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A.2 Configuration space

A.2.1 System (R, Z,ϕ )

Definition

The coordinates (R,Z, ϕ) are defined on the space

0 ≤ R < ∞
-∞≤ Z < ∞
0 ≤ ϕ < 2π (A.60)

and is related to (x,y,z ) by

R = ∘ ------- x2 + y2 (A.61)
Z = -z
ϕ = arctan(y∕x ) + πH(- x) [2π]

which is inverted to

x = R cosϕ (A.62)
y = R sinϕ
z = -Z

Position Vector

The position vector then becomes

^ ^ X = R R + Z Z
(A.63)

where we define a local orthonormal basis ( ^ ^ ^) R,Z, ϕ as

^R = cosϕ^x + sinϕ^y (A.64)
^Z = -^z (A.65)
^ϕ = ^R ×^Z = -sinϕ^x + cosϕy^ (A.66)

Covariant Basis

The covariant vector basis is defined in (A.1), which becomes here

eR = ∂X-- ∂R = ^R (A.67)
eZ = ∂X-- ∂Z = ^ Z (A.68)
eϕ = ∂X ---- ∂ ϕ = R∂ ^R --- ∂ϕ = Rϕ^ (A.69)

so that we have the covariant basis

( ) (eR, eZ,eϕ) = R^, ^Z,R ^ϕ
(A.70)

the scaling factors

(hR,hZ ,hϕ) = (1,1,R )
(A.71)

and the normalized tangent basis

( ) ^ ^ ^ (e^R, ^eZ,^eϕ) = R,Z, ϕ
(A.72)

Contravariant Basis

The Contravariant vector basis is defined in (A.9), which becomes here

eR = ∇R = ^R (A.73)
eZ = ∇Z = ^ Z (A.74)
eϕ = ∇ϕ = ^ϕ -- R (A.75)

The relations (A.10-A.11) are here readily verified. The normalized reciprocal basis is

( ) ( ) R Z ϕ ^ ^ ^ e^ ,^e ,^e = R, Z,ϕ
(A.76)

which here coincides with the normalized tangent basis, since both bases are orthogonal.

Metric Coefficients

They are defined in (A.12) and become here

gij = ( ) 1 0 0 ( 0 1 0 ) 0 0 R2 (A.77)
gij = ( ) 1 0 0 ( 0 1 0 ) 0 0 1∕R2

As a result

2 g = R
(A.78)

and the Jacobian is

J = R
(A.79)

Differential elements

dl(R) = dR (A.80)
dl(Z) = dZ (A.81)
dl(ϕ) = Rdϕ (A.82)

dS(R ) = RdZdϕ^R (A.83)
dS(Z ) = RdRdϕ^Z (A.84)
dS(ϕ) = dRdZ^ϕ (A.85)
3 d X = RdRdZd ϕ
(A.86)

Christoffel Symbols

They are defined in (A.49) and are all zero here except

{ } { } ϕ ϕ 1-ϕϕ ∂gϕϕ- 1- ϕ R = R ϕ = 2g ∂R = R

{ } R 1-RR ∂gϕϕ- ϕ ϕ = - 2g ∂R = - R
(A.87)

Differential Operations

Gradient

∇f = ∂f-R^+ ∂f-^Z + 1-∂f-^ϕ ∂R ∂Z R ∂ϕ
(A.88)

Divergence

( ) ( ) ( ) ∇ ⋅A = 1--∂- RA ⋅ ^R + -∂- A ⋅Z^ + 1--∂- A ⋅ϕ^ R ∂R ∂Z R ∂ϕ
(A.89)

Curl

(∇ × A ) ⋅^R = -∂- ∂Z( ) A ⋅ϕ^-1- R∂-- ∂ϕ( ) A ⋅Z^ (A.90)
(∇ × A ) ⋅^ Z = -1 R∂-- ∂ϕ( ^) A ⋅R-1- R-∂- ∂R( ^) RA ⋅ϕ (A.91)
(∇ × A ) ⋅^ϕ = ∂ --- ∂R( ) A ⋅Z^- ∂ --- ∂Z( ) A ⋅ ^R (A.92)

