Considering plasma states which are not in perfect thermodynamic equilibrium (no exact Maxwellian distribution), even though they represent equilibrium states in the sense that the force balance is equal to 0 and a stationary solution exists, means their entropy is not at the maximum possible and hence free energy appears available which can excite perturbations to grow. Such an equilibrium
state is unstable. The stability of a plasma confined by a toroidal and poloidal magnetic configuration is therefore seen to be determined by the free energies associated with eventual currents parallel to В (force-free currents) and with the plasma pressure. The ratio of these two free energies turns out to be (3P. Apparently, the gradients of plasma current magnitude and pressure, Vj and Vp, are the destabilizing forces in connection with the ‘bad’ magnetic field curvature discussed in Sec. 9.4 and necessarily inherent to a tokamak.

The occurring instabilities can be divided into three types: (a) Ideal MHD Modes:

Amongst all types, the ideal MHD instabilities are the most virulent due to their fast growth and the possible extension over the entire plasma. Thus, on the short time scale of relevance (microseconds) the resistivity of the plasma is negligible.

A toroidally confined plasma sees ‘bad’ convex curvature of the helical magnetic field lines on the outboard side of the torus. Consequently, such a configuration is subject to the onset of flute-type interchange instabilities of which the driving mechanisms have already been demonstrated in Sec. 9.4. However, because the magnetic field lines in a tokamak are more concentrated on the inboard side where there is ‘good’ curvature (concave as seen from the plasma), the average curvature of В-field lines over a full poloidal rotation is ‘good’ for windings with a rotational transform l < 2n, i. e., q > 1. Therefore, interchange perturbations do not grow in normal (q > 1) tokamaks. It is however observed that the perturbations can locally grow or ‘balloon’ in the outboard ‘bad’ curvature region. In that case a high local pressure gradient is responsible for driving the so-called ballooning instability. By establishing appropriate pressure profiles and appropriate magnetic field line windings, those modes can be suppressed almost everywhere in the plasma.

Another instability which represents the most dramatic one in ideal MHD is the kink instability already discussed in Sec. 9.6. It causes a contortion of the helical plasma column and consequently of the magnetic flux surfaces. Preferably it occurs in tokamak plasmas at low pressures and is driven by the radial gradient of the toroidal current. Fortunately these instabilities are bounded to small intervals of q(a) lying close below integer values. For the current profile distributions typical of tokamak fusion plasmas, such unstable modes arise mainly when q(a) is just a little less than 2. The associated kink distortion of the plasma column can be stabilized by an enhanced toroidal magnetic field strength such that the Kruskal-Shafranov condition

Be

is fulfilled, which we had used previously and extended to a more stringent criterion at the plasma edge, Eq. (10.42).

As seen in the preceding section, adjusting the safety factor q to the
appropriate value is associated with a limitation of the plasma beta. In order to avoid the major MHD unstable activities the overall P is limited by the maximum critical beta

ftri, [%] = CT^~ (10.50)

aB

with the so-called Troyon factor CT (dimensionless) ranging from 2.8 to 5 depending on the nature of the instabilities, if the plasma current I is in MA, the minor radius a in m and the confining field in Tesla. Since the plasma cross section need not necessarily be of circular shape (actually, most tokamak plasmas of today’s experiments feature an elliptic, bean or D-shaped cross section which permit optimization of energy confinement and plasma pressure profiles), parameters which account for the actual cross-sectional contour will also enter the Troyon factor. Note that the rough analysis in the preceding section, Eqs.(10.40) — (10.48), can also provide for the functional relation given in Eq. (10.50).

(b) Resistive MHD Modes

Additional types of macroscopic instabilities are attributable to the electrical resistivity of a tokamak plasma, which makes the instability grow more slowly. Characteristically, the growth times are of the order of 10’4 to 10’2 s which, however, is still short compared with the energy confinement time Tt (seconds). Resistive MHD modes result from the diffusion or tearing of magnetic field lines relative to the plasma fluid and can thereby destroy the nested topology of the magnetic flux surfaces. For helically resonant B-perturbations, magnetic field diffusion may preponderate the ideal MHD effects in thin boundary layers around surfaces having a rational q. Then the magnetic field lines can reconnect in these layers thereby producing nonaxisymmetric helical islands as suggested in Fig. 10.12. The tearing mode instability in a tokamak is driven by the radial gradient of the equilibrium current density. Upon formation of a magnetic island filament it grows until it acquires all the accessible free energy of the current. Outside these so-called resonant surfaces (q is rational) the plasma undergoes a sequence of MHD equilibria.

For low-(3 plasmas it is possible that these resistive modes couple nonlinearly amongst each other (different rational q), which leads to a disruption of the plasma current. Increasing the plasma current, which-according to Eq. (10.48)-is associated with lowering the safety factor q, is seen to diminish the existence of current distributions which are stable against tearing modes.

Another effect brought about by the nonlinear evolution of resistive MHD modes and observed in tokamak plasmas is the ‘sawtooth’ behaviour of some plasma parameters such as the electron temperature and the current density at the centre of the plasma column where q can drop below 1. Sawtooth oscillations can be delayed or prevented by modifying the current profile near the q = 1 magnetic surface. These sawtooth oscillations are not deemed catastrophic since their activity is constrained within a small region internal to the q = 1 surface, where it comes to a redistribution of the plasma energy.

(c) Microinstabilities

Microinstabilities are often associated with non-Maxwellian velocity distributions. The deviation from thermodynamic equilibrium means that there is free energy which can drive instabilities, often evolving into plasma turbulence. Also, nonuniformity and anisotropy of distributions can give rise to instabilities. Hence it is the particle kinetic effects that play an important role here, and the plasma cannot be expected to behave as a simple fluid and therefore cannot be treated as such anymore. It rather requires a kinetic description.

The electron velocity distribution becomes increasingly anisotropic as the plasma density decreases since, in order to carry a given plasma current, the individual electrons are further required to align their velocities with the current density direction. This drift velocity tends to make the velocity distribution function more and more asymmetric and hence unstable. On the other hand, the electrons will encounter collisions which randomize their velocities and thus reestablishes symmetry in the velocity distribution. However, as the density is further decreased, this stabilizing effect due to collisions is ultimately overcome by the destabilizing effect of the increasing drift velocity.

Further, anisotropy occurs in a plasma when it is confined by mirror fields, since particles having a large V||-component will escape and are therefore lost from the distribution. Hence, also the trapped particles in a tokamak, which bounce back and forth in the local mirror fields, can constitute a source of instabilities, preferably when the perturbation frequency is less than the bounce frequency. These then are classified as trapped particle instabilities and are still being investigated for effects on increased cross-field diffusion in tokamaks.

As a current is driven through the plasma or a beam of high energetic particles is injected, the different species will drift relative to one another. The drift energy can excite waves in the plasma. Since oscillation energy may be gained at the expense of the drift energy, the disturbance can grow. Such an instability is called a ‘two-stream’ or beam-plasma instability.

Where there is a steep density gradient in the plasma, an instability may appear due to the electron drift caused by the gradient and is called the drift instability. It can be stabilized by appropriate magnetic shear.

Microinstabilities are a large and complex field in plasma physics, which is still under investigation and many theoretical predictions are still to be proven by experiments.

Generally, there are three effective ways to prevent plasma instabilities: (i) magnetic shear, (ii) minimum-B configuration and (iii) dynamic stabilization by oscillating E — or В-fields or by proper-phase force feedback.