In deriving the relative requirements of the magnetic field strength and the plasma pressure for effective confinement-Sec. 9.1-we considered, in the fluid model, the case of magnetohydrodynamic equilibrium. However, we did not examine whether this equilibrium state is stable or not. In an equilibrium state all forces are balanced allowing thus for a steady-state solution of the set of magnetohydrodynamic equations discussed at the end of Sec. 6.3. The equilibrium is labeled stable if small perturbations are inherently damped and it is unstable if small deviations from the equilibrium state are amplified, that is, if perturbations propagate and grow with time; this then is called an instability.

The unperturbed state requires perfect thermodynamic equilibrium in which the plasma particles have Maxwellian velocity distributions, while the plasma density and the magnetic field is uniform. Note that in the magnetic confinement configurations of interest to nuclear fusion reactors these requirements are not met. In the example of a mirror field, the isotropy of the Maxwell distribution is significantly disturbed by the particles lost through the mirror throats, which had obviously a dominant V|rcomponent. Further, a mirror-device will feature VB and VNj and thus be non-uniform. Though all forces can be balanced in a steady state, this state is not in perfect thermodynamic equilibrium and possesses so — called ‘free’ energy which can drive instabilities. Even periodic motions of the plasma fluid elements, e. g. plasma oscillations or, alternatively, waves, can thus be induced. An instability constitutes a motion which reduces the free energy and brings the plasma closer to perfect thermodynamic equilibrium. There exists a wide range and variety of possible plasma waves and instabilities, the discussion of which is far beyond the scope of this textbook. Hence we restrict ourselves to a few demonstrative examples of interest here.

The instability most relevant to mirror machines is the so-called flute-type instability. In the simple mirror geometry of Fig.9.3, the curvature of the magnetic field-except for the end-regions-is seen to be convex. Any outward perturbation of the confined plasma, i. e. a ripple on its boundary surface where all magnetically confined plasmas appear to have an energy density gradient, takes the plasma into regions of lower magnetic induction and lower kinetic pressure; hence, such a displacement to regions of reduced energy density will provide for free kinetic energy to let the perturbation grow. This can lead to flutes of plasma moving across magnetic field lines, Fig. 9.6, and result in particle loss from the containment region.

Further insight into the onset and mechanism of the flute instability is provided if we recall the drifts and forces associated with a so-called "bad" convex В-field curvature, where the curvature drift, Eq.(5.61), will lead to charge separation occurring perpendicular to the magnetic field and the radius of curvature, Fig. 9.7. This polarization creates an azimuthal electric field causing an additional ExB drift, which transports both ions and electrons in the radially outward direction thus forming the flute-like bumps on the plasma column.

This type of instability can be avoided by generating a so-called ‘minimum-B’ field configuration in which the field lines are (almost) everywhere concave into the plasma. Here the charged particle then senses an increasing В-field in every direction and therefore finds itself in a magnetic well; the term minimum-B is thus commonly used for such a magnetic topology.

The simplest means of producing such a minimum-B field configuration is to locate four current carrying bars on the periphery of a magnetic device with their positions suggested in Fig. 9.8. These bars are called Ioffe bars with the current in adjacent bars flowing in opposite directions.

Another means of generating a minimum-B magnetic field configuration is by using a coil having the shape of the seam of a baseball. If the coil of the baseball configuration is suitably flattened and oriented in opposition with another similar coil, one obtains again a minimum-B configuration, called a "Yin-Yang" coil configuration, Fig. 9.9.

Further examination of these minimum-B configurations makes it clear that they all provide a central circular region for the plasma but that in two opposite directions the flattened fan-shaped magnetic fields are open and thereby still form a magnetic mirror. This feature can be extended by adding such devices to the ends of mirror solenoidal fields to form more effective mirrors because the magnetic well can in principle be deeper, and further, the particles contained

Mirror devices are basically appropriate for steady state operation, in which the particle injection rate balances the diffusion leakage rate. The diffusion occurs dominantly through the open ends and constitutes also a diffusion in velocity space, since-as previously mentioned-the velocity distribution in a mirror plasma is no longer Maxwellian due to the preferred loss of particles with large V|| / vj_; one rather deals with a so-called loss-cone distribution excluding all particles in the loss-cone as resulting from Eq.(9.38), Fig. 9.5. This deviation from the Maxwellian distribution drives so-called velocity-space instabilities, here specifically the ‘loss-cone’ instability, which can enhance the velocity-space diffusion into the loss-cone. It has been observed that such instabilities are less harmful to plasma confinement when the mirror device is short in dimension.