Grad-Shafranov equation in (r,𝜃,ϕ) coordinates

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A.13 Grad-Shafranov equation in (r,𝜃,ϕ) coordinates

Deﬁnition of (r,𝜃,ϕ) coordinates

Deﬁne (r,𝜃,ϕ) coordinates by

R = R0 + rcos𝜃,

(567)

Z = rsin 𝜃,

(568)

where (R,ϕ,Z) are the cylindrical coordinates and R₀ is a constant. The above transformation is shown graphically in Fig. 40.

pict

Fig. 40: The relation between coordinates (r,𝜃) and (R,Z).

The Jacobian of (r,𝜃,ϕ) coordinates can be calculated using the deﬁnition. Using x = R cosϕ, y = R sinϕ, and z = Z, the Jacobian (with respect to the Cartesian coordinates (x,y,z)) is written as

| | | | ||∂∂xr ∂∂x𝜃 ∂∂xϕ|| ||∂R∂corsϕ ∂Rc∂o𝜃sϕ-∂R∂coϕsϕ|| 𝒥 = ||∂y ∂y ∂y||= ||∂R-sinϕ ∂Rsinϕ-∂R-sinϕ|| ||∂∂rz ∂∂𝜃z ∂∂ϕz|| || ∂∂rZ- ∂∂Z𝜃- ∂∂ϕZ- || |∂r ∂𝜃 ∂ϕ ∂r ∂𝜃 ∂|ϕ ||cos𝜃cosϕ − rsin𝜃cosϕ − R sin ϕ|| = ||cos𝜃sinϕ − rsin𝜃sinϕ R cosϕ || | sin𝜃 r cos 𝜃 0 | = sin 𝜃(− Rr sin𝜃 cos2ϕ − Rrsin𝜃sin2ϕ) + rcos𝜃(− R sin2 ϕcos𝜃− R cos2ϕcos𝜃) 2 2 = − Rrsin 𝜃− Rr cos𝜃 = − Rr. (569)

Toroidal elliptic operator Δ^⋆Ψ in (r,𝜃,ϕ) coordinate system

Next, we transform the GS equation from (R,Z) coordinates to (r,𝜃) coordinates. Using the relations R = R₀ + r cos𝜃 and Z = r sin𝜃, we have

∘-------------- ∂r Z r = (R − R0)2 + Z2 ⇒ ---= --= sin𝜃 ∂Z r

(570)

-------Z------- ∂𝜃- r−-Z-Zr- sin 𝜃 = ∘ (R-−-R-)2 +-Z2-⇒ cos𝜃 ∂Z = r2 . 0

(571)

∂𝜃 r− ZZ- 1− sin2 𝜃 cos2𝜃 ⇒ ∂Z-= r(R-−-Rr) = -R-−-R--= R-− R-- 0 0 0

(572)

∂ sin𝜃 r − ZZ- 1− sin2𝜃 cos2𝜃 ------= ---2-r-= --------= ----- ∂Z r r r

(573)

The GS equation in (R,Z) coordinates is given by

2 ( ) ∂-Ψ2 + R ∂- -1∂-Ψ = − μ0R2dP-− -dgg(Ψ). ∂Z ∂R R ∂R dΨ dΨ

(574)

The term ∂Ψ∕∂Z is written as

∂ Ψ ∂Ψ ∂r ∂Ψ ∂ 𝜃 --- = ------+ ------ ∂Z ∂r ∂Z ∂𝜃∂Z 2 = ∂Ψ-sin 𝜃+ ∂Ψ--cos-𝜃. (575) ∂r ∂ 𝜃R − R0

Using Eq. (575), ∂²Ψ∕∂Z² is written as

∂2Ψ ∂ ( ∂Ψ ) ∂ (∂Ψ cos2 𝜃 ) --2-= --- ---sin 𝜃 + --- ---------- ∂Z ∂Z ∂r( ) ∂Z ∂ 𝜃R − R0 ( ) ( ) = sin 𝜃 ∂- ∂Ψ- + ∂Ψ--∂-(sin𝜃)+ cos2𝜃--∂- ∂Ψ- + ∂Ψ-∂- -cos2-𝜃- ∂Z ∂r ∂r ∂Z R− R0 ∂Z ∂𝜃 ∂𝜃∂Z R − R0 (∂2Ψ ∂2Ψ cos2 𝜃) ∂Ψ cos2𝜃 cos2𝜃 ( ∂2Ψ ∂2Ψ cos2𝜃 ) = sin 𝜃 -∂r2 sin𝜃+ ∂r∂𝜃R-−-R0- + -∂r--r-- + R-−-R0- ∂𝜃∂r-sin𝜃 + ∂𝜃2-R-− R0- 2 − ∂Ψ----1--2 cos𝜃sin 𝜃 cos-𝜃- (576) ∂ 𝜃R − R0 R − R0

∘ -------------- -∂r = -∂- (R − R0)2 + Z2 = R-−-R0-= cos𝜃 ∂R ∂R r

(577)

sin𝜃 =

∘------Z--------
(R − R0)2 + Z2

.

cos𝜃

= −Z

R − R
---3-0-
r

∂𝜃-= − Z-. ∂R r2

(578)

cos𝜃 =

R − R0
∘--------2----2-
(R − R0 ) +Z

∂ r− R−R0(R − R0) 1− cos2𝜃 sin2𝜃 ---cos𝜃 = -----r-2--------= -------- = ----- ∂R r r r

(579)

( ) [ ( )] R-∂- 1-∂Ψ- = R ∂-- 1- ∂Ψ-∂r-+ ∂Ψ-∂𝜃- ∂R R ∂R ∂R [R ( ∂r ∂R ∂𝜃 ∂R )] = R ∂-- 1- ∂Ψ-cos𝜃 − ∂Ψ-Z- ∂R R ∂r ∂𝜃 r2 ∂ ( ∂Ψ ∂Ψ Z ) (∂Ψ ∂Ψ Z )( 1 ) = ∂R- ∂r-cos𝜃− -∂𝜃r2 + R -∂r cos𝜃 − ∂𝜃-r2 − R2- ( 2 2 ) ( 2 2 ) ( ) ( ) = ∂-Ψ2-∂r + ∂-Ψ--∂𝜃 cos𝜃+ ∂Ψ--∂-cos𝜃− ∂-Ψ--∂r-+ ∂-Ψ2-∂𝜃- -Z2 − ∂Ψ--∂- Z2 − 1- ∂Ψ-cos𝜃 − ∂Ψ-Z2 ( ∂r ∂R ∂r∂𝜃∂R ) ∂r ∂R ( ∂𝜃∂r ∂R ∂𝜃 ∂R) r ∂𝜃 ∂R r R( ∂r ∂𝜃 r) ∂2Ψ- ∂2Ψ--Z ∂-Ψsin2𝜃 ∂2Ψ-- ∂2Ψ-Z- Z- ∂Ψ- 1- -1 ∂Ψ- ∂-Ψ-Z = ∂r2 cos𝜃 − ∂r∂𝜃r2 cos𝜃+ ∂r r − ∂𝜃∂r cos 𝜃− ∂𝜃2 r2 r2 + ∂𝜃Z r32cos𝜃− R ∂r cos𝜃− ∂𝜃r2 ∂2Ψ ∂2Ψ sin2 𝜃 ∂2Ψ Z ∂ Ψsin2𝜃 ∂Ψ 1 = --2-cos2𝜃+ ---2--2--− 2------2 cos𝜃+-------- + --Z -32cos𝜃 (580) ∂r( ∂ 𝜃 r ) ∂r∂𝜃 r ∂r r ∂𝜃 r − 1- ∂Ψ-cos𝜃− ∂Ψ-1 sin𝜃 (581) R ∂r ∂ 𝜃r

Summing the the right-hand-side of Eq. (576) and the expression on line (580) yields

∂2Ψ- ∂Ψ-1 1-∂2Ψ- ∂r2 + ∂rr + r2∂ 𝜃2 .

(582)

Using these, the GS equation is written as

+

+

−

----1-----
R0 + rcos𝜃

( )
∂Ψ- ∂Ψ-1
∂r cos𝜃− ∂𝜃 r sin𝜃

= −μ₀(R₀+r cos𝜃)²

−

g(Ψ),

which can be arranged in the form

( 2) ( ) 1 ∂-r-∂-+ 1--∂- Ψ− -----1---- ∂Ψ-cos𝜃− ∂-Ψ1 sin 𝜃 = − μ0(R0+r cos 𝜃)2dP-− dgg(Ψ), r ∂r ∂r r2∂𝜃2 R0 + rcos𝜃 ∂r ∂𝜃r dΨ dΨ

(583)

which agrees with Eq. (3.6.2) in Wessson’s book[27], where f is deﬁned by f = RB_ϕ∕μ₀, which is diﬀerent from g ≡ RB_ϕ by a 1∕μ₀ factor.

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