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E.2 Expression of δB∥ in terms of δA

δB∥ = e∥ ⋅∇ × δA = e∥ ⋅∇ × (δA ⊥ + δA∥e∥) (288)
Accurate to O(λ1), δB∥ in the above equation is written as (e∥ vector can be considered as constant because its spatial gradient combined with δA will give O(λ2) terms, which are neglected)
δB ∥ ≈ e∥ ⋅∇ × δA ⊥ + e∥ ⋅(∇ δA∥ ×e∥) = e ⋅∇ × δA (289) ∥ ⊥
[Using local cylindrical coordinates (r,ϕ,z) with z being along the local direction of B0, and two components of δA⊥ being δAr and δAϕ, then ∇× δA⊥ is written as
( ) ( ) [ ] ∂δA-ϕ ∂δAr- 1 ∂-- ∂δAr- ∇ × δA⊥ = − ∂z er + ∂z er + r ∂r(rδAϕ)− ∂ϕ e∥
(290)

Note that the parallel gradient operator ∇∥≡ e∥⋅∇ = ∂∕∂z acting on the the perturbed quantities will result in quantities of order O(λ2). Retaining terms of order up to O(λ), equation (285) is written as

[ ] 1 ∂-- ∂-δAr- ∇ ×δA ⊥ ≈ r ∂r(rδAϕ)− ∂ ϕ e∥,
(291)

Using this, equation (289) is written as

[ ] 1 ∂-- ∂δAr- δB∥ = r ∂r(rδA ϕ)− ∂ϕ .
(292)

However, this expression is not useful for GEM because GEM does not use the local coordinates (r,ϕ,z).]

 

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