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12.4 Metric elements

Let ψ = r and define hψR = ∇ψ ⋅Rˆ, hαR = ∇α⋅ˆR, etc. Explicit expressions for these elements can be written as

ψR h = ∇ψ ⋅ ˆR = ∇r ⋅ ˆR R- ˆ ˆ ˆ = −𝒥 (Z𝜃R − R 𝜃Z )⋅R R- = −𝒥 Z 𝜃 = cos 𝜃 (375)
hψZ = ∇ψ ⋅ ˆZ ˆ = ∇r ⋅Z = − R(Z𝜃Rˆ− R 𝜃ˆZ)⋅ ˆZ 𝒥 = R-R𝜃 𝒥 = sin𝜃
ψϕ ˆ h = ∇ψ ⋅ϕ = 0
h𝜃R = ∇ 𝜃⋅ ˆR = R-(ZψRˆ− R ψˆZ)⋅Rˆ 𝒥 = R-Z 𝒥 ψ 1 = − r sin 𝜃
h 𝜃Z = ∇ 𝜃⋅ ˆZ R- ˆ ˆ ˆ = 𝒥 (ZψR − R ψZ)⋅Z R- = − 𝒥 Rψ 1 = r cos𝜃
h𝜃ϕ = ∇𝜃 ⋅ ˆϕ = 0
h αR = ∇ α ⋅ ˆR [ ˆ ( - - ) ( - - ) ] = ϕ-+ -∂δR-Z𝜃 − ∂δR-Zψ Rˆ+ ∂δ-RR ψ − ∂δR-R 𝜃 ˆZ ⋅ ˆR R ∂ψ 𝒥 ∂𝜃𝒥 ∂𝜃𝒥 ∂ψ𝒥 ∂δ R ∂δ R = ∂ψ𝒥-Z 𝜃 − ∂𝜃-𝒥 Z ψ - - = ∂δ-Rr cos𝜃 − ∂δ-R sin𝜃 ∂ψ𝒥 ∂𝜃𝒥 ∂δ- ∂-δ1 = − ∂ψ cos𝜃+ ∂ 𝜃r sin𝜃 ∂δ 1 = − ---cos 𝜃+ ˆq-sin𝜃 ∂r r
hαZ = ∇[ α ⋅ ˆZ ] ˆϕ ( ∂δR ∂δ R ) ( ∂δ R ∂δ R ) = R-+ ∂-ψ𝒥-Z 𝜃 − ∂𝜃𝒥-Z ψ ˆR + ∂𝜃-𝒥-Rψ − ∂ψ-𝒥 R 𝜃 Zˆ ⋅ ˆZ - - ∂δR- ∂-δR- = ∂𝜃𝒥 R ψ − ∂ψ 𝒥 R𝜃 1 ∂δ = − ˆq- cos𝜃 − ---sin𝜃 r ∂r
1 hαϕ = R-.
(376)

Using expression (370), dδ∕dr can be evaluated analytically, yielding

- ( ( ) ) ( ) [ ] dδ = 2dqarctan (∘R0-−-r)-tan 𝜃 + 2q---(-----1-------)--tan 𝜃 -d (∘R0-−-r)- dr dr R20 − r2 2 (√R0−r)- ( 𝜃) 2 2 dr R20 − r2 1+ R20−r2 tan 2 ( ( ) ) = 2dqarctan (∘R0-−-r)-tan 𝜃 dr R20 − r2 2 1 ( 𝜃) − R0 + 2q---(----------(--))2-tan 2 (R--+r)∘R2--−-r2 1+ (√RR02−−rr)2 tan 𝜃2 0 0 0
where use has been made of
d --- dxarctan(x) = 1 ----2- 1+ x

(I did not remember this formula and I use SymPy to obtain this.) These expressions are used to benchmark the numerical code that assume general flux surface shapes. The results show that the code gives correct result when concentric circular flux surfaces are used.

 

Taking the 𝜃 derivative of δ, equation (370) is written as (using Sympy)

∂δ ( tan2(𝜃) 1) 1 --= 2qA -----2-+ - -2---2(𝜃)---- ∂𝜃 2 2 A tan 2 + 1
(377)

where

A = ∘(R0-− r) R20 − r2
(378)

Equation (367) should be equal to qˆ given by Eq. (367). This was verified numerically.

 

Taking the r derivative of Eq. (350), we obtain

dΨp= -1-∘-2πg0r-- dr q(r) R20 − r2
(379)

i.e.,

dΨ-= -1--∘--g0r---. dr q(r) R20 − r2
(380)

dψ-= ----1------1--∘--g0r--. dr Ψ(a)− Ψ(0)q(r) R20 − r2
(381)

 

 

pict pict

 

Fig. 28: ∂δ∕∂𝜃 and ∂δ∕∂ψ. Numerical and analytical results are plotted, which are so close to each other that they can not be distinguished by eyes. ∂δ∕∂𝜃 is equal to the local safety factor ˆq. Note that ∂δ∕∂ψ is discontinuous at the 𝜃 cut, which in this case is on the high-field-side.

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