Magnetic Field Line
((3 + cos(ι · u + t)) · cos(u), sin(ι · u + t), (3 + cos(ι · u + t)) · sin(u))
Open in the app The dials and keys named below are the app's.
What it draws
Read 3 + cos(ι u + t) as a distance from the vertical axis: it runs between 2 and 4. The cos(u) and sin(u) it multiplies place the point at that distance and at the angle u around the axis. The second expression, sin(ι u + t), is the height, between −1 and 1.
Distance and height share the angle ι u + t, so the point never leaves a torus: a central circle of radius 3 lying flat, and a tube of radius 1. The angle the long way round is u, and the angle round the tube is ι u + t. Each turn the long way carries the line ι of a turn round the tube.
Rotational transform
In a fusion machine the plasma is held by a magnetic field whose lines wind round a torus like this one. The number of turns round the tube per turn the long way is the rotational transform, written ι. A field running only the long way lets the plasma drift out to the wall; the twist carries each line between the top and the bottom of the tube, and the drift cancels on average.
When ι is a fraction m/n, the line closes after n turns the long way. Here ι = 0.7 = 7/10, so the line closes after ten turns, seven times round the tube, which is why Turns of u opens at 10. When ι is irrational the line never closes, and it passes as near as you like to every point of the torus. That torus is a flux surface, and a machine nests many of them, one inside the next.
The clock only shifts the tube angle. Since ι u + t = ι(u + t/ι), the line slides along itself, and the whole curve turns about the vertical axis at 1/0.7 ≈ 1.43 radians per second at Speed 1.
History
Lyman Spitzer proposed the stellarator at Princeton in 1951. Its field gets its twist from the shape of the coils alone, where a tokamak drives a current through the plasma for it.
Try
- Set ι to 0.5: the line closes after two turns the long way, and the other eight turns only retrace it.
- Set ι to 0: the line is a flat circle round the axis, and the clock walks it round the tube. There is no twist left to carry it between top and bottom.
- Change each ι u to u/√2 and set Turns of u to 24: the transform is now irrational, the line never closes, and its strands creep over the whole surface.
- Set Turns of u to 3: the line stops a little past two turns round the tube, the start of the covering.