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6 Magnetic surface coordinates (ψ,𝜃,ϕ)

A coordinate system (ψ,𝜃,ϕ), where ϕ is the usual cylindrical toroidal angle, is called a magnetic surface coordinate system if Ψ is a function of only ψ, i.e., ∂Ψ∕∂𝜃 = 0 (we also have ∂Ψ∕∂ϕ = 0 since we are considering axially symmetrical case). In terms of (ψ,𝜃,ϕ) coordinates, the contravariant form of the magnetic field, Eq. (149), is written as

B = − Ψ ′∇ ϕ× ∇ ψ +g 𝒥-∇ ψ ×∇ 𝜃, R2
(153)

where Ψ′≡ dΨ∕dψ. The covariant form of the magnetic field, Eq. (150), is written as

( 𝒥 ) ( 𝒥 ) B = Ψ′R2∇ ψ ⋅∇𝜃 ∇ψ + − Ψ′R2-|∇ ψ|2 ∇ 𝜃+ g∇ ϕ.
(154)

  6.1 Local safety factor
  6.2 Global safety factor
  6.3 Relation between Jacobian and poloidal angle 𝜃
  6.4 Calculating poloidal angle
   Normalized poloidal angle
   Jacobian for equal-arc-length poloidal angle
   Jacobian for equal-volume poloidal angle (Hamada coordinate)
   Jacobian of Boozer’s poloidal angle
   Jacobian for PEST poloidal angle (straight-field-line poloidal angle)
   Comparison between different types of poloidal angle
   Verification of Jacobian
   Radial coordinate

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