Global safety factor

8.2 Global safety factor

The global safety factor deﬁned in Eq. (37) is actually the poloidal average of the local safety factor, i.e.,

∫ 2π q(ψ) ≡ -1- ˆqd𝜃 (159) 2π 0 1 g ∫ 2π 𝒥 = −2π-Ψ′ 0 R2d𝜃. (160)

Note that q and

deﬁned this way can be negative, which depends on the choice of the positive direction of ϕ and 𝜃 coordinates (note that the safety factor given in G-eqdsk ﬁle is always positive, i.e. it is the absolute value of the safety factor deﬁned here).

Next, let us transform the 𝜃 integration in expression (160) to a curve integral in the poloidal plane. Using the relation dℓ_p and d𝜃 [Eq. (168)], expression (160) is further written

1--g ∮ -dℓp-- q(ψ) = − 2πΨ ′ sign(𝒥 )R|∇ψ | 1 sign(𝒥) ∮ dℓ = − --g------′ ---p--. (161) 2π sign(Ψ ) R |∇ Ψ|

Expression (161) is used in the GTAW code to numerically calculate the value of q on magnetic surfaces (as a benchmarking of the q proﬁle speciﬁed in the G-eqdsk ﬁle). Expression (161) can also be considered as a relation between q and g. In the equilibrium problem where q is given (ﬁxed-q equilibrium), we can use expression (161) to obtain the corresponding g (which explicitly appears in the GS equation):

( ∮ --dℓp--)−1 sign(Ψ′) g = − 2πq R |∇ Ψ| sign(𝒥 ).

(162)

We note that expression (162) involves magnetic surface averaging, which is unknown before we know Ψ. Therefore iteration is usually needed in solving the ﬁxed-q equilibrium (i.e., we guess the unknown Ψ, so that the magnetic surface averaging in expression (162) can be performed, yielding the values of g.)

Using B_p = |∇Ψ|∕R and B_ϕ = g∕R, the absolute value of q in expression (161) is written

∮ |q| =-1-g --dℓp-. (163) 2π R |∇ Ψ| 1 ∮ 1|Bϕ| = 2π- R--Bp-dℓp (164)

which is the familiar formula we see in textbooks.

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