**3. The MHD model**

6 Will-be-set-by-IN-TECH

dissipation which is ubiquitous on real plasmas causes the currents and the magnetic fields to decay, so the configuration would be lost in the resistive time scale. It is then imperative to apply adequate methods to drive currents and heat the plasma. Some common methods that

• Current induction by a primary coil. A primary coil is located at the center of the torus and plays the role of the primary of an electric transformer while the plasma itself is the secondary (the primary coil was not sketched in Fig. 1). This is the usual approach to induce the toroidal current in tokamaks and RFP's. It does not allow to operate in truly

• Radio frequency waves. Energy can be transferred to the plasma from an external source of electromagnetic waves (antenna). The electric field of the waves transfers momentum to the particles inducing currents and heating the plasma by collisions. Within a multi-species plasma there exists a number of resonant frequencies that enhance the coupling between the plasma and the antenna (ionic and electronic cyclotron resonances, hybrid resonances,

• Neutral beam injection. Neutral atoms injected are not deflected by the magnetic field and can penetrate the plasma until they become ionized through collisions. Once ionized these particles follow orbits determined by the magnetic field and their energy. This process

• Rotating magnetic fields. Plasma electrons may be dragged, and thus a current may be

• Helicity injection. When a current is established along the magnetic field some amount of magnetic helicity (see Sec. 4) is injected in the magnetic configuration. The driven current may destabilize the configuration triggering a relaxation process that redistributes the current. This is the main method used in spheromak sustainment and is the subject of

Early experiments in toroidal pinch configurations exhibited, under certain conditions, the spontaneous reversal of the toroidal field near the wall of the chamber. This unexpected feature was succesfully explained in terms of the relaxation theory proposed by Taylor (1974). According to this theory, MHD fluctuations cause the plasma to minimize its magnetic energy

Some years later, it was realized that the minimum energy state, for a given amount of magnetic helicity, inside a sphere is a system of nested toroidal magnetic flux surfaces (Rosenbluth & Bussac, 1979). The idea of a configuration relevant for fusion research that would be self produced (or self-organized) inside a simply connected volume attracted the attention of the scientific community. Several experiments were designed in order to check this theoretical prediction. The success of these experiments was considered a proof of the

Despite the initial enthusiasm, it was later realized that the relaxation process involves MHD fluctuations that strongly degrade the confinement. Because of these fluctuations the confinement peformance of the spheromak is much lower than that of the tokamak or the

have already been successfully implemented are:

etc.).

study of this Chapter.

**2.4 The spheromak configuration**

steady state and it can not be used in compact tori.

heats the plasma and drives localized currents.

while conserving the total magnetic helicity (see Sec. 4).

remarkable robustness of the relaxation theory (Bellan, 2000).

induced, by externally applied rotating magnetic fields.

The MHD model describes the macroscopic behavior of a plasma in many situations of interest in a relatively simple manner. Its validity relies, however, in a number of assumptions that have to be borne in mind in order to understand what kind of phenomena can be explained by the model and what effects lie outside this description.
