# How to plot the lines of the magnetic fields of a Tokamak using StreamPlot3D?

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

## Introduction

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 $$2$$):

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:

Manipulate[
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/

• 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. Jan 8 at 7:43
• 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, Jan 8 at 19:14