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Symmetric Solov'ev Equilibrium

ElectromagneticFields.SolovevSymmetric — Module

Symmetric Solov'ev equilibrium in cartesian (x,y,z) coordinates. Based on McCarthy, Physics of Plasmas 6, 3554, 1999.

The covariant components of the vector potential are given by

\[A (x, y) = \frac{B_0}{2} \, \bigg( 0 , \, 0 , \, - \frac{\alpha}{4} (R_0 + x)^4 - \beta y^2 \bigg)^T ,\]

Parameters:

  • R₀: major radius, which places the magnetic axis at x = -R₀
  • B₀: B-field at magnetic axis
  • α, β: free constants
source

Constructing the Field

The parameters are the major radius $R_0$, the field strength $B_0$, and the two coefficients $\alpha$ and $\beta$ of the flux function. The flux function depends on $x$ through $(R_0 + x)^4$, so $R_0$ shifts the magnetic axis to $x = -R_0$; taking $R_0 = 0$ puts it at the origin:

using CairoMakie
using ElectromagneticFields

equ = SolovevSymmetric.init(0.0, 1.0, 2.0, 0.5)
Quadratic Solovev Equilibrium with
  R₀ = 0.0
  B₀ = 1.0
  α  = 2.0
  β  = 0.5

Plotting

Unlike the general Solov'ev equilibrium, this one is written in cartesian coordinates. Its flux surfaces are the contours of $A_z$, and because the flux function involves $x$ and $y$ only as $(R_0 + x)^4$ and $y^2$, they are symmetric both up/down and left/right about the magnetic axis:

plot_equilibrium(equ)
Example block output

$\alpha$ weights the quartic dependence on $x$, $\beta$ the quadratic one on $y$, so it is their ratio that decides the shape of the surfaces — the larger $\beta / \alpha$, the flatter they get:

fig = Figure(size = (1200, 400))

for (n, (α, β)) in enumerate(((2.0, 0.5), (1.0, 1.0), (0.5, 2.0)))
    plot_equilibrium!(fig[1,n], SolovevSymmetric.init(0.0, 1.0, α, β);
        title = "α = $α, β = $β")
end

fig
Example block output