Tearing Modes

Consider a magnetic perturbation of a tokamak plasma equilibrium that has $m$ periods in the poloidal direction, and $n$ periods in the toroidal direction. Here, $m$ is termed the poloidal mode number, wheras $n$ is termed the toroidal mode number. Such a perturbation resonates with the plasma (i.e., satisfies ${\bf k}\cdot{\bf B}=0$, where ${\bf k}$ is the wavenumber of the perturbation, and ${\bf B}$ is the equilibrium magnetic field) at the magnetic flux-surface, of minor radius $r_s$, at which [63]

$$q(r_s) = \frac{m}{n}.$$ (1.82)

Such a surface is termed a rational magnetic flux-surface, for obvious reasons. Now, the well-known flux-freezing constraint [21] of ideal magnetohydrodynamics forbids any change in the topology of magnetic field-lines due to the perturbation [26]. In particular, the constraint requires the perturbed radial magnetic field at the rational surface to be zero.

An ideal kink mode is an instability of a tokamak plasma that is unstable even when the perturbed radial magnetic field is constrained to be zero at the associated rational surface (assuming that the surface lies within the plasma). As we have seen, such instabilities grow on the very short Alfvén time. Fortunately, ideal kink modes are relatively easy to avoid. In fact, ideal kink modes can be avoided by not allowing the edge safety-factor to fall significantly below 3, and not allowing the plasma beta to approach the Troyon limit. It is also necessary to prevent the plasma density from exceeding the so-called Greenwald limit [28] that is associated with the radiative collapse of the plasma current profile.

A tearing mode [27] is an instability of a tokamak plasma that is driven by the same free energy source as an ideal kink mode (namely, radial current and pressure gradients within the plasma [26]), but changes the topology of the magnetic field by tearing and reconnecting magnetic field-lines at the rational surface. This implies that the mode is not subject to the constraint that the perturbed radial magnetic field at the rational surface must be zero. Consequently, a tearing mode can be unstable even when the corresponding (i.e., possessing the same poloidal and toroidal mode numbers) ideal kink mode is stable [63].

Magnetic reconnection is made possible by finite plasma electrical resistivity [21]. Now, magnetic flux diffuses through a tokamak plasma, due to the action of resistivity, on the characteristic resistive time

$$\tau_R = \mu_0\,a^2\,\sigma,$$ (1.83)

where the plasma electrical conductivity is specified in Equation (1.61) [21]. As is shown in Table 1.5, the resistive time in a tokamak fusion reactor exceeds the Alfvén time by many orders of magnitude. To be more exact, the ratio of the two timescales,

$$S = \frac{\tau_R}{\tau_A},$$ (1.84)

is known as the Lundquist number [21], and is of order $10^9$ in a tokamak fusion reactor.

Figure 1.6: Schematic diagram of an $m=2/n=1$ magnetic island chain in a tokamak plasma.

It is not surprising that a tearing mode grows and saturates on a timescale that is related to the resistive time [51]. In fact, the saturation time is typically a few percent of the resistive time [29]. This implies that a tearing mode is an extremely slowly growing instability. Nevertheless, the pulse duration of a tokamak plasma is generally sufficiently long for tearing instabilities to develop fully. As illustrated in Figure 1.6, a tearing mode changes the topology of the magnetic field in the vicinity of the rational magnetic flux-surface to produce a magnetic island chain with $m$ periods in the poloidal direction and $n$ periods in the toroidal direction. The chain typically has a radial width (defined as the width of the magnetic separatrix that separates reconnected from unreconnected magnetic field-lines) that is a few percent of the plasma minor radius. The presence of the island chain is significant because heat and particles are able to get from one (radial) side of the magnetic separatrix to the other by rapidly streaming along magnetic field-lines, rather than having to slowly diffuse across magnetic flux-surfaces. Consequently, the plasma pressure profile is flattened within the magnetic separatrix [19], giving rise to a degradation of the energy and particle confinement properties of the plasma [9]. Incidentally, the flattening of the electron temperature within the separatrix of a magnetic island allows the island structure to be imaged by an electron cyclotron emission diagnostic [43,47].