Representation of ideal magnetohydrodynamic modes

One of the most fundamental properties of ideal magnetohydrodynamics is the condition that plasma motion cannot change magnetic topology. The conventional representation of ideal magnetohydrodynamic modes by perturbing a toroidal equilibrium field through $δB→=∇×(ξ→×B→)$ ensures that $δB→·∇ψ=0$ at a resonance, with ψ labelling an equilibrium flux surface. Also useful for the analysis of guiding center orbits in a perturbed field is the representation $δB→=∇×αB→$⁠. These two representations are equivalent, but the vanishing of $δB→·∇ψ$ at a resonance is necessary but not sufficient for the preservation of field line topology, and a indiscriminate use of either perturbation in fact destroys the original equilibrium flux topology. It is necessary to find the perturbed field to all orders in $ξ→$ to conserve the original topology. The effect of using linearized perturbations on stability and growth rate calculations is discussed.