Nonlinear tearing mode interactions and mode locking in reversed-field pinches

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Abstract
The nonlinear interaction of a set of tearing instabilities and plasma flow is studied in a cylindrical plasma. An analytic theory of mode locking is developed, which includes the effects of the localized electromagnetic torques, plasma inertia, and cross‐field viscosity. The calculation is specialized for the case of mode locking on the Madison Symmetric Torus (MST) reversed‐field pinch [R. N. Dexter et al., Fusion Technol. 19, 131 (1991)]. In MST plasmas, a set of m=1 tearing instabilities become phase locked and form a toroidally localized, rotating magnetic disturbance. An evolution equation for the phase velocity of this magnetic disturbance is derived that accounts for two types of electromagnetic torques. The external