The maximum ratio of plasma pressure to magnetic pressure a tokamak can sustain before magnetohydrodynamic instabilities terminate the discharge.
ReviewedLast reviewed: 9 Aug 2026·Category: Glossary
Defining Beta
In magnetic confinement fusion, beta (β) is the ratio of plasma kinetic pressure to magnetic pressure:
β = 2μ0 ⟨p⟩ / B2 where ⟨p⟩ is the volume-averaged plasma pressure and B is typically the vacuum toroidal field at the magnetic axis. Beta is usually expressed as a percentage.
Because a fusion reactor's power output scales with β2B4, achieving high beta is critical for economic viability—it means extracting more fusion power from a given magnet investment.1
The Troyon Scaling
In 1984 Troyon and collaborators at EPFL showed, through systematic ideal-MHD stability calculations, that the maximum achievable toroidal beta in a tokamak follows a simple linear scaling:2
βmax (%) = βN · Ip / (a BT) where βN is the normalized beta (the "Troyon coefficient"), Ip is in MA, a in metres, and BT in tesla. The original Troyon limit gives βN ≈ 2.8 for conventional profiles; optimized profiles and wall stabilization can push βN to 3.5–5 or beyond.
The dominant instabilities enforcing the beta limit are ballooning modes (pressure-driven, localized on the outboard side) and external kink modes (global current-driven deformations). When the plasma pressure gradient exceeds the stabilizing magnetic field line bending, these modes grow on Alfvénic timescales and can cause major disruptions.3
Pushing Beyond the Limit
Several strategies extend accessible beta. A nearby conducting wall can stabilize the external kink, but resistive-wall modes then require active feedback or plasma rotation for sustained stabilization. Advanced tokamak scenarios with reversed magnetic shear and bootstrap-current-dominated profiles have achieved βN > 4 transiently in DIII-D.4
Spherical tokamaks are inherently high-beta devices: their low aspect ratio and high natural elongation yield βT values of 15–40%, far above conventional tokamak values of 3–5%. This is a central argument for the compact-reactor pathway.3
ITER's baseline scenario targets βN ≈ 1.8, well below the stability boundary, to ensure robust operation. Advanced ITER scenarios may explore βN up to 2.5–3.0.
Sources
Troyon, F. et al., 'MHD-limits to plasma confinement,' Plasma Physics and Controlled Fusion 26 (1984) 209-215