Minimum-energy state of an axi-symmetric toroidal plasma

A constrained condition of the toroidal flux function I is proposed for equilibria with a minimum energy state of an axi-symmetric plasma. Following Bhattacharjee, the authors use a modified helicity Kw triple-bond ∫v ω(ψ)A · Bdv as a constraint to plasma motion, where ω(ψ) is an arbitrary function of poloidal flux function ψ. For non-ideal motions ζ of a slightly resistive plasma, the magnetic flux is not conserved completely. The authors may, therefore, define the variation as those for which δψ = -ζ · ∇ψ, δp = -ζ · ∇p - γp∇ · ζ, whereas I varies independent on ζ. And by minimizing the potential energy subject to the invariant Kw, they derive Grad-Shafranov equation and dI/dψ = -λ((ψ - ψc) dω/dψ + 2ω) as Euler equations, where λ is a Lagrange multiplier. Next they analyze the equilibrium of a cylindrical plasma as an infinite aspect ratio approximation. In case that dp/dψ ≠ 0 near the magnetic axis, for example p = p0(ψ/ψc)2, it is shown that there exists a diamagnetic equilibrium like a tokamak configuration as a minimum energy state that field reversal ratio F > 1 and pinch parameter θ << 1. Calculated F - θ diagram, also, shows that the θ value of F = 0 becomes higher as β increasing or as α decreasing in which case the current channel shrinks to the magnetic axis. The proposed constrained condition of I is demonstrated to simulate the another equilibrium models of cylindrical RFP and to have more degrees of freedom