Posted on: 2nd September 2013

Andy Warhol elevated the humble banana to an icon of pop art, with his 1967 album cover for the band the Velvet Underground. Proving its versatility, the banana is now also an icon of magnetically confined fusion research, thanks to the curious shapes traced out by particles zigzagging around the tokamak; every student of fusion physics has to learn about banana orbits.

Often the food analogies applied to tokamaks centre around doughnuts, due to the shape of magnetic field that confines the hot fusion plasma. But as one delves deeper into the complicated world of gyrokinetics, the simplistic doughnut transforms into a more complex banana orbit in a journey from the ideal to the real world.

#### Simple spirals

In the simplistic, doughnut-shaped magnetic field, charged particles are constrained to spiral along the magnetic field by one of the Universe’s strangest phenomena, the Lorentz force, which governs the interaction of charged particles with a magnetic field. This interaction happens when the particle is moving at an angle to the magnetic field. Bizarrely the field neither attracts nor repels the particle, but applies to it a sideways force, at right angles to both the direction of travel, and the magnetic field lines. For example, a particle travelling horizontally through a vertical magnetic field might experience a force to turn left (depending on the respective polarities of the particle’s charge and the magnetic field). Having turned left, it’s still travelling horizontally – at right angles to the vertical field – so it continues to feel a force to the left. And so it turns some more, and more, and more – trapped in a never-ending circle.

If the original angle of travel of the particle was not exactly at right angles to the magnetic field (in the example above, perhaps travelling slightly upwards) then it will still circulate around the field line, but moving along it as well, tracing out a spiral path. The closer the particle’s path is to parallel with the magnetic field, (in our example vertical) the more stretched out the spiral is – but always following the field line.

#### Banana realities

The premise of the tokamak is to construct a doughnut shaped magnetic field and then the plasma particles will merrily spiral around it for ever. Enter an uncomfortable reality of geometry; as you can see in the main image above, the magnets are closer together in the centre of the torus (the hole of the doughnut) than they are around the outside. This means the magnetic field is not uniform: it is stronger in the inside part of the ring.

This means that the helical path the particle follows is not symmetrical. A tighter turn on the high field (inner) side of the line, and looser on the outside leads to a drift either upwards or downwards (depending on the direction of rotation). This is the beginning of our banana orbit , as shown in the projected cross-section at the left-hand side of the figure. As an example, let’s follow a particle on the inside of the banana halfway up, gradually creeping downwards to trace the banana’s inner edge.

If this downward drift continued unchecked, the particles would escape the plasma all too soon. Countering this drift was one of the master strokes of tokamak design, via the inclusion of a solenoid in the hole of the doughnut. Driving a changing current through this solenoid drives a plasma current around the doughnut, serving to generate a field at right angles to the ring of magnet coils. This additional right-angled field gives a twist to the doughnut’s magnetic field, thereby circulating the particles that would otherwise drift away back into the middle. This gives the banana its curve.

#### The pointy end of the banana

As the particles circulate around into the higher field area their progress along the  field line slows, because the field lines are no longer parallel, but converge slightly as the field increases. In fact for particles moving slowly along the field but rapidly around it – i.e. in more tightly wound spirals – this field convergence is enough to stop them altogether, and bounce them back towards to where they came from. This defines the apex of the banana orbit.

However, the particles do not retrace their steps exactly, because of the continuous drifting below the field line. Instead they trace the outer part of the banana’s outline, back to the middle and then continue downwards to its lower apex. However, beyond the centreline – the part of the banana orbit the furthest from the centre – our example particle’s downward drift now brings it back towards the centre of the plasma. Its path traces the lower half of the banana’s outline, to the lower apex in the higher magnetic field and then back up to its original starting place. In addition, the magnetic field line angle is different on the outer part of the banana from that on the inner part, producing the zig-zag shown in the figure.

#### A tangled tale

The cross section of this orbit is banana shaped, although the reality for the many particles in the plasma is much more of a tangle… the initial speed and angle of the particle can radically vary the exact number and distribution of bounces, and indeed whether any particle stays within the plasma or escapes.

The shape can change too, close to the centre of the plasma the cross section of the orbit is more like a potato, or a kidney bean; the other particles that do not bounce – called the passing particles – follow orbits that are just slightly outwardly displaced with respect to the magnetic field lines.

Put them all together, it all begins to merge into one giant bowl of spaghetti. Endlessly complex, but delicious nonetheless!

Thanks for all the help on this article and image, from Sean Conroy for orbit data, Dave Cooper for data processing, Russell Perry and Chad Heys for graphics flair, and Tom Todd for technical discussions.

This is the last picture of the week for 2013, as EFDA undergoes a restructure, and will be the last one from this correspondent. The last two years have been a fascinating journey through the many aspects of fusion and I hope I have succeeded in conveying some of my fascination for the extraordinary quest that fusion research is.

Special thanks are due to Culham Studio team – Russell, Mat, Chad, Mark, Katie, Holly, and Stuart – for taking my crazy ideas and making them reality, and never once – even if I deserved it – thumping me!

- Phil Dooley, Sept 2, 2013