![[ ( ) ( ) ]
μ0J = − Ψ′ 𝒥-2|∇ ψ|2 + Ψ′ 𝒥2∇ ψ ⋅∇𝜃 ∇ψ × ∇ 𝜃− g′∇ ϕ × ∇ψ, (472)
R ψ R 𝜃](tokamak_equilibrium609x.png)
Let us derive the contravariant expression for the current density. Using
| μ0J = ∇× B, |
along with magnetic field expression (154) and the curl formula (116), we obtain
which is the contravariant form of the current density vector. Next, for later use, calculate the parallel current. By using Eqs. (154) and (472), the parallel current density is written as![[( ) ( ) ] ( )
μ J⋅B = − Ψ′ 𝒥-|∇ ψ|2 + Ψ ′ 𝒥-∇ ψ⋅∇ 𝜃 g∇ψ × ∇𝜃 ⋅∇ϕ − g′ − Ψ′ 𝒥-|∇ψ |2 ∇ϕ × ∇ψ ⋅∇ 𝜃
0 R2 ψ R2 𝜃 R2
[( ) ( ) ]
= − Ψ′ 𝒥-|∇ ψ|2 + Ψ ′ 𝒥-∇ ψ⋅∇ 𝜃 g𝒥−1 + g′𝒥 −1Ψ′ 𝒥-|∇ ψ|2
R2 ψ R2 𝜃 R2
{ ( ) ( ) ′ }
= − g2𝒥− 1 1 Ψ ′ 𝒥2|∇ψ |2 + 1 Ψ′ 𝒥-2∇ψ ⋅∇𝜃 − g2Ψ ′ 𝒥2|∇ψ|2
g R ψ g R 𝜃 g R
[(Ψ ′𝒥 ) ( Ψ′𝒥 ) ]
= − g2𝒥− 1 -- -2|∇ψ |2 + ----2∇ψ ⋅∇𝜃 . (473)
g R ψ g R 𝜃](tokamak_equilibrium610x.png)