Expression of ∇α

11.3 Expression of ∇α

Next, let us calculate the gradient of the generalized toroidal angle α, which is deﬁned by Eq. (299), i.e., α = ϕ −δ, where δ = ∫ ₀^𝜃 d𝜃 = ∫ ₀^𝜃B ⋅∇ϕ∕(B ⋅∇𝜃)d𝜃. The gradient of α is written as

∇ α = ∇ ϕ− ∇ δ ˆ - - = ϕ-− ∂δ∇ ψ − ∂δ∇ 𝜃. (301) R ∂ψ ∂𝜃

Using Eqs. (208) and (209), the above expression is written as

- - ˆϕ- ∂δ-R- ˆ ˆ ∂-δR- ˆ ˆ ∇ α = R + ∂ψ 𝒥 (Z 𝜃R − R𝜃Z)− ∂ 𝜃𝒥 (ZψR − Rψ Z) ˆϕ ( ∂δR ∂δR ) ( ∂δ R ∂δR ) = --+ -----Z𝜃 − ----Zψ Rˆ+ ----R ψ −-----R 𝜃 ˆZ. (302) R ∂ψ 𝒥 ∂𝜃𝒥 ∂𝜃𝒥 ∂ ψ𝒥

(Note that ∂δ∕∂ψ is discontinuous across the 𝜃 cut.) Then

( ) [ ˆ ( - - ) ( - - ) ] ∇ψ ⋅∇ α = − RZ 𝜃ˆR + R-R𝜃ˆZ ⋅ ϕ- + ∂δ-R-Z𝜃 − ∂δR-Zψ ˆR + ∂δR-R ψ − ∂-δR-R𝜃 ˆZ 𝒥 𝒥 R ∂ψ 𝒥 ∂𝜃𝒥 ∂𝜃𝒥 ∂ψ 𝒥 R ( ∂δ R ∂δR ) R ( ∂δR ∂δ R ) = − 𝒥-Z𝜃 ∂ψ-𝒥-Z𝜃 − ∂𝜃𝒥-Zψ + 𝒥-R 𝜃 ∂𝜃𝒥-Rψ − ∂ψ-𝒥-R𝜃 . (303)

[ ] ϕˆ ( ∂δ R ∂δR ) ( ∂δR ∂ δR ) ( R R ) ∇α ⋅∇ 𝜃 = R- + ∂ψ-𝒥-Z𝜃 − ∂𝜃𝒥-Zψ ˆR + ∂𝜃𝒥-R ψ − ∂ψ-𝒥-R𝜃 ˆZ. ⋅ 𝒥-ZψRˆ− 𝒥 R ψˆZ ( - - ) ( - - ) ∂δ-R- ∂δ-R- R- ∂δ-R- ∂δ-R- R- = ∂ψ 𝒥 Z 𝜃 − ∂𝜃 𝒥 Zψ 𝒥 Zψ − ∂𝜃 𝒥 Rψ − ∂ψ 𝒥 R𝜃 𝒥 Rψ. (304)

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Fig. 14: The value of 𝜃, α and their gradients on an annulus on the poloidal plane (R,Z). Note that ∇𝜃 is single-valued while ∂α∕∂R and ∇α are multi-valued and thus there is a jump near the branch cut when a single branch is chosen.

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Fig. 15: The same as Fig. 14, but gradients are computed in cylindrical coordinates. The results agrees with those of Fig. 14, which provides the conﬁdence in the correctness of the numerical implementation.

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