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A.8 Relation of plasma current density to pressure gradient

Due to the force balance condition, the plasma current is related to the plasma pressure. Using the equilibrium constraint in the R direction, the toroidal current density Jϕ given by Eq. (54) can be written as

dP 1 dg Jϕ = R dΨ-+ μ-R-dΨ-g. (545) 0
The parallel (to the magnetic field) current density is written as
J-⋅B- J∥ ≡ B JϕB-ϕ +-Jp ⋅Bp = B ( dP -1-dg ) g- 1-dg(∇-Ψ)2 = -R-dΨ +-μ0R-dΨg--R-+-μ0dΨ--R--- [ B ] gdP + 1-dg (g-)2 + (∇Ψ-)2 = -dΨ---μ0dΨ---R------R---- dP 1 dg B2 gdΨ-+-μ0dΨB-- = B . dP-1- -1-dg- = gdΨ B + μ0 dΨB. (546)
For later use, define
σ ≡ J∥ B = gdP--1-+ -1-dg. (547) dΨ B2 μ0 dΨ
Equation (547) is used in GTAW code to calculate J∥∕B (actually calculated is μ0J∥∕B)[15]. Note that the expression for J∥∕B in Eq. (547) is not a magnetic surface function. Define σps as
σps ≡ σ − ⟨σ⟩ dP-[-1- ⟨ 1-⟩ ] = g dΨ B2 − B2 (548) Jps ≡ -∥-, (549) B
where J∥ps is called Pfirsch-Schluter (PS) current. In cylindrical geometry, due to the poloidal symmetry, the Pfiersch-Schluter current is zero. In toroidal geometry, due to the poloidal asymmetry, the PS current is generally nonzero. Thus, this quantity characterizes a toroidal effect.

Another useful quantity is μ0⟨J ⋅ B⟩, which is written as

μ0⟨J⋅B ⟩ = μ0gdP-+-dg⟨B2⟩, (550) dΨ dΨ
where ⟨…⟩ is flux surface averaging operator.

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