Numerical simulation of equilibrium plasma configurations in toroidal traps based on the Morozov-Solovyov equations

The well-known Grad-Shafranov equation has traditionally been used for a long time to study equilibrium configurations in magnetic traps. This is a two-dimensional semi-linear elliptic equation. To close the problem, you 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 replacement $P(\Psi)+\mathrm{const}$ and, therefore, the absolute values of 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, in particular, a general system of hydrodynamic equations of a quasi-neutral two-component ideal plasma for stationary flows is 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 for a resting plasma 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 Morozov-Solovyov equations provide much more information about 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.