General toroidal angle ζ

In Sec. 8.1, we introduced the local safety factor

(ψ,𝜃). Equation (156) indicates that if the Jacobian is chosen to be of the particular form 𝒥 = h(ψ)R², then the local safety factor is independent of 𝜃, i.e., magnetic line is straight in (𝜃,ϕ) plane. On the other hand, if we want to make ﬁeld line straight in (𝜃,ϕ) plane, the Jacobian must be chosen to be of the speciﬁc form 𝒥 = h(ψ)R². We note that, as mentioned in Sec. 8.3, the poloidal angle is fully determined by the choice of the Jacobian. The speciﬁc choice of 𝒥 = α(ψ)R² is usually too restrictive for achieve a desired poloidal resolution (for example, the equal-arc poloidal angle can not be achieved by this choice of Jacobian). Is there any way that we can make the ﬁeld line straight in a coordinate system at the same time ensure that the Jacobian can be freely adjusted to obtain desired poloidal angle? The answer is yes. The obvious way to achieve this is to deﬁne a new toroidal angle ζ that generalizes the usual toroidal angle ϕ. Deﬁne a new toroidal angle ζ by[10]

where δ = δ(ψ,𝜃) is a unknown function to be determined by the constraint of ﬁeld line being straight in (𝜃,ζ) plane. Using Eq. (157), the new local safety factor in (ψ,𝜃,ζ) coordinates is written as

ˆqnew ≡ B-⋅∇-ζ B ⋅∇ 𝜃 = (∇-ϕ×-∇-ψ+-ˆq∇-ψ-×∇-𝜃)⋅∇-ζ (∇ ϕ× ∇ ψ+ ˆq∇ ψ ×∇ 𝜃)⋅∇ 𝜃 ∇-ϕ×-∇-ψ-⋅∇ζ-+qˆ∇-ψ-×-∇𝜃-⋅∇ζ = (∇ϕ × ∇ψ )⋅∇𝜃 ∇ ϕ× ∇ ψ ⋅∇(− qδ)+ ˆq∇ ψ ×∇ 𝜃⋅∇ ϕ = ---------(∇ϕ-×-∇ψ-)⋅∇𝜃--------- = − q∂δ+ ˆq. (251) ∂𝜃

where c(ψ) can be an arbitrary function of ψ. A convenient choice for c(ψ) is c(ψ) = q, i.e., making the new local safety factor be equal to the original global safety factor, i.e.,

_new = q. In this case, equation (252) is written as

where 𝜃_ref is an starting poloidal angle arbitrarily chosen for the integration, and δ(ψ,𝜃_ref) is the constant of integration. In the following, both 𝜃_ref and δ(ψ,𝜃_ref) will be chosen to be zero. Then the above equations is written

Substituting the above expression into the deﬁnition of ζ (Eq. 250), we obtain

which is the formula for calculating the general toroidal angle. If 𝜃 is a straight-ﬁeld line poloidal angle, then ζ in Eq. (256) reduces to the usual toroidal angle ϕ.

In summary, magnetic ﬁeld line is straight in (𝜃,ζ) plane with slope being q if ζ is deﬁned by Eq. (256). In this method, we make the ﬁeld line straight by deﬁning a new toroidal angle, instead of requiring the Jacobian to take particular forms. Thus, the freedom of choosing the form of the Jacobian is still available to be used later to choose a good poloidal angle coordinate. Note that the Jacobian of the new coordinates (ψ,𝜃,ζ) is equal to that of (ψ,𝜃,ϕ). [Proof:

−1 𝒥 new = ∇ψ × ∇ 𝜃⋅∇ ζ = ∇ψ × ∇ 𝜃⋅∇ (ϕ − qδ) = ∇ψ × ∇ 𝜃⋅∇ ϕ− ∇ ψ× ∇ 𝜃⋅∇ (qδ) = ∇ψ × ∇ 𝜃⋅∇ ϕ− 0 − 1 = 𝒥 .

∫ δ(ψ,𝜃+ 2π) = 1 𝜃+2πˆqd𝜃− (𝜃− 𝜃 )− 2π q 0 ref 1∫ 𝜃 1∫ 𝜃+2π = q ˆqd𝜃 − (𝜃 − 𝜃ref)− 2π + q qˆd𝜃 0 ∫ 𝜃+2π 𝜃 = δ(ψ,𝜃)− 2π + 1 ˆqd𝜃 q 𝜃 = δ(ψ,𝜃). (257)

[In numerical implementation, the term ∫ ₀^𝜃 BB-⋅∇⋅∇ϕ𝜃

d𝜃 appearing in δ is computed by using

∫ 𝜃 ∫ 𝜃 ∫ 𝜃 B-⋅∇-ϕd𝜃 = -g-𝒥-d𝜃 = g--1 -R--dlp 0 B ⋅∇ 𝜃 0 R2 Ψ′ 0 R2Ψ ′|∇ ψ| ∫ 𝜃 g-1--- = 0 R |∇ Ψ|dlp ∫ 𝜃 = 1-Bϕdℓp. (258) 0 R Bp

∫ ∫ ∂(δq) -d-(-g ) 𝜃-𝒥- -g ∂-- 𝜃𝒥-- dq- ∂ψ = −dψ Ψ ′ 0 R2 d𝜃− Ψ ′∂ψ 0 R2 d𝜃− dψ 𝜃. (259)

12.1 General toroidal angle ζ