Description
The ballooning stability threshold is known to manifest itself differently in tokamaks and stellarators. While in tokamaks, the stability threshold often sets a hard limit for plasma pressure, stellarators can operate beyond the ballooning stability threshold. The latter instead causes a soft beta limit associated with progressively worse confinement, but the device can still operate significantly above the threshold.
From a local perspective, when a tokamak crosses the ballooning stability threshold, all field lines on a flux surface become unstable simultaneously. This is due to every field line on a tokamak flux surface being a toroidally shifted copy of any other field line on the same flux surface. However, in a stellarator, different field lines are distinct, which makes them cross the stability threshold at different times. This means that, when a stellarator plasma first becomes unstable, one should expect to see marginal ballooning modes localized around individual field lines. Only a small fraction of the plasma, or even the most unstable flux surface, will then be affected by the instability. This is in stark contrast to the situation in a tokamak, where the instability covers the entire region of bad curvature.
Using Wendelstein 7-X as a representative stellarator, we solve both the local ballooning mode eigenvalue problem, scanning over different field lines, and the global linear MHD eigenvalue problem. The latter is done using the CAS3D linear ideal MHD code. The code results confirm that marginal ballooning modes are localized around individual field lines, suggesting that stellarator ballooning modes have a milder onset simply thanks to the circumstance that they initially only affect a small portion of a flux surface.
While we have found important qualitative differences between linear marginal ballooning modes in tokamaks and stellarators, further work is needed to understand the differences in their nonlinear behavior.