A three-dimensional magnetohydrodynamic equilibrium in an axial coordinate with a constant curvature

Abstract

Motivated by the superior confinement observed in the relaxed three dimensional (3D) states in the reversed field pinch, 3D plasma equilibria in coordinate systems based on space curves with a constant curvature as the axial coordinates are studied by using the method of metric perturbation. Constancy of the curvature allows the development of magnetohydrodynamic equilibrium with asymptotic good 2D flux surfaces near the coordinate axis. The perturbation parameter is the product of the torsion variation along the coordinate axis and the distance from it. The lowest order equilibrium with good 2D flux surfaces is symmetric with respect to translation along the space curve. It embodies the approximate toroidal-helical symmetry and is determined by a generalized Grad–Shafranov equation which includes information of the constant curvature and the average torsion of the space curve. Based on this fundamental equilibrium, a formal scheme is developed that allows us to find the ideal MHD equilibrium taking into account the full metric variation of the torsion along the spatial axis. In this limit, the flux surfaces are shown to exist for the full plasma. Numerical examples are given for the lowest order equilibria. It is suggested that equilibria based on this type of coordinates can allow easier evaluation of plasma shapes and magnetic boundary conditions that are more compatible with the relaxed central core. This could be the needed requirement for strongly self-organized equilibria with a large good confinement region.

Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis

“…This method allowed for several significant analytical results in the context of stellarator equilibria. An estimate for equilibrium and stability β-values using the direct approach was first given in Lortz & Nuhrenberg (1976, 1977 by carrying out the expansion up to third order in ρ. Higher-order formulations of the direct approach were also used in Bernardin, Moses & Tataronis (1986), Salat (1995) to prove important geometric properties of MHD equilibria and, more recently, in Chu et al (2019), to obtain a generalized Grad-Shafranov equation for near-axis equilibria with constant axis curvature. Finally, we note that the direct method can also be used to derive a Hamiltonian formulation for the magnetic field lines and obtain adiabatic invariants to successively higher order in ρ (Bernardin & Tataronis 1985).…”