One of the simplest systems for magnetic containment is the pinch concept; here the plasma carries an electric current and is confined by the magnetic field induced by this current. As the current is increased, the larger magnetic field compresses the plasma and also raises its temperature by Joule-heating. Hence, confinement and heating is simultaneously provided. For that, extremely large currents (some 105 A) are needed thus rendering pinches to operate only in short pulses. The two principle configurations, denoted as z-pinch and 0-pinch, are sketched in Fig. 9.10.

V

b) 6 — pinch

Fig. 9.10: Representation of pinch geometries.

The confinement requirements are particularly obvious in the case of a z — pinch. In order that the plasma particles be retained, the electric currents must be high enough to generate a magnetic field of such strength that the magnetic energy density is able to balance the plasma pressure, that is

where T = Ті = Te has been assumed. The magnetic field induced by j can be found from Maxwell’s equation

VX —= j.

К

Using Stoke’s integral theorem, we then relate

= I

where ds is the path element along the circumference of the plasma column of cross section area A with dA denoting the oriented differential area element, and I defines the total electric current in the column. At the plasma surface, r = a, the path integral is

—2mlree = —Be(a) (9.49)

і к к

where e0 is the unit vector in the azimuthal direction.

Since В possesses only an azimuthal component, B0, the absolute value of the magnetic induction at the surface, is here given by

B(a) = ^~ . (9.50)

2na

Upon insertion of Eq. (9.50) in Eq.(9.46) and considering the particle numbers as referring to a plasma column of unit length, i. e. in the volume a27t X 1 m,

N*= N*+ Nl = a2n(Ni + Ne), (9.51)

we arrive at the requirement

2

=N*kT, (9.52)

8k

which is commonly referred to as the Bennett pinch condition.

Magnetic pinches are also troubled by instabilities. Two types of instabilities discussed here can occur in a cylindrical plasma carrying a large current along its axis.

One is the so-called ‘sausage’ instability, Fig. 9.11, which arises due to axial perturbations in the plasma column diameter and can be elucidated as follows. Consider the initial equilibrium state to be disturbed by some expansion of the plasma column to a larger radius. Since B(a) ~ 1/a, Eq. (9.50), the magnetic pressure at ai > a0 is weaker than that at the equilibrium surface (a0) and subsequently reinforces the expansion process. As these perturbations grow they make the plasma column look like a string of linked sausages and finally disrupt the plasma column. A restabilization of this pinch instability is possible by applying a sufficiently strong magnetic field along the axis.

Another characteristic instability in a cylindrical pinch is associated with

perturbations of the linear axial geometry, i. e., when the plasma column exhibits an axial curvature as illustrated in Fig. 9.12. It is seen therein that the magnetic field lines are closer together at the inside of a helical bend and hence the magnetic induction is larger there than at the outside of the bend. This difference between the magnetic pressure at the inside and outside of the bend causes any small disturbance of the straightness of the equilibrium plasma cylinder to grow until the plasma column scrapes on the surrounding wall. This unstable behaviour against bending is called a helical kink instability and can as well be controlled by an additional strong axial magnetic field.

weak B^)

weaker В Fig. 9.12: Helical kink instability resulting from perturbations of the linear axial geometry.

Though a fusion device magnetically pinching the plasma is also seen to suffer instabilities which are mainly caused by the extreme electrical currents supplied, a 0-pinch can turn out to be remarkably stable. However, as for all open systems, linear pinches suffer from end leakage. To diminish end losses, a pinch usually features a magnetic mirror configuration, whereby also a minimum B — field structure is desirable. On the other hand, closed pinches can be realized when the plasma column is bent into a torus. Such a topology for magnetic confinement fusion will be discussed in the following chapter.

Problems

9.1 What assumptions are needed to make the previously stated magnetic confinement requirement, Eq. (4.12), consistent with Eq. (9.5)? Consider that the kinetic plasma pressure is largest at the centre and negligible at the plasma surface.

9.2 A magnetic mirror field, confining a fusion plasma with fuel particle densities Nd = Nt = 4xl019 m‘3 at Tj = 30 keV, varies from Btmx = 6 T at its throats to Вmin = 1 T at the mid-plane. Find:

(a) the number of ions escaping through the loss cones.

(b) an approximate value of the mirror confinement time.

9.3 Depict the trapping and loss domains of deuterium and tritium ions along their E0 = constant lines, akin to those shown in Fig. 9.4, for the mirror configuration of problem 9.2.

9.4 Attempt to sketch a loss cone distribution function accounting for the absence of particles with large Уц / vj_.

9.5 Discuss the development of the flute instability.

9.6 For each of the following open magnetic confinement fusion reactor concepts, draw a sketch of the concept, label the major components, and describe their purpose. Describe how fusion fuel ions are confined, and list what the energy and particle losses are in the system, and where they occur. Describe all the different electrical currents, magnetic fields, and their purpose. You may have to perform a literature search for some of the concepts.

(a) Magnetic Mirror

(b) Ying-Yang Coils

(c) Z-pinch

(d) Theta-Pinch

(e) Reversed Field Mirror

(f) Combined: Magnetic Mirror with Ying-Yang Coils