A.2.2 System (r,θ,ϕ )

Definition

The coordinates (r,θ,ϕ) are defined from the origin (R ,Z ) p p on the space

0 ≤ r < ∞ (A.93)
0 ≤ θ < 2π

and is related to (R, Z,ϕ) by

r = --------------------- ∘ 2 2 (R - Rp) + (Z - Zp ) (A.94)
θ = arctan((Z - Zp)∕ (R - Rp)) + πH(Rp - R) [2π]

which is inverted to

R = Rp + r cosθ (A.95)
Z = Zp + r sinθ (A.96)

Position Vector

The position vector then becomes

^ ^ X = Rp R + ZpZ + r^r
(A.97)

where we define a local orthonormal basis ( ^ ^) ^r,θ,ϕ as

^r = cosθR^ + sinθZ^ (A.98)
^θ = ^ϕ×^r = -sinθ^R + cosθ^Z (A.99)

since

^ϕ×^r = ( ) ^R × Z^×( ) cosθR^+ sinθZ^ (A.100)
= [( ) ] cosθR^+ sinθZ^ ⋅ ^R^Z -[( ) ] cosθR^+ sinθ ^Z ⋅ ^Z^R (A.101)
= cosθ^Z - sinθ^R (A.102)

Covariant Basis

The covariant vector basis is defined in (A.1), which becomes here

er = ∂X ---- ∂r = ^r (A.103)
eθ = ∂X-- ∂ θ = r∂-^r ∂ θ = r^θ (A.104)
eϕ = ∂X ∂-ϕ- = Rp∂ ^R ∂ϕ- + r∂^r ∂ϕ- = (Rp + r cosθ)∂R^ ∂-ϕ = R^ϕ (A.105)

so that we have the covariant basis

( ) ^ ^ (er,eθ,eϕ) = ^r,rθ,Rϕ
(A.106)

the scaling factors

(h ,h ,h ) = (1,r,R ) r θ ϕ
(A.107)

and the normalized tangent basis

( ) (e^,^e ,^e ) = ^r,^θ, ^ϕ r θ ϕ
(A.108)

Contravariant Basis

The Contravariant vector basis is defined in (A.9), which becomes here

er = ∇r = ^r (A.109)
eθ = ∇θ = 1- r^θ (A.110)
eϕ = ∇ϕ = -^ϕ R (A.111)

The relations (A.10-A.11) are here readily verified. The normalized reciprocal basis is

( r θ ϕ) ( ^ ^) e^,^e ,^e = ^r,θ,ϕ
(A.112)

which here coincides with the normalized tangent basis, since both bases are orthogonal.

Metric Coefficients

They are defined in (A.12) and become here

gij = ( ) 1 0 0 ( 0 r2 0 ) 0 0 R2 (A.113)
gij = ( ) 1 0 0 ( 0 1∕r2 0 ) 0 0 1∕R2

As a result

2 2 g = r R
(A.114)

and the Jacobian is

J = rR
(A.115)

Differential elements

dl(r) = dr (A.116)
dl(θ) = dθ (A.117)
dl(ϕ) = Rdϕ (A.118)

dS(r) = rRdθdϕ^r (A.119)
dS(θ) = Rdrdϕ^ θ (A.120)
dS(ϕ) = rdrdθ^ϕ (A.121)
3 d X = rRdrd θdϕ
(A.122)

Christoffel Symbols

They are defined in (A.49) and are all zero here except

{ } { } θ θ 1-θθ∂gθθ 1- θ r = r θ = 2g ∂r = r
(A.123)

{ } r 1- rr ∂gθθ θ θ = -2 g ∂r = - r
(A.124)

