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5 Covariant/contravariant representation of equilibrium magnetic field

The axisymmetric equilibrium magnetic field is given by Eq. (67), i.e.,

B = ∇ Ψ × ∇ϕ + g∇ϕ.
(147)

In a general coordinate system (ψ,𝜃,ϕ) (not necessarily magnetic surface coordinates), the above expression can be written as

B = − Ψψ∇ ϕ× ∇ ψ − Ψ𝜃∇ϕ × ∇𝜃 + g∇ϕ,
(148)

where the subscripts denote the partial derivatives with the corresponding subscripts. Note that Eq. (148) is a mixed representation, which involves both covariant and contravariant basis vectors. Equation (148) can be converted to the contravariant form by using the metric tensor, giving

𝒥-- B = − Ψψ∇ ϕ ×∇ ψ − Ψ𝜃∇ϕ × ∇𝜃 + gR2∇ ψ × ∇𝜃.
(149)

Similarly, Eq. (148) can also be transformed to the covariant form, giving

( 𝒥-- 𝒥-- 2) ( -𝒥- 2 -𝒥- ) B = Ψ ψR2∇ ψ ⋅∇𝜃 + Ψ𝜃R2|∇ 𝜃| ∇ψ + − ΨψR2 |∇ ψ| − Ψ𝜃R2 ∇𝜃 ⋅∇ψ ∇ 𝜃+ g∇ ϕ.
(150)

For the convenience of notation, define

𝒥 hαβ = R2∇ α⋅∇ β,
(151)

then Eq. (150) is written as

ψ𝜃 𝜃𝜃 ψψ ψ𝜃 B = (Ψψh + Ψ𝜃h )∇ ψ + (− Ψψh − Ψ𝜃h )∇𝜃 + g∇ ϕ.
(152)

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