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8.1 Local safety factor

The local safety factor ˆq is defined by

B ⋅∇ ϕ ˆq = B-⋅∇-𝜃,
(155)

which characterizes the local pitch angle of a magnetic field line in (𝜃,ϕ) plane of a magnetic surface. Substituting the contravariant representation of the magnetic field, Eq. (153), into the above equation, the local safety factor is written

ˆq(ψ,𝜃) = − g-𝒥-. R2 Ψ′
(156)

Note that the expression ˆq in Eq. (156) depends on the Jacobian 𝒥 . This is because the definition of ˆq depends on the definition of 𝜃, which in turn depends on the the Jacobian 𝒥 .

In terms of ˆq , the contravariant form of the magnetic field, Eq. (153), is written

B = − Ψ′(∇ϕ × ∇ψ + ˆq∇ψ × ∇ 𝜃).
(157)

and the parallel differential operator B0 ⋅∇ is written as

( ) B0 ⋅∇ = − Ψ′(∇ ϕ× ∇ ψ + ˆq∇ ψ × ∇𝜃)⋅∇ = − Ψ′𝒥 −1 -∂-+ ˆq-∂- . ∂ 𝜃 ∂ϕ
(158)

If ˆq happens to be independent of 𝜃 (i.e., field lines are straight in (𝜃,ϕ) plane), then the above operator becomes a constant coefficient differential oprator (after divided by 𝒥−1). This simplification is useful because different poloidal harmonics are decoupled in this case. We will discuss this issue futher in Sec. 12.

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