{ ϕ } { ϕ } 1 ∂g 1 = = --gϕϕ--ϕϕ-= --cos θ ϕ r r ϕ 2 ∂r R
(A.125)

{ } r = - 1grr∂gϕϕ-= - R cosθ ϕ ϕ 2 ∂r
(A.126)

{ } { } ϕ = ϕ = 1gϕϕ∂gϕϕ-= - -rsinθ ϕ θ θ ϕ 2 ∂θ R
(A.127)

{ } θ 1-θθ∂gϕϕ- R- ϕ ϕ = - 2g ∂θ = r sinθ
(A.128)

Differential Operations

Gradient

∂f 1∂f 1 ∂f ∇f = ∂r-^r + r∂-θ ^θ + R-∂ϕϕ^
(A.129)

Divergence

-1--∂- -1- ∂-( ^) 1-∂--( ^) ∇ ⋅A = rR ∂r (rRA ⋅^r)+ rR ∂θ RA ⋅ θ + R ∂ϕ A ⋅ϕ
(A.130)

Curl

(∇ × A ) ⋅^r = 1-- rR∂-- ∂θ( ) RA ⋅ ^ϕ-1- R-∂- ∂ϕ( ) A ⋅ ^θ (A.131)
(∇ × A ) ⋅^ θ = 1- R∂-- ∂ϕ(A ⋅^r) --1 R-∂- ∂r( ^) RA ⋅ϕ (A.132)
(∇ × A ) ⋅^ϕ = 1 -- r ∂ --- ∂r( ) rA ⋅θ^-1 -- r ∂ --- ∂θ(A ⋅^r) (A.133)

A.2.3 System (ψ, s,ϕ)

Definition

The coordinates (ψ,s,ϕ), used to parametrize closed flux-surfaces, are defined from the origin (Rp, Zp) on the (closed) space

min(ψ0,ψa) ≤ ψ ≤ max(ψ0,ψa ) (A.134)
0 ≤ s ≤ smax (A.135)

and is related to (r,θ,ϕ) by

ψ = ψ(r,θ) (A.136)
s = s(r,θ)

which is inverted to

r = r(ψ, s)
θ = θ(ψ, s)

Note that ψ(r,θ) must be a monotonic function of r from ψ0 at the center (Rp, Zp) to ψa at the edge. It is the case for nested flux-surfaces.

We define a local orthonormal basis ( ) ψ^, ^s, ^ϕ as

^ ψ = ∇-ψ-- |∇ ψ| (A.137)
^s = ϕ^×^ψ

The transformation from ( ) ^r,θ^ to ( ) ^ψ,^s is a rotation of angle α such that

( ) ( ) ( ) ^ψ = cosα - sinα ⋅ ^r ^s sin α cos α ^θ
(A.138)

Position Vector

The position vector remains

^ ^ X = Rp R + ZpZ + r^r
(A.139)

Covariant Basis

The covariant vector basis is defined in (A.1), which becomes here

eψ = ∂X-- ∂ψ = | ∂r-|| ∂ψ |s^r+ r | ∂^r-|| ∂ψ |s = | -∂r|| ∂ ψ|s^r+ r | -∂θ|| ∂ ψ|s^θ (A.140)
es = ∂X ---- ∂s = ∂r|| --|| ∂sψ^r+ r∂^r|| --|| ∂sψ = ∂r || ---|| ∂sψ^r+ r∂ θ|| ---|| ∂sψ^θ (A.141)
eϕ = ∂X-- ∂ ϕ = Rp∂ ^R ∂ϕ + r∂^r- ∂ϕ = (Rp + r cosθ)∂R^ ∂ ϕ = R^ ϕ (A.142)

so that we have the covariant basis

( ) ∂r|| ∂θ || ∂r|| ∂θ|| (eψ,es,eϕ) = --|| ^r + r ---||θ^, --|| ^r + r--|| ^θ,R^ϕ ∂ψ s ∂ψ s ∂s ψ ∂s ψ
(A.143)

