Linear Pinches

Let us examine a particularly simple magnetic confinement device. Suppose that the plasma is contained in a cylindrical vacuum vessel of radius and finite length, capped by conducting end plates. See Figure 1.4. Let , , be a conventional cylindrical coordinate system whose axis corresponds to that of the vessel. Suppose that a uniform axial current, , is driven through the plasma by electrically biasing the end plates. In other words, suppose that the electric current density within the plasma is

(1.51) |

(1.52) |

(1.54) |

(1.56) |

(1.57) |

The previous two equations yield

Table 1.5 uses the information in Table 1.2 to estimate the critical plasma current needed to achieve self-sustaining nuclear fusion. It can be seen that, in both low-field and high-field confinement devices, the critical current is about 9 MA. The critical poloidal magnetic field-strength needed to achieve nuclear fusion is Note from Table 1.5 that the critical poloidal field-strength is much less than the total field-strength. This is because another much larger component of the magnetic field (in this case, an axial component) is needed to stabilize the plasma [26]. (See Section 1.10.)As well as generating a poloidal magnetic field, the axial current that passes through the plasma heats it ohmically. The ohmic heating rate per unit volume is

(1.60) |

(1.62) |

(1.63) |

(1.64) |

(1.65) |

(1.66) |

There are two serious problems with a linear pinch magnetic confinement device. The first problem is that the plasma is ideally unstable. A so-called *ideal* plasma
instability is one that does not change the topology of the magnetic field (i.e., it does not require the reconnection of magnetic field-lines). For the case of a linear
pinch, the relevant instabilities are the “sausage” mode and the “kink” mode [26]. These instabilities are global in nature, cause distortions in the shape of
the plasma that are consistent with their names, and lead to the complete disruption of the plasma discharge [50]. Ideal instabilities arise because the force balance criterion described in Equation (1.53) breaks down. Consequently,
the electromagnetic pinch force is balanced by plasma inertia, rather than by plasma pressure. In other words,

(1.67) |

(1.68) |

The second problem with a linear pinch is associated with the need to keep the impurity content of the plasma within acceptable limits. Suppose, for the sake of example, that the inner surface of the vacuum vessel is lined with graphite tiles, as is the case in many magnetic confinement devices. The mass density of graphite is . Moreover, the mass of a carbon atom is kg. Thus, the number density of carbon atoms within the tiles is . Note that this number density exceeds that of the particles in the plasma by nine orders of magnitude. (See Table 1.2.) Now, we need to keep the impurity content of the plasma such that , otherwise the fuel ion dilution and radiation losses become unacceptable. According to Table 1.3, this implies that the number density of carbon atoms within the plasma, , cannot exceed 1.5% of the electron number density. Suppose that, as a consequence of plasma-wall interaction, a layer of carbon atoms of thickness is ablated into the plasma. It is easily demonstrated that

(1.69) |

(1.70) |

In a linear pinch, the interaction of the plasma with the curved surface of the vacuum vessel is moderated by the small gyro-radii of charged particles within the plasma, which ensure that particles cannot freely stream to the surface, but instead have to slowly diffuse across magnetic field-lines in order to reach it. (See Section 2.6.) However, in the presence of an axial magnetic field, charged particles can reach the end plates by moving along magnetic field-lines. This process is not moderated by the small particle gyro-radii. In fact, the only possible moderation mechanism is collisions. The mean-free-path between collisions of an electron moving parallel to a magnetic field-line is [21]

(1.71) |

(1.72) |

(1.73) |

(1.74) |

(1.75) |