It is well known that the Newcomb equation, the inertia free linear ideal MHD equation, plays fundamental roles in the MHD stability analysis of a tokamak. Let (r, θ, φ) be a flux coordinates system where the magnetic filed lines are straight; r is a flux label, θ the poloidal angle, and φ the toroidal angle. If the radial displacement ξ(r,θ, φ) is expressed as a sum of its poloidal Fourier harmonics (n: toroidal mode number of the perturbation under consideration) ξ(r,θ, φ) = Σ _l ξ_l (r)exp(ilθ - inφ), then the Newcomb equation reads
N ξ = - d/dr (Ldξ/dr) - d/dr(M ^tξ) + Mdξ/dr + Kξ= 0, (1)
where the vector function ξ(r) is defined by ξ(r) = (ξ_-Lf(r), ..., ξ_Lf(r))^t , and the matrices L and K are hermitian (^t: transpose of a vector or a matrix). The feature of Eq.(1) is that rational surfaces are regular singular points of the equation. A useful eigenvalue problem associated with Eq.(1) is given by Nξ = - λ Rξ, where R is a diagonal matrix whose element is R _m,m ∝ (m/q(r) - n)² , and q(r) is the safety factor.
Equations (1) and the eigenvalue problem have wide applications in the MHD stability analysis of a tokamak. The eigenvalue problem provides a useful tool of ideal MHD stability analysis, and it is still effective in the stability analysis of finite-n ballooning-peeling modes near the plasma edge. Equation (1) is applied for constructing a quadratic form expressing the change of the potential energy against external modes, the form which is essential in the stability analysis of ideal kink and resistive wall modes. Finally, the solution of Eq.(1) including the so called big solutions around rational surfaces gives the matching data in the resistive or tearing mode stability analysis.
We report the mathematical property of the Newcomb equation and the eigenvalue problem including the aspect of numerical methods. We also discuss the applications of the Newcomb equation to MHD stability issues to be resolved in future tokamaks such as ITER.