the scaling factors

( ∘ ---------------∘ --------------- ) ∂r ||2 ∂θ ||2 ∂r||2 ∂ θ||2 (hψ,hs,hϕ) = ( ---|| + r2 ---||, --|| + r2---|| ,R) ∂ψ s ∂ψ s ∂s ψ ∂s ψ
(A.144)

and the normalized tangent basis

( | | [ | | ] ) 1 [ ∂r | ∂θ | ] 1 ∂r| ∂θ| (^eψ,^es,^eϕ) = h-- ∂ψ-|| ^r + r ∂ψ-|| ^θ ,h- ∂s|| ^r + r ∂s|| θ^ , ^ϕ ψ s s s ψ ψ
(A.145)

Contravariant Basis

The Contravariant vector basis is defined in (A.9), which becomes here

eψ = ∇ψ = |∇ ψ |ψ ^ (A.146)
es = ∇s = ^s (A.147)
eϕ = ∇ϕ = ^ϕ -- R (A.148)

The relations (A.11) then give

eψ = es × eϕ -ψ---s----ϕ e ⋅e × e = ψ^ ----- |∇ψ | (A.149)
es = --eϕ-×-eψ-- es ⋅eϕ × e ψ = ^s
eϕ = eψ × es -ϕ---ψ----s e ⋅e × e = Rϕ^

so that we have the following tangent basis

( ψ^ ) (eψ,es,eϕ) = ----,^s,R ^ϕ |∇ ψ |
(A.150)

the scaling factors

( ) (h ,h ,h ) = -1--,1,R ψ s ϕ |∇ ψ|
(A.151)

the normalized tangent basis

( ) (^e ,^e ,^e ) = ^ψ,^s, ^ϕ ψ s ϕ
(A.152)

the reciprocal basis

( ) ( ) ^ϕ eψ,es,eϕ = |∇ ψ|ψ^, ^s,-- R
(A.153)

and the normalized reciprocal basis

( ) ( ) ^eψ,^es,^eϕ = ψ^, ^s, ^ϕ
(A.154)

which here coincides with the normalized tangent basis, since both bases are orthogonal.

By comparing (A.143) with (A.149), we also find that

| ∂r-|| ∂ψ |s = cosα- |∇ ψ| (A.155)
∂θ || ---|| ∂ψs = - sinα ------- r|∇ ψ | (A.156)
| ∂r|| ∂s|ψ = (^s ⋅^r) = sinα (A.157)
| ∂θ|| ∂s|ψ = ( ^) --^s⋅θ-- r = cos-α- r (A.158)

Metric Coefficients

They are defined in (A.12) and become here

gij = ( ) 1∕ |∇ ψ|2 0 0 ( 0 1 0 ) 0 0 R2 (A.159)
gij = ( 2 ) |∇ψ | 0 0 ( 0 1 0 ) 0 0 1∕R2

As a result

-R2--- g = |∇ ψ |2
(A.160)

and the Jacobian is

R J = ----- |∇ ψ|
(A.161)

Differential elements

dl(ψ ) = -dψ-- |∇ψ | (A.162)
dl(s) = ds (A.163)
dl(ϕ ) = Rdϕ (A.164)

dS(ψ) = Rdsdϕ^ψ (A.165)
dS(s) = R ----- |∇ψ |dψdϕ^s (A.166)
dS(ϕ) = --1-- |∇ψ |dψdsϕ^ (A.167)
3 R d X = |∇ψ-|dψdsdϕ
(A.168)

Christoffel Symbols

They are defined in (A.49) and are here

Differential Operations

Gradient

∂f ∂f 1 ∂f ∇f = |∇ψ |---^ψ + ---^s+ ----ϕ^ ∂ψ ∂s R ∂ϕ
(A.169)

