Tokamak Safety Factor Calculator

The safety factor q is a dimensionless number describing the helical pitch of the magnetic field lines in a tokamak: it equals the number of times a field line travels the long way around the torus (toroidally) for each single trip the short way (poloidally). Low q means tightly wound, highly twisted field lines; high q means gently twisted, nearly straight field lines.

In the simple cylindrical (large aspect ratio) approximation, q(r) = rB_t/(RB_p), where r is the minor-radius position, R is the major radius, B_t is the toroidal field, and B_p is the poloidal field (produced mainly by the plasma current) at that position.

Because its value determines how magnetohydrodynamically stable the plasma is against resonant instabilities. Low-order rational values of q, most importantly q=1 and q=2, correspond to magnetic surfaces where field lines close on themselves after only a few turns, which resonantly drives the growth of magnetic islands and can trigger disruptive instabilities.

A magnetic surface with q=1 inside the plasma core is associated with the internal kink mode, which drives the well-known 'sawtooth' oscillation: the core temperature and density periodically crash and rebuild as the q=1 surface reconnects, a routinely observed and generally tolerable instability in most tokamaks.

A q=2 rational surface can drive the m=2, n=1 tearing mode, which forms magnetic islands that degrade confinement and, if they grow large enough or lock to the vacuum vessel wall, can trigger a full plasma disruption, an abrupt, potentially damaging loss of the entire plasma current.

Most tokamak operating scenarios target an edge safety factor q_a in the range of roughly 3 to 4, comfortably above the dangerous q=2 boundary, to provide adequate stability margin while still allowing enough plasma current for good confinement.

For fixed field ratio B_t/B_p, the safety factor scales with r/R, the inverse of the local aspect ratio. Since B_p itself is generated by the plasma current, which is also tied to the machine's overall aspect ratio and size, q is ultimately set by a combination of the toroidal field strength, plasma current, and geometry.

No. This calculator uses the classic cylindrical (large aspect ratio, circular cross-section) approximation, the standard textbook introduction to the concept. Real tokamak q-profiles depend on the detailed current density distribution and plasma shape (elongation, triangularity) and are computed numerically with equilibrium codes such as those solving the Grad-Shafranov equation.

Yes, q(r) generally increases from the core to the edge in a typical tokamak, since the poloidal field from the enclosed plasma current builds up differently than the roughly uniform toroidal field. The full q(r) profile, not just a single value, is what determines the complete MHD stability picture.

Disruptions triggered by low-q instabilities can dump the plasma's full thermal and magnetic energy onto the vessel walls in milliseconds, a serious engineering concern for large reactors. Maintaining an adequate safety factor margin throughout the discharge is therefore a core requirement of tokamak operational scenarios, not just an academic detail.