The Troyon Beta Limit
The empirical maximum on normalized beta set by ideal MHD kink and ballooning stability.
The scaling
Troyon and colleagues found from many stability computations that the maximum stable beta in a tokamak follows a simple scaling in terms of plasma current, minor radius, and field:
beta_max (percent) = beta_N * I_p / (a * B)with I_p in mega-amperes, a in meters, B in tesla, and the Troyon coefficient beta_N typically around 2.8 to 3.5 for conventional current profiles. Above this beta, ideal external kink and ballooning modes become unstable.
Beta_N as a figure of merit
Because the limit is nearly constant in beta_N, that normalized value is quoted as the achievement metric. Advanced scenarios with tailored current and pressure profiles, wall stabilization, or strong shaping can push beta_N well above the no-wall limit, entering the resistive-wall-mode regime that requires active control.
How the limit is computed
- Construct a family of equilibria at increasing beta with a fixed profile shape
- Test each with an ideal-MHD stability code (kink and ballooning)
- Identify the marginal beta and infer beta_N
The no-wall and ideal-wall limits bracket what is achievable passively versus with a conducting wall and feedback control.
Design implications
The Troyon limit ties achievable pressure to current and field. Spherical tokamaks like the Hyperion breeder reach high toroidal beta at modest field; the design must then keep beta_N below the stability boundary or provide active control of the resistive wall mode. Verifying the operating point against the Troyon limit is a standard step in qualifying the design.