particle trajectories

3) - Magnetic confinement (p 1 - 2 - 3 - 4 )

How do we bring together particles that at first sight have no reason to stay together? In the sun, gravitation takes care of it. On earth, we use powerful magnetic fields.

a) particle trajectories

Plasma confinement in a tokamak is based on the property that charged particles have of describing a helical trajectory around magnetic field line. Look at the movement of a charged particle around a right magnetic field line.

Watch the film "straight trajectory" (mpg, 493 kb)

The particle, represented in blue, describes a helical movement around the magnetic field line, which follows the guide centre of the trajectory, represented in green.

The gyration radius of the particle, called Larmor radius, depends on the intensity of the magnetic field, the mass and charge of the particle and its energy. The stronger the magnetic field, the smaller the Larmor radius, the particle staying "stuck" near the field line. Moreover, the electrons, much lighter than the ions, have a much smaller Larmor radius for the same energy. Finally, very energetic particles have a much larger Larmor radius than low energy particles, and are therefore more difficult to confine. The Larmor radius may typically vary from several millimetres for not very energetic particles with an intense magnetic field to tens of centimetres for very energetic particles.

The confinment solution thus consists in closing the magnetic field line on itself to trap the particle, as you may see below.

Watch the film " cylindrical trajectory " (mpg, 750 kb)

We are now in a configuration where the path of the magnetic field is solely toroidal.

Unfortunately, on a simple circular trajectory of this type, the particle undergoes a slow cross drift, due to the drift gradient of the magnetic field and centrifugal force, depending on the sign of its charge. For example, the ions will drift up (as illustrated on the diagram opposite) and the electrons down.

To compensate this effect, the idea is to stabilise the configuration by adding a poloidal component to the toroidal magnetic field. This is the magnetic configuration used in the tokamak. The field lines become helixes twisted round stacked toric surfaces, called magnetic surfaces.

The particle then spends half its time head upwards, where the vertical drift, which we suppose to be towards the top as in the example opposite, moves it away from the magnetic surface, and the other half head down, where the vertical drift pulls it back to the magnetic surface. The drift effect is thus on average compensated.

Particle following a helical field line

In a tokamak, the toroidal magnetic field is produced by external coils, whereas the poloidal magnetic field is induced by a current flowing toroidally in the plasma . This current is generated by transformer effect, from a primary circuit of which the secondary is the plasma. Tore Supra is outstanding in being equipped with supra-conducting magnets, which enable it to guarantee a permanent toroidal field (the machines equipped with conventional magnets are limited in duration by heating of the copper coils). The pulse duration is thus limited by the capacity of the primary circuit generating the plasma current inducting the poloidal field.

Finally, there exists another configuration, called a stellarator , in which the magnetic field is provided completely by external toroidal as well as poloidal coils. The fact of not having an intense current flowing in the plasma is an advantage in the event of plasma disruption, but the drawback is the complexity of the necessary magnetic coils. This may be seen on the diagram of the German stellarator project W7X , where the coils are represented in blue and the plasma in orange.

Principle of the stellarator (source : Euratom-IPP )

The pitch of the helix on each magnetic surface is called safety factor (that is the number of large toroidal turns necessary to complete 1 small poloidal turn). In a tokamak configuration this safety factor typically varies from 1 in the centre of the plasma to several units on the edge. It is worth noting that, in general, if we follow the field line, it will entirely describe the magnetic surface around which it winds in the course of its successive journeys. This is true except in the case of a rational safety factor (i.e. equal to the ratio of two whole numbers). In this special case, the field line closes in on itself after a whole number of turns, resulting in specific properties for this (local modification of transport, triggering of instabilities, and so on)

Finally, we should note that in a first approximation, the macroscopic features (density, temperature, pressure and so on) are homogenous on a magnetic surface. We may thus describe them in a poloidal section simply as functions the plasma radius, for example by taking their value on each white circle showing a magnetic surface in the figure below. We talk in terms of radial profile (only depending on the radius), which for density, temperature and pressure is maximum in the centre of the plasma, decreasing towards the edge of the plasma, as illustrated on the figure below.