Divergence

|∇ ψ| ∂ ( ) |∇ψ |∂ ( R ) 1 ∂ ( ) ∇ ⋅ A = -------- RA ⋅ ^ψ + -------- -----A ⋅^s + ----- A ⋅ ^ϕ R ∂ψ R ∂s |∇ ψ| R ∂ϕ
(A.170)

Curl

(∇ × A ) ⋅ ^ ψ = -1 R-∂- ∂s( ^) RA ⋅ϕ-1- R∂-- ∂ϕ(A ⋅^s) (A.171)
(∇ × A ) ⋅^s = 1 -- R ∂ --- ∂ ϕ( ) A ⋅ ^ψ-|∇ ψ| ----- R ∂ --- ∂ψ( ) RA ⋅ ^ϕ
(∇ × A ) ⋅ϕ^ = |∇ ψ| ∂ ∂-ψ(A ⋅^s) -|∇ ψ| ∂ ∂s-( ) A ⋅ψ^ |∇ψ-|

A.2.4 System (ψ, θ,ϕ)

Definition

The coordinates (ψ,θ,ϕ) are defined from the origin (Rp, Zp) on the space

min (ψ0, ψa) ≤ ψ < max (ψ0,ψa)

and is related to (r,θ,ϕ) by

ψ = ψ (r,θ)

which is inverted to

r = r (ψ, θ)

Note that ψ(r,θ) must be a monotonic function of r from ψ0 at the center (Rp, Zp) to ψa at the edge. It is the case for nested flux-surfaces.

Position Vector

The position vector then becomes

^ ^ X = Rp R + ZpZ + r (ψ,θ)^r
(A.172)

Covariant Basis

The covariant vector basis is defined in (A.1), which becomes here

eψ = ∂X-- ∂ψ = | ∂r-|| ∂ψ |θ^r (A.173)
eθ = ∂X ---- ∂ θ = ∂r|| --|| ∂θψ^r+ r∂ ^r --- ∂ θ = ∂r || ---|| ∂ θψ^r+ r^θ (A.174)
eϕ = ∂X-- ∂ ϕ = Rp∂ ^R ∂ϕ + r∂^r- ∂ϕ = (Rp + r cosθ)∂R^ ∂ ϕ = R^ ϕ (A.175)

so that we have the covariant basis

( ) ∂r|| ∂r|| (eψ, eθ,eϕ) = ---|| ^r, --|| ^r + r^θ,Rϕ^ ∂ ψ θ ∂θ ψ
(A.176)

the scaling factors

(| | | ∘ ---|------ ) ||-∂r|| || ∂r-||2 2 (h ψ,hθ,hϕ) = |∂ ψ| |, ∂ θ| + r ,R θ ψ
(A.177)

and the normalized tangent basis

( [ | ] ) (^eψ,^eθ,^eϕ) = ^r, 1-- ∂r|| ^r + r^θ , ^ϕ hθ ∂θ|ψ
(A.178)

Contravariant Basis

The Contravariant vector basis is defined in (A.9), which becomes here

eψ = ∇ψ = |∇ ψ |ψ ^ (A.179)
eθ = ∇θ = ^ θ- r (A.180)
eϕ = ∇ϕ = -^ϕ R (A.181)

The relations (A.11) then give

eψ = --eθ ×-eϕ-- eψ ⋅eθ × eϕ = ----^r----- |∇ ψ|cosα (A.182)
eθ = ϕ ψ --e-×-e---- eθ ⋅eϕ × eψ = ^ (-rθ)-- ^θ ⋅^s = ^ -rθ-- cosα (A.183)
eϕ = --eψ ×-eθ-- eϕ ⋅e ψ × eθ = R^ϕ (A.184)

since ^s = ^ ϕ× ^ ψ , so that we have the following tangent basis

( ) ^r r^θ (eψ,eθ,eϕ) = |∇-ψ|cosα-,cosα-,R^ϕ
(A.185)

the scaling factors

( ) (hψ, hθ,hϕ) = ----1-----,-r--,R |∇ ψ|cosα cosα
(A.186)

