How do you bring together particles that have no apparent reason to want to stay together? In the sun, gravity takes care of that. 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 a magnetic field line. Look at the movement of a charged particle around a straight magnetic field line.
The particle, shown in blue, describes a helix around the field line, which is followed by the guide centre of the trajectory, shown in green.
The particle’s radius of gyration, known as the return Larmor radius, depends on the strength of the magnetic field, the mass and charge of the particle, and its energy. The stronger the magnetic field, the smaller the Larmor radius, as the particle remains “stuck” in the vicinity of the field line. In addition, electrons, which are much lighter than ions, have a much smaller Larmor radius at the same energy. Finally, very energetic particles have a larger Larmor radius than low-energy particles, and are therefore more difficult to confine. The Larmor radius can typically vary from millimetres for low-energy particles with an intense magnetic field to tens of centimetres for very energetic particles.
The trick is to close the field line on itself to trap the particle, as you can see opposite.
This gives us a configuration where the direction of the magnetic field is purely toroidal.
Unfortunately, on a simple circular trajectory of this type, the particle undergoes a slow transverse drift, linked to the magnetic field gradient and the centrifugal force, the direction of which depends on the sign of its charge. For example, ions will drift upwards (as shown in the diagram opposite) and electrons downwards.
To compensate for 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 tokamaks.
The field lines become helices wrapped around interlocking toroidal surfaces, known as magnetic surfaces.
The particle then spends half its time with its head up, where vertical drift, assumed to be upwards as in the example opposite, takes it away from the magnetic surface, and the other half with its head down, where vertical drift brings it closer to the magnetic surface. The effect of drift is then on average compensated for.
In a tokamak, the toroidal magnetic field is generated by external windings, while the poloidal magnetic field is induced by a current flowing toroidally in the plasma. This current is generated by a transformer effect, from a primary circuit whose secondary is the plasma. Tore Supra/WEST has the unique feature of being equipped with superconducting magnets, which enable it to ensure a permanent toroidal field (machines equipped with conventional magnets are limited in duration by the heating of the copper coils). The duration of the discharge is then limited by the capacity of the primary circuit generating the plasma current that induces the poloidal field.
Finally, there is another configuration, called a stellarator, in which the magnetic field is provided entirely by external windings, both toroidal and poloidal. The fact that there is no intense current circulating in the plasma is an advantage in the event of plasma disruption, but it comes at a price in terms of the complexity of the magnetic windings required. This can be seen in this diagram of the German W7-X stellarator, where the winding is shown in blue and the plasma in orange.
The pitch of the propeller on each magnetic surface (i.e. the number of large toroidal turns needed to complete one small poloidal turn) is called the ‘safety factor’. In a tokamak configuration, this safety factor typically varies from 1 at the centre of the plasma to a few units at the edge. It should be noted that, in the general case, if we follow the field line, it will completely describe the magnetic surface around which it wraps as it passes. This is true except in the case of a rational safety factor (i.e. equal to the ratio of two integers). In this particular case, the field line closes in on itself after a whole number of turns, which gives the magnetic surface specific properties (local modification of transport, triggering of instabilities, etc.).
Finally, it should be noted that, to a first approximation, macroscopic quantities (density, temperature, pressure, etc.) are homogeneous on a magnetic surface. They can therefore be described in a poloidal section simply as a function of the plasma radius, for example by taking their value on each white circle illustrating a magnetic surface in the diagram below. This is referred to as a radial profile (depending only on the radius), which is in the case of density, temperature and pressure maximum at the centre of the plasma and decreasing towards the edge of the discharge, as illustrated in the figure below.
