**2.3 Formation and sustainment**

Once an MHD equilibrium with good stability properties has been devised it is necessary to find appropriate methods to form and sustain the configuration. The formation methods depend on the configuration under consideration. In fact, a given configuration can be obtained using different formation schemes. In most cases, after the formation process the plasma has a temperature sensibly lower than that required for fusion. Moreover, the resistive

RFP. Little is known about the dynamics of these fluctuations since the relaxation theory is only able to predict the final state of the plasma but it can not provide any detail on how this

Dynamics of Magnetic Relaxation in Spheromaks 91

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

The MHD model regards the plasma as a quasi-neutral electrically conducting fluid. The first and most fundamental assumption of this description is to regard the ensemble of ions and electrons conforming the plasma as a single continuum medium. This is valid when the length scales associated with the magnetic field gradients is much larger than the internal length scales of the plasma (such as the ionic and electronic gyroradii). This condition holds

The second important assumption is to consider that the plasma is in thermodynamic equilibrium so the particles have a Maxwellian distribution of velocities. This is a good approximation as long as the shortest time scale of the process under consideration is much longer than the collision time and the shortest length scale of the system is larger than the mean free path of the particles. In other words, the plasma should be in a collisional regime (this condition is required to derive the fluid equations from the kinetic equations (Braginskii, 1965)). The collisionality hypothesis is usually not satisfied at the highest temperatures obtained in modern tokamak experiments. However, spheromak plasmas are much colder (*<sup>T</sup>* <sup>∼</sup> 102 eV) so that this assumption is still reasonable. Moreover, there are several arguments supporting the validity of the MHD model even in collisionless systems (Friedberg, 1987);

Finally, in the context of the MHD model the plasma is assumed to be electrically neutral (or quasi-neutral since the charges are present but exactly balanced). This is approximately true when the length scales under consideration are larger than the Debye shielding of electrons.

Now we seek for the equations that describe the evolution of the two main quantities that govern the dynamics of such an MHD system: the velocity field and the magnetic field. The equation for the evolution of the plasma velocity **u**, expresses the balance of linear momentum

where *ρ* is the mass density and *p* is the thermodynamic pressure. The second term on the right hand side is the Lorentz force, where **J** is the current density and **B** is the magnetic field. We note that due to quasi-neutrality the current density is produced by the relative motion

= −∇*p* + **J** × **B** + *μ*∇ · Π (7)

state is attained (Jarboe, 2005).

by the model and what effects lie outside this description.

in virtually every laboratory plasma dedicated to fusion research.

**3.1 Basic assumptions of the MHD model**

**3. The MHD model**

(Priest & Forbes, 2000).

**3.2 MHD equations**

*ρ ∂***u**

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> **<sup>u</sup>** · ∇**<sup>u</sup>**

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 have already been successfully implemented are:

