On quasisymmetric plasma equilibria sustained by small force

We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the magnetohydrostatic (steady three-dimensional incompressible Euler) equations with a small force. The solutions are also ‘nearly’ quasisymmetric. The primary idea is, given a desired quasisymmetry direction $\xi$ , to change the smooth structure on space so that the vector field $\xi$ is Killing for the new metric and construct $\xi$ –symmetric solutions of the magnetohydrostatic equations on that background by solving a generalized Grad–Shafranov equation. If $\xi$ is close to a symmetry of Euclidean space, then these are solutions on flat space up to a small forcing.

Exact force-free plasma equilibria with axial and with translational symmetries

Exact force-free plasma equilibria satisfying the nonlinear Beltrami equation are derived. The construction is based on a nonlinear transformation that allows to get from any solution to the linear Beltrami equation a one-parametric family of exact solutions to the nonlinear one. Exact force-free plasma equilibria connected with the Sine-Gordon equation are presented.

Safety Factor for the New Exact Plasma Equilibria

An exact formula for the limit of the safety factor q at a magnetic axis is derived for the general up-down asymmetric plasma equilibria possessing axial symmetry, generalizing Bellan’s formula for the up-down symmetric ones. New exact axisymmetric plasma equilibria depending on arbitrary parameters α, ξ, bkn, zkn, where k = 1, ⋯, M, n = 1⋯, N, are constructed (α ≠ 0 is a scaling parameter), which are up-down asymmetric in general. The equilibria are not force-free if ξ ≠ 0 and satisfy Beltrami equation if ξ = 0. For some values of ξ the magnetic field and electric current fluxes have isolated invariant toroidal magnetic rings, for another ξ they have invariant spheroids (blobs) and for some values of ξ both invariant toroidal rings and spheroids (blobs). A generalization of the Chandrasekhar – Fermi – Prendergast magnetostatic model of a magnetic star is presented where plasma velocity V(x) is non-zero.