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The tokamak

The good confinement in a tokamak is achieved by winding the "guiding rails" the particles follow around the torus chamber. This is achieved with current-carrying coils creating the required combination of a poloidal (Bp) and a toroidal (Bt) magnetic field. Click on the image to enlarge.

The most obvious way to confine a plasma magnetically is to immerse it in a purely cylindrical plasma configuration with current-carrying coils wound poloidally around it to create the (axial or longitudinal) confining magnetic field. However, such a machine trivially suffers from end losses: as particles are roughly glued to magnetic field lines, they hit the machine's wall at either end of the straight cylinder because the field lines are intersecting the surfaces at both ends of the tube. One way that was explored in the past to solve this problem was the creation of a "magnetic mirror": when entering regions with increased magnetic field strength, charged particles gyrating around magnetic field lines slow down parallel to the field, while accelerating perpendicular to it to conserve their total energy. A sufficiently strong magnetic field thus manages to stop the parallel motion and to throw the particles back into the machine. Since very high magnetic fields or very long machines are needed to achieve this sufficiently efficiently for particle populations that are fast enough to allow fusion, this solution proved to be unpractical. But another solution comes to mind when looking at the graph of a "linear" cylindric device suffering from end losses: gluing the two ends of the linear machine through which particles are otherwise lost together should solve the confinement problem! The resulting doughnut or torus shape and the circularly bent "toroidal" rather than straight axial magnetic field are two of the characteristics of the "tokamak". The name tokamak is derived from the Russian "toroidalnaya kamera magnitnaya" which signifies toroidal magnetic chamber. This type of machine is the most successful magnetic confinement device presently known. The magnetic configuration of the tokamak requires more than just the "toroidal" magnetic field component (which, for its existence, necessitates so-called "poloidal" current-carrying coils) along the torus. Since a magnetic field bent on itself is inhomogeneous (it is stronger at the inside of the doughnut than at the outside of it), the particle's gyro-motion is not characterized by a single gyro-radius: At the part of the trajectory closest to the symmetry axis of the machine, the gyro-radius is somewhat tighter than that at the outermost part. Hence the motion perpendicular to the magnetic field is not just a circle with a fixed radius and center, but a circle of which the radius changes periodically. As a consequence, a slow drift, directed perpendicular to both the magnetic lines and to the direction in which the field changes, results. This drift of the charged particles in a toroidal device with only toroidal magnetic field destroys the confinement required for fusion.

Orbits of a passing (red) and a trapped (blue) particle. Click on the image to enlarge.

The trajectory of the center of the gyro-oscillation is known as the "guiding center". The gyration around this guiding center is referred to as the Larmor gyration. Adding a poloidal component to the toroidal magnetic field results in guiding centers that not only spiral around the machine in the toroidal but also in the poloidal direction. Such a configuration is stable as it solves the problem of the drifts (which are compensated on average by spinning around the machine poloidally) and thus manages to confine the very energetic particles, or at least to keep them sufficiently long in the machine for fusion to take place. Two classes of particles exist. Particles for which the velocity component perpendicular to the total magnetic field is significant have guiding centers that reflect from the central high magnetic field regions as their parallel velocity goes through zero there. Because the poloidal cross section of such "trapped" orbits has the shape of a banana, they are often referred to as "banana" particles. Particles with a sufficiently large velocity along the magnetic field have guiding centers that never reach a stagnation point. The latter are known as "passing" or "circulating" particles, their parallel velocity never going through zero.