Numerical Simulation of Equilibrium Plasma Configurations in Toroidal Traps Based on the Morozov–Solovyov Equations

Abstract

The well-known Grad–Shafranov equation has traditionally been used for many years to study equilibrium configurations in magnetic traps. This is a two-dimensional semi-linear elliptic equation. To close the problem, we need to set two functions: the plasma pressure (as a function of the magnetic flux) and the total current function. Having solved the problem, we get a magnetic field and a pressure distribution. The magnetic field is invariant with respect to the replacement of $P(\Psi ) + \operatorname{const}$ and, therefore, the absolute values of the plasma concentration and temperature cannot be determined. In 1974, A.I. Morozov and L.S. Solovyov published an article “Stationary plasma flows in a magnetic field.” In this paper, a general system of hydrodynamic equations of a quasi-neutral two-component ideal plasma for stationary flows was written out. For the case of axial symmetry, the authors managed to write this system in a more visible form by introducing three flow functions (magnetic field, electrons, and ions). This very complex system of equations is somewhat simplified for the case of a resting plasma: now two flow functions are sufficient: the magnetic field and electrons. In this paper, the Morozov–Solovyov equations (MS equations) for a plasma at rest in their most general form will be used for the first time to study stationary plasma configurations in a toroidal magnetic trap with a Z-elongated cross-section shape. The geometric parameters correspond to two operating tokamaks JET and JT60. The main conclusion is that the MS equations provide much more information on the properties of equilibrium configurations than the Grad–Shafranov equation. In particular, it is possible to find the absolute values of the concentration of the retained plasma.