The Suydam Criterion

The cylindrical local stability condition against pressure-driven interchange in a screw pinch.

Statement

The Suydam criterion is a necessary condition for ideal-MHD stability of a cylindrical screw pinch against localized interchange modes near a rational surface. It requires that the stabilizing effect of magnetic shear exceed the destabilizing pressure gradient: (r/8)(q'/q)^2 B_z^2/mu0 + r dp/dr > 0, where q'/q is the shear and dp/dr the pressure gradient. Violation guarantees instability.

The Physical Balance

The first term is the energy cost of bending field lines with finite shear, which resists a localized interchange. The second term is the free energy of the pressure gradient, negative where pressure falls outward. Suydam's inequality states that the shear term must dominate; strong shear stabilizes, and steep pressure gradients destabilize, precisely where the shear is weak.

Relation to Mercier

Suydam is the straight-cylinder limit of the more general Mercier criterion for a torus. The toroidal version adds the average field-line curvature and the magnetic well through flux-surface averages, which can either help or hurt depending on geometry. Both are local, necessary but not sufficient conditions: passing them does not ensure stability against global modes, which require solving the full Newcomb or ballooning equation across the profile. The historical importance of Suydam is that it first made explicit the trade between shear and pressure gradient that governs every later interchange criterion.

Relevance

Historically important for pinch and stellarator analysis, Suydam remains a quick screen for whether a pressure profile is locally over-steep for the available shear. For the Hyperion breeder concept the toroidal Mercier criterion is the working tool, but the Suydam picture supplies the intuition that pressure gradients must be shear-limited, an input to profile design in simulation-stage modeling.