the limiting case of axial symmetry. An example of a solution of the Hill vortex type is presented under the assumption of weak axial asymmetry of the problem. The paper is dedicated to the memory of the brilliant physicist Vitaly D. Shafranov, who could think simply and very precisely.