Concentric-circular magnetic conﬁguration with a given safety factor proﬁle

12 Concentric-circular magnetic conﬁguration with a given safety factor proﬁle

Assume magnetic surfaces of a magnetic conﬁguration are known and given by

R (r,𝜃) = R0 + rcos𝜃,

(342)

Z(r,𝜃) = rsin 𝜃,

(343)

where (r,𝜃) are two parameters and r is magnetic surface label (i.e., ∂Ψ∕∂𝜃|_r = 0). The above parametric equations specify a series of concentric-circular magnetic surfaces.

Assume the toroidal ﬁeld function g(r) = RB_ϕ is given. Then the toroidal magnetic ﬁeld is determined by B_ϕ = g∕R. Further assume the safety factor proﬁle q(r) is given, then the magnetic ﬁeld is fully determined. Next, let us derive the explicit form of the poloidal magnetic ﬁeld B_p, which is given by

Bp = ∇ Ψ × ∇ ϕ = 1-∇ Ψp × ∇ ϕ, 2π

(344)

which involves the poloidal magnetic ﬂux Ψ_p. Therefore our task is to express Ψ_p in terms of q and g. Using q(r) = dΨ_t∕dΨ_p, we obtain

dΨ_p =

dΨ_t,

Integrate the above equation over r, we obtain

∫ r ∫ r1 dΨp = -dΨt, 0 0 q

(345)

which an be written as

∫ r (∫ r∫ π ) Ψ (r)− Ψ (0) = -1--d B rdrd𝜃 , p p 0 q(r) 0 − π ϕ

(346)

where use has been made of Ψ_t = ∫ ₀^r ∫ _−π^πB_ϕrdrd𝜃. Using B_ϕ = g∕R and R = R₀ + r cos𝜃, the above equation is written

∫ r ( ∫ r∫ π ) Ψp (r)− Ψp(0) = -1-d ----g-----rdrd𝜃 0 q(r) 0 −π R0 + r cos𝜃

(347)

Using maxima (an open-source computer algebra system), the above integration over 𝜃 can be performed analytically, giving

∫ π 1 2π ----------d𝜃 = ∘--2---2. −π R0 + r cos𝜃 R 0 − r

(348)

Using this, equation (347) is written as

∫ (∫ ) r-1-- r --2πgr--- Ψp(r)− Ψp(0) = 0 q(r)d 0 ∘R2-−-r2dr , 0

(349)

which can be simpliﬁed as

∫ r Ψp(r) − Ψp(0) = -1--∘-2πgr--dr. 0 q(r) R20 − r2

(350)

This is what we want—the expression of the poloidal magnetic ﬂux in terms of q and g. [Another way of obtaining Eq. (350) is to use Eq. (160), i.e.,

dΨ g ∫ 2π 𝒥 --p= − - -2d𝜃, dr q 0 R

(351)

where 𝒥 is the Jacobian of the (r,𝜃,ϕ) coordinates and is given by 𝒥 = R(R_𝜃Z_r −R_rZ_𝜃) = −Rr. Then Eq. (351) is simpliﬁed as

∫ 2π ∫ 2π dΨp-= g rd𝜃 = g ----r-----d𝜃 = g r∘-2π---, dr q 0 R q 0 R0 + r cos𝜃 q R20 − r2

(352)

which, after being integrated over r, gives Eq. (350).]

Using Eq. (350), the poloidal magnetic ﬁeld in Eq. (344) is written as

( ) ∇r×-∇-ϕ-∂-- ∫ r-1----2πgr--- Bp = 2π ∂r 0 q(r) ∘R2-−-r2dr 0 = ∇r × ∇ ϕ-1--∘--gr---. (353) q(r) R20 − r2

[Using the formulas ∇r = − R𝒥-

(Z_𝜃

−R_𝜃

) and 𝒥 = R(R_𝜃Z_r −R_rZ_𝜃), where 𝒥 is the Jacobian of the (r,𝜃,ϕ) coordinates, we obtain 𝒥 = −Rr and ∇r = cos𝜃

+ sin𝜃

, ∇ϕ =

∕R. Then Eq. (353) is written as

cos𝜃ˆZ − sin 𝜃ˆR 1 gr Bp = ------------- ---∘---2---2 R q(r) R 0 − r

(354)

This is the explicit form of the poloidal magnetic ﬁeld in terms of g and q. The magnitude of B_p is written as

-----1-------1-----gr---- Bp = (R0 + rcos𝜃)q(r) ∘R2-−-r2 0

(355)

Note that both B_p and B_ϕ depend on the poloidal angle 𝜃.]

I use Eq. (350) to compute the 2D data of Ψ (Ψ = Ψ_p∕2π) on the poloidal plane when creating a numerical G-eqdsk ﬁle for the above magnetic conﬁguration (Fortran code is at /home/yj/project_new/circular_configuration_with_q_given).

Assume that the poloidal plasma current is zero, then g(r) = RB_ϕ is a constant independent of r. This is always assumed by the authors who use concentric-circular conﬁguration but is seldom explicitly mentioned.

In analytical work, 1∕R dependence on (r,𝜃) is often approximated as

≈

(1 − 𝜀cos𝜃),

where 𝜀 = r∕R₀ is the local inverse aspect ratio.

  12.1 Is the above B divergence-free?
  12.2 Is the above B a solution to the GS equation?
  12.3 Explicit expression for the generalized toroidal angle
  12.4 Metric elements
  12.5 Safety factor proﬁle

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