Tearing Modes in Toroidal Plasmas

Figure 14.1 shows contours of the poloidal magnetic flux for a typical plasma discharge in the KSTAR tokamak [42]. It can be seen that there are a number of differences from the idealized tokamak pictured in Figure 1.5. The first main difference is that the flux-surfaces in Figure 14.1 do not have circular cross-sections. In particular, the flux-surfaces are highly vertically elongated. It turns out that this feature allows the toroidal plasma current driven in a tokamak discharge to be increased without violating the Kruskal-Shafranov criterion. (See Section 1.10.) Hence, all modern tokamaks have strongly-shaped, vertically-elongated cross-sections. The second main difference is that the edge of the plasma is defined by a last closed magnetic flux-surface that features a magnetic X-point (i.e., a hyperbolic null in the poloidal magnetic field). Plasma that crosses the last closed flux-surface is rapidly conducted along magnetic field-lines, in a thin scrape-off layer, to divertor plates located below the plasma. The purpose of this feature is to mitigate the interaction of the plasma with the plasma-facing components, and, thereby, help limit the flux of impurities into the plasma [44]. Hence, all modern tokamaks have magnetic X-points. Given that the poloidal magnetic field-strength at the X-point is zero, the safety-factor value on the last closed flux-surface is infinite. [See Equation (1.76).] Experimentally, it is found that the safety-factor value on the magnetic flux-surface that encloses 95% of the poloidal magnetic flux enclosed by the last closed flux-surface, which is known as $q_{95}$, plays an analogous role to the edge safety-factor value, $q_a$, in a tokamak without a magnetic X-point [64]. (See Section 14.4.) Thus, the rule of thumb for safe operation, $q_a\gtrsim 3$, is replaced by $q_{95}\gtrsim 3$. (See Section 1.9.)

In the type of highly-shaped plasma equilibrium pictured in Figure 14.1, tearing modes with the same toroidal mode number, but different poloidal mode numbers, are coupled together [13,23]. Somewhat confusingly, this effect is known as toroidal mode coupling. Toroidal mode coupling allows magnetic island chains resonant on different rational surfaces within the plasma to interact. The interaction is expected to be mutually destabilizing [23]. However, sheared plasma rotation acts to prevent such interaction. (See Chapter 14.)