How engineers sculpt the cross-section of a tokamak plasma—elongation, triangularity, and squareness—to unlock higher pressure limits and suppress destructive instabilities.
Left to its own devices, a tokamak plasma would settle into a roughly circular cross-section. That shape is magnetically simple but thermodynamically limiting: the maximum pressure the plasma can sustain before erupting in a disruption scales directly with how much the cross-section departs from a circle. Plasma shaping is the deliberate use of poloidal-field coils to stretch, indent, and otherwise sculpt that cross-section so the plasma can carry more current, store more energy, and resist the magnetohydrodynamic (MHD) modes that would otherwise tear it apart.1
Three dimensionless numbers capture most of the geometry. Elongation (κ) measures vertical stretch: a plasma with κ = 1 is circular, while modern designs run κ ≈ 1.7–1.9. Higher elongation raises the ideal-MHD beta limit roughly linearly, but it also invites vertical displacement events (VDEs), so active feedback coils are essential.2
Triangularity (δ) describes how much the D-shaped cross-section leans inward or outward. Positive triangularity (δ > 0) pushes the tips of the D toward the high-field side, stabilising ballooning modes at the outboard edge. Negative triangularity (δ < 0) reverses that lean and has recently attracted intense interest because it can suppress edge-localised modes (ELMs) without requiring an H-mode pedestal, simplifying divertor design.3
Squareness (ζ) refines the profile between the top and outboard midplane. Although less discussed, squareness affects the volume of the plasma at a given elongation and can influence the distribution of bootstrap current.
Shaping improves performance through several coupled mechanisms. A more elongated plasma increases the effective safety factor at the edge, raising the current limit before disruptions. Favourable triangularity steepens the pressure gradient that the edge can support, boosting the pedestal height in H-mode. Together, elongation and triangularity raise the normalised beta (βN) ceiling, which directly determines how much fusion power a given magnetic field can confine.4
Modern reactor concepts—from ITER to compact spherical tokamaks—rely on aggressive shaping to meet their performance targets. The trend toward high-temperature superconductors has relaxed some coil-placement constraints, allowing designers to explore shaping regimes that were previously inaccessible.