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Axisymmetric Tokamak (Toroidal)

ElectromagneticFields.AxisymmetricTokamakToroidal — Module

Axisymmetric tokamak equilibrium in (r,θ,ϕ) coordinates with covariant components of the vector potential given by

\[A (r, \theta, \phi) = \frac{B_0 R_0}{2} \, \bigg( \frac{Z}{R} \cos (\theta) - \ln \bigg( \frac{R}{R_0} \bigg) \sin (\theta) , \, - r \, \bigg[ \frac{Z}{R} \sin (\theta) + \ln \bigg( \frac{R}{R_0} \bigg) \cos (\theta) \bigg] , \, + \frac{r^2}{q_0 R_0} \bigg)^T ,\]

resulting in the magnetic field with covariant components

\[B (r, \theta, \phi) = \frac{B_0}{q_0} \, \bigg( 0 , \, \frac{r^2}{R}, \, q_0 R_0 \bigg)^T ,\]

where $R = R_0 + r \cos \theta$ and $Z = r \sin \theta$.

Parameters:

  • R₀: position of magnetic axis
  • B₀: B-field at magnetic axis
  • q₀: safety factor at magnetic axis
source

Constructing the Field

using CairoMakie
using ElectromagneticFields

equ = AxisymmetricTokamakToroidal.init()
Axisymmetric Tokamak Equilibrium in Toroidal Coordinates with
  R₀ = 1.0
  B₀ = 1.0
  q₀ = 2.0

Plotting

The coordinates $(r, \theta, \phi)$ are the natural ones for this field — the flux surfaces are the surfaces of constant $r$. For the plot the poloidal plane is sampled in $(R,Z)$ and mapped to $(r,\theta)$ first, so the result is directly comparable with the cartesian and cylindrical versions of the same equilibrium:

plot_equilibrium(equ)
Example block output

Evaluating the Field

AxisymmetricTokamakToroidal.@code()

The chart maps $(r, \theta, \phi)$ to the cartesian coordinates, and to_cartesian and from_cartesian move between the two:

t = 0.0
ξ = [0.5, π/4, 0.0]

x = to_cartesian(t, ξ)
3-element Vector{Float64}:
 1.3535533905932737
 0.0
 0.35355339059327373
roundtrip = from_cartesian(t, x) ≈ ξ
roundtrip
true

Because the toroidal chart is left-handed, the determinant of its Jacobian is minus the Jacobian determinant J:

using LinearAlgebra

signed = det(DF(t, ξ)) ≈ orientation() * J(t, ξ)
orientation(), signed
(-1, true)