The Mercier Criterion

The local stability condition against pressure-driven interchange modes on a single flux surface.

What It Tests

The Mercier criterion is a necessary condition for ideal MHD stability against localized interchange (flute-like) modes. Derived by minimizing the energy functional for displacements localized near a rational surface, it balances the destabilizing pressure gradient against the stabilizing effects of magnetic shear and the average magnetic well or hill of the flux surface. Violating Mercier guarantees instability; satisfying it does not by itself guarantee stability against non-local modes.

The Ingredients

The criterion combines three quantities on a flux surface: the magnetic shear s, the pressure gradient dp/dr, and the field-line curvature averaged over the surface. Strong shear is stabilizing, a favorable average curvature (a magnetic well, where the field weakens toward the plasma) is stabilizing, and the pressure gradient is destabilizing where curvature is unfavorable. The combination must satisfy an inequality often written as D_I < 1/4.

Relation to Suydam

The Mercier criterion is the toroidal generalization of the Suydam criterion for a cylindrical screw pinch. Both express the same physics: sufficient magnetic shear must overcome the pressure-gradient drive of interchange modes. In a torus the average curvature and the toroidal geometry enter through flux-surface averages that the cylindrical version omits.

Relevance

Mercier stability is checked at every flux surface as a baseline in equilibrium and stability workflows, flagging surfaces where the pressure gradient is locally too steep for the available shear and well. For the Hyperion breeder concept, Mercier evaluation across the profile is a routine design-stage screen preceding fuller ballooning and kink analysis; the device is a simulation study.