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14.3 Explicit expression for the generalized toroidal angle

For the above magnetic field, the toroidal shift involved in the definition of the generalized toroidal angle can be expressed in simple analytical form. The toroidal shift is given by

- ∫ 𝜃 δ = ˆqd𝜃, 0
(367)

where the local safety factor qˆ can be written as

g0𝒥 ˆq = −R2-Ψ-′.
(368)

Using 𝒥 = −Rr and

′ -1--∘-g0r---- Ψ = q(r) R20 − r2
(369)

The local safety factor ˆq in Eq. (368) is written as

∘ --2---2 1- ˆq = q R 0 − r R .
(370)

Using this, expression (367) is written

- ∘ -------∫ 𝜃1 δ = q R20 − r2 R-d𝜃 0
(371)

Assume 𝜃 ∈ (−π,π), then the integration ∫ 0𝜃1∕Rd𝜃 can be analytically performed (using maxima), yielding

( ) ∫ 𝜃 2arctan ---sin𝜃(R0√−r2)-2 -1d𝜃 = -------∘(cos𝜃+1)-R0−r---. 0 R R20 − r2
(372)

Then expression (371) is written

( ) - (R0 − r) (𝜃 ) δ = 2q arctan ∘--2---2-tan 2 , R0 − r
(373)

where use has been made of sin𝜃∕(cos𝜃 + 1) = tan(𝜃∕2). Using this, the generalized toroidal angle can be written as

- α = ϕ− δ ( ) (R0 − r) (𝜃) = ϕ− 2qarctan ∘R2-−-r2-tan 2 . (374) 0
The results given by the formula (373) are compared with the results from my code that assumes a general numerical configuration. The results from the two methods agree with each other, as is shown in Fig. 27, which provides confidence in both the analytical formula and the numerical code.

pict

Fig. 27: The results of δ = ∫ 0𝜃qˆ d𝜃 computed by using formula (373) and the numerical code agree with each other. The different lines correspond to values of δ on different magnetic surfaces. In the numerical code, two kinds of poloidal angles can be selected: one is the equal-volume poloidal angle, and another is the equal-arch-length angle. Make sure that the latter is selected when doing the comparison because the the poloidal angle 𝜃 appearing in the analytical formula is the equal-arc-length poloidal angle.

In passing, we note that the straight-field-line poloidal angle 𝜃f can also be considered to be defined by

α = ϕ − q𝜃f,
(375)

i.,e,

- 1 δ 𝜃f = q(ϕ − α ) = q.
(376)

Then using Eq. (374), 𝜃f is written as

( ) (R0 − r) (𝜃 ) 𝜃f = 2 arctan ∘--2---2-tan 2 , R0 − r
(377)

which agrees with Eq. (A2) in Gorler’s paper[13].

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