the normalized tangent basis

( ) (^e ,^e ,^e ) = ^r,^θ, ^ϕ ψ θ ϕ
(A.187)

the reciprocal basis

( ) ( ) ^θ ϕ^ eψ,eθ,eϕ = |∇ ψ|ψ^, -,-- r R
(A.188)

and the normalized reciprocal basis

( ) ( ) ^eψ,^eθ,^eϕ = ψ^, ^θ, ^ϕ
(A.189)

which here does not coincide with the normalized tangent basis, since both bases are not orthogonal.

By comparing (A.176) with (A.185), we also find that

| ∂r| ∂-ψ||θ = 1 |∇-ψ|cosα- (A.190)
| ∂r-|| ∂ θ|ψ = r┌ ----------- ││ ---1--- ∘ (^ )2 - 1 θ ⋅^s = r tanα (A.191)

Metric Coefficients

They are defined in (A.12) and become here

( ) 1∕ [|∇ψ |cosα]2 rtan α∕[|∇ ψ |cos α] 0 gij = ( rtan α∕ [|∇ψ |cosα] r2∕cos2α 0 ) 0 0 R2
(A.192)

or equivalently

( ∂r∕∂ψ |2θ ∂r∕∂ ψ|θ ∂r∕∂θ |ψ 0 ) g = ( ∂r∕∂ψ | ∂r∕∂θ| ∂r∕∂ θ|2 + r2 0 ) ij θ ψ ψ 2 0 0 R

and

( |∇ψ |2 - |∇ ψ |sinα ∕r 0 ) ij ( 2 ) g = - |∇ψ |sin α∕r 1∕r 0 2 0 0 1∕R
(A.193)

As a result

2 2 g = ---R--r---- |∇ ψ |2cos2 α
(A.194)

and the Jacobian is

---Rr----- J = |∇ψ |cosα
(A.195)

Differential elements

dl(ψ) = ---d-ψ---- |∇ψ |cosα (A.196)
dl(θ) = --r-- cosαdθ (A.197)
dl(ϕ) = Rdϕ (A.198)

dS(ψ ) = -Rr-- cosαdθdϕ ^ ψ (A.199)
dS(θ) = R ---------- |∇ ψ|cosαdψdϕ^θ (A.200)
dS(ϕ ) = r ---------- |∇ ψ|cosαdψdθϕ^ (A.201)
d3X = ---Rr----dψd θdϕ |∇ ψ|cos α
(A.202)

Christoffel Symbols

They are defined in (A.49) and are here

Differential Operations

Gradient

∇f = |∇ ψ| ∂f-ψ^+ 1-∂f-^θ +-1∂f-^ϕ ∂ψ r ∂θ R ∂ ϕ
(A.203)

Divergence

( ) ( ) ( ) ∇ ⋅A = |∇ψ-|cosα-∂-- -Rr-A ⋅ ^ψ + |∇-ψ|cosα--∂- ---R-----A ⋅θ^ + 1-∂-- A ⋅ϕ^ Rr ∂ψ cosα Rr ∂θ |∇ ψ|cos α R ∂ϕ
(A.204)

Curl

(∇ × A ) ⋅ψ^ = cosα- Rr-∂- ∂θ( ) RA ⋅ ^ϕ--1 R-∂- ∂ ϕ( ) A ⋅ ^θ (A.205)
(∇ × A ) ⋅^θ = 1- R-∂- ∂ϕ(A ⋅^r) -|∇ψ-|cosα- R∂-- ∂ψ( ) RA ⋅ ^ϕ (A.206)
(∇ × A ) ⋅^ϕ = |∇ψ-|cosα- r∂-- ∂ψ( ) rA-⋅ ^θ cosα-|∇-ψ-|cos-α- r-∂- ∂ θ( ) --A--⋅^r--- |∇ψ |cosα (A.207)

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