Toroidal elliptic operator in magnetic surface coordinate system

If (ψ,𝜃,ϕ) are magnetic surface coordinates, i.e., ∂Ψ∕∂𝜃 = 0, then the toroidal elliptic operator in Eq. (389) is reduced to

2[( ) ( ) ] △ ⋆Ψ = R-- Ψψ 𝒥-|∇ ψ|2 + Ψψ-𝒥-∇ψ ⋅∇ 𝜃 , 𝒥 R2 ψ R2 𝜃

(565)

and the GS equation, Eq. (392), is reduced to

[ ] R2- ( 𝒥-- 2) ( 𝒥-- ) 2dP- dg- 𝒥 Ψψ R2|∇ψ | + Ψ ψR2 ∇ψ ⋅∇𝜃 = − μ0R dΨ − dΨg(Ψ). ψ 𝜃

(566)