The Newcomb Equation
The marginal ideal-MHD equation whose solutions test stability and yield the tearing index in the outer region.
Marginal stability
Newcomb's equation is the Euler-Lagrange equation obtained by setting the ideal-MHD growth rate to zero (marginal stability) for a cylindrical or toroidal plasma. It is a second-order ordinary differential equation for the radial displacement perturbation:
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d/dr [ f(r) d(xi)/dr ] - g(r) xi = 0The coefficients f and g depend on the equilibrium profiles, the mode numbers m and n, and the safety factor q. Because it is the marginal equation, its solutions reveal whether a mode is stable or unstable by the oscillation theorem: a zero crossing of the solution between boundaries signals instability.
Singular surfaces
The coefficient f vanishes at rational surfaces where q = m/n, making those points singular. Newcomb showed how to treat these singularities and formulated the stability criterion in terms of the behavior of solutions near them. This is where the tearing index Delta' is defined, by matching solutions across the singular layer.
How it is solved numerically
- Integrate the ODE outward and inward from the boundaries to each rational surface
- Match the logarithmic derivatives to extract Delta' for tearing analysis
- Apply the oscillation (Newcomb) criterion to detect ideal instability
For toroidal geometry the single equation becomes a coupled system over poloidal harmonics, solved as a boundary-value problem.
Its role
The Newcomb equation is the backbone of ideal internal-kink and tearing-stability analysis and supplies the Delta' input to the Rutherford island equation. It is a standard part of the stability toolkit applied to equilibria such as the Hyperion breeder.