The Newcomb Equation

The marginal ideal-MHD equation governing the perturbed flux outside a resistive layer.

The Marginal Problem

The Newcomb equation is the Euler-Lagrange equation obtained by minimizing the ideal-MHD energy functional for a given helical perturbation at marginal stability (zero growth rate). In a cylindrical screw pinch it is a second-order ordinary differential equation for the radial displacement or, equivalently, the perturbed poloidal flux, with coefficients built from the equilibrium profiles and the mode numbers m and n.

Singular Surfaces

The equation has regular singular points at the rational surfaces where q = m/n. There the coefficient of the highest derivative vanishes and the ideal solution is singular. Newcomb's analysis specifies how to treat these surfaces: an ideal-stable equilibrium requires that no solution has a zero (a marginal point) between singular surfaces, a small-solution criterion that generalizes Suydam's local test to the global mode.

Connection to Tearing

The jump in the logarithmic derivative of the Newcomb solution across a rational surface defines the tearing stability index Delta'. Thus the Newcomb equation is the ideal outer-region equation whose matching to the thin resistive inner layer determines tearing-mode stability. Solving it for the equilibrium of interest is the first step in any tearing analysis, and the same solution reveals whether an ideal external kink is present when a marginal point appears in the vacuum region outside the plasma. Because the equation is only second order, it is inexpensive to solve repeatedly across a scan of equilibria.

Relevance

The Newcomb equation is the practical bridge between ideal external-kink stability and resistive tearing modes, both central to tokamak operating limits. For the Hyperion breeder concept, Newcomb solutions provide the Delta' inputs to tearing-stability assessment across candidate equilibria at the design stage; these are computations for a machine in simulation.