Axisymmetric Tokamak (Toroidal)

ElectromagneticFields.AxisymmetricTokamakToroidalEquilibrium — Type

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

AxisymmetricTokamakToroidalITER returns this equilibrium with ITER's parameters.

source

Constructing the Field

using CairoMakie
using ElectromagneticFields

equ = AxisymmetricTokamakToroidalEquilibrium()

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)

Evaluating the Field

field = FieldFunctions(AxisymmetricTokamakToroidalEquilibrium())

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(field, t, ξ)

3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
 1.3535533905932737
 0.0
 0.35355339059327373

roundtrip = from_cartesian(field, 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(field, t, ξ)) ≈ orientation(field) * J(field, t, ξ)
orientation(field), signed

(-1, true)

The Regularized Gauge

ElectromagneticFields.AxisymmetricTokamakToroidalRegularizationEquilibrium — Type

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

\[A (r, \theta, \phi) = B_0 \, \bigg( 0 , \, \bigg( \frac{R_0}{\cos (\theta)} \bigg)^2 \, \ln \bigg( \frac{R}{R_0} \bigg) - \frac{r R_0}{\cos (\theta)} , \, + \frac{r^2}{2 q_0} \bigg)^T ,\]

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 $R = R_0 + r \cos \theta$.

This is the same magnetic field as AxisymmetricTokamakToroidalEquilibrium, in a gauge whose poloidal vector potential is regular on the magnetic axis.

Parameters:

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

source

The vector potential is fixed only up to a gauge, and the two gauges are genuinely different equilibria here, because the generated code follows whatever was written down. The poloidal vector potential of the chart above is singular on the magnetic axis; this variant carries the same magnetic field in a gauge that is regular there.

regular = FieldFunctions(AxisymmetricTokamakToroidalRegularizationEquilibrium())

same_field = B♭(regular, t, ξ) ≈ B♭(field, t, ξ)
same_field

true

The vector potentials differ, as they must:

[A♭(field, t, ξ) A♭(regular, t, ξ)]

3×2 StaticArraysCore.SMatrix{3, 2, Float64, 6} with indices SOneTo(3)×SOneTo(2):
 -0.0146829   0.0
 -0.0996909  -0.10164
  0.0625      0.0625