Pressure and toroidal ﬁeld function proﬁle

13.4 Pressure and toroidal ﬁeld function proﬁle

The function p(Ψ) and g(Ψ) in the GS equation are free functions which must be speciﬁed by users before solving the GS equation. Next, we discuss one way to specify the free functions. Following Ref. [16], we take P(Ψ) and g(Ψ) to be of the forms

P(Ψ) = P0 − (P0 − Pb)ˆp(Ψ),

(412)

1 1 -- 2g2(Ψ) = 2 g20(1 − γ ˆg(Ψ )),

(413)

with and ĝ chosen to be of polynomial form:

ˆp(Ψ) = Ψ-α,

(414)

-- --β ˆg(Ψ) = Ψ ,

(415)

where

-- Ψ − Ψa Ψ ≡ Ψ-−-Ψ--, b a

(416)

with Ψ_b the value of Ψ on the boundary, Ψ_a the value of Ψ on the magnetic axis, α, β, γ, P₀, P_b, and g₀ are free parameters. Using the proﬁles of P and g given by Eqs. (412) and (413), we obtain

dP-(Ψ-)= − 1-(P − P )αΨα−1 dΨ Δ 0 b

(417)

where Δ = Ψ_b − Ψ_a, and

-- g dg-= − 1g20γ 1-βΨβ−1. dΨ 2 Δ

(418)

Then the term on the r.h.s (nonlinear source term) of the GS equation is written

dP dg [ 1 -α−1] 1 1 -β−1 − μ0R2--- − --g = μ0R2 --(P0 − Pb)α Ψ + -g20γ--βΨ . dΨ dΨ Δ 2 Δ

(419)

The value of parameters P₀, P_b, and g₀ in Eqs. (412) and (413), and the value of α and β in Eqs. (414) and (415) can be chosen arbitrarily. The parameter γ is used to set the value of the total toroidal current. The toroidal current density is given by Eq. (545), i.e.,