Numerical solutions of Grad-Shafranov equation in a field-reversed configuration

The solution of Grad-Shafranov equation in field-reversed configuration (FRC) is a basic problem. The solution of Grad-Shafranov equation can help to understand most of physical processes in FRC plasma, such as magnetohy-drodynamic (MHD) instabilities and plasma transport. In the present paper, based on the FRC asymptotic theory by Barnes D C, the code for solving the two-dimensional Grad-Shafranov equation in FRC is developed. By using the code, the equilibriums of FRC with different elongations and separatrix radii are investigated in the present paper. The one-dimensional numerical results show that the plasma density gradient increases linearly with magnetic flux increasing in the FRC center, while, it steepens due to the high magnetic field distribution at the separatrix. The results also show that the plasma density in the closed field region increases with the density at the separatrix increasing, which implies that FRC embodies the strong confinement ability. It is a key problem to choose equations determining the shape of the separatrix in a two-dimensional numerical investigation. In the present paper, the shape equation is described as r(s) = r(s) (max) (1- z(2a)), in which a is the shaping parameter. When a = 1, the separatrix shape is elliptical, and when a > 1, the separatrix shape is like a racetrack. The geometry character of the separatrix appears in the one-order equations (in one-order equations: partial derivative psi(0)/partial derivative z = partial derivative psi(0)/partial derivative r(s) partial derivative r(s)/partial derivative r(z), where partial derivative psi(0)/partial derivative r(s) is determined by lead equations and partial derivative r(s)/partial derivative z is given by separatrix equation). The two-dimensional numerical results show that O-point moves outward as the sparatrix radius increases. The curvature radius of magnetic flux surface increases with the separatrix radius increasing. The O-point of magnetic flux surface is just at the curvature center. Thus O-point moves outward as the sparatrix radius increases.