I would like to plot the magnetic field lines of a tokamak.


Since the magnetic field is solenoidal: $\nabla \cdot \vec{B} = 0$, it is possible to decompose it as:

$\vec{B} = \vec{B}_{T} + \vec{B}_{P} \tag{1}$

where $\vec{B}_{T}$ is the Toroidal magnetic field and $\vec{B}_{P}$ is the Poloidal magnetic field. This vector decomposition is called Toroidal-Poloidal decomposition $[1]$.

Now, in toroidal fusion reactors, or Tokamaks, the intuitive picture of the magnetic field lines of $\vec{B}$, $\vec{B}_{T}$ and $\vec{B}_{P}$ is given by Figure $1$:

Figure $1$: The magnetic fields of a Tokamak system.

My question

An illustrative example:

Well, if you want to plot the magnetic field lines of a magnetic dipole, you simply take the field $[2]$:

$\vec{B}_{\mathrm{dip}} = \frac{\mu_{0}}{4\pi r^{3}}2cos\theta \hat{r} + \frac{\mu_{0}}{4\pi r^{3}}sin\theta \hat{\theta} \tag{2}$

and perform a coordinate transformation $(r,\theta,\phi) \to (x,y,z)$ to get $[2]$, $[3]$:

$\vec{B}_{\mathrm{dip}} = \frac{3 \mu m x z}{4 \pi \left(x^2+y^2+z^2\right)^{5/2}}\hat{x}+\frac{3 \mu m y z}{4 \pi \left(x^2+y^2+z^2\right)^{5/2}}\hat{y}-\frac{\mu m \left(x^2+y^2-2 z^2\right)}{4 \pi \left(x^2+y^2+z^2\right)^{5/2}} \hat{z} \tag{3}$

with $(3)$ you can use your favorite software to plot the magnetic configuration. Using mathematica you should get (Figure $2$):

enter image description here

Figure $2$: The magnetic field lines of theoretical magnetic dipole.

For the present case,

The question:

The magnetic field of a tokamak is far more complicated than that of a magnetic dipole. Assuming an axisymmetric ideal MHD equilibrium leads to the Grad-Shafranov equation. This can be solved numerically for a specified toroidal current density, $\mathbf{j}_{\phi}$. Assuming the current density is linearly related to the poloidal flux yields the Solov’ev solution. For large aspect ratio, Wesson's Tokamaks provides the approximate large aspect ratio solution:

$B_{\phi} = \frac{B_{\phi 0} R_0}{R} \tag{4}$

$B_{\theta} = B_{\theta a} \frac{1 - \left (1 - \frac{r^2}{a^2} \right )^{\nu+1}}{r / a} \tag{5}$

where its assumed current $j \propto \left (1 - \frac{r^2}{a^2} \right )^{\nu}$ and pressure $p \propto 1 - \frac{r^2}{a^2} $. Here the simple toroidal coordinates are used. Also, $B_{\phi0}$ is the toroidal magnetic field at $R=R_{0}$ (center of the cross section) and $B_{\theta a}$ is the poloidal magnetic field at $r=a$ (edge of plasma). $R$ and $R_{0}$ are respectively distance from axis of symmetry and torus major radius ( simple toroidal coordinates ).

The code that I've written is:

 StreamPlot3D[{0, (c*f)/Sqrt[x^2 + y^2], (
   a (1 - (1 - Sqrt[x^2 + y^2 + z^2]/a^2)^(1 + ν)) *d)/
   Sqrt[x^2 + y^2 + z^2]}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}], {a, 1,
   5}, {ν, 1, 5}, {d, 1, 5}, {c, 1, 5}, {f, 1, 5}]

But I simply do not get the right results.

So: how can I plot the magnetic field lines of $(4)$ and $(5)$ using the StreamPlot3D?


$[1]$ https://en.wikipedia.org/wiki/Poloidal%E2%80%93toroidal_decomposition

$[2]$ GRIFFITHS.D. Introduction to Electrodynamics. page 255.

$[3]$ https://pages.vassar.edu/magnes/mathematica/

  • 1
    $\begingroup$ Can you please add the code you use for defining c, d, and f ? This would help others to solve your problem. Be also a little careful with using subscripts for parameter names, sometimes this can lead to problems, also avoid using Capital letters for the first letter of a parameter as this might clash with internal symbols. $\endgroup$
    – Dunlop
    Jan 8 at 7:43
  • $\begingroup$ Are expressions (4) and (5) in Toroidal-poloidal coordinates? en.wikipedia.org/wiki/Toroidal_and_poloidal_coordinates Are they being converted to Cartesian correctly? You could check the function by taking slice by setting x, y or z to zero and doing a 2D plot, $\endgroup$ Jan 8 at 19:14

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