Resistive instabilities in magnetically confined fusion plasmas with velocity shear and rotation
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Using a resistive generalization of the Frieman-Rotenberg formalism for Lagrangian magnetohydrodynamic stability with equilibrium velocity, the leading-order effects of velocity shear and rotation on linear tearing layer stability are studied for tokamak equilibria. The separation-of-time-scales formalism needed for a proper formulation of ideal and resistive stability calculations is presented. Using this formalism, a dispersion relation is first obtained for marginal ideal modes in plane-symmetric equilibria. It is demonstrated how resistive modes arise as a natural continuation of marginal ideal modes. The dispersion relation for resistive modes in slab geometry is derived and used to demonstrate the resistive stability boundary. The widely misrepresented constant-Ψ limit is explained in detail, and used to obtain a dispersion rela- tion for tearing modes. Nyquist techniques are used to compare the Glasser effect in slab and cylindrical models. The resistive layer equations are also obtained in cylindrical geometry, allowing direct verification of the limited validity of gravity-curvature equivalence heuristic for resistive modes. Numerical complications that arise from velocity shear are discussed. Layer equations are also derived in the constant-Ψ limit. The constant-Ψ dispersion relation is obtained for cylindrical equilibria, and used to study the leading-order effects of rotation and velocity shear on the critical value of ∆′ required for tearing instability. It is found that rotation and velocity shear can couple with the parallel current and the current gradient in the layer to reduce ∆′[subscript crit]. If parallel currents are sufficiently weak to compete with second-order effects, velocity shear can be stabilizing, while rotation is found to have a destabilizing effect. Second-order coupling of velocity shear and rotation can have either sign, and thus can affect stability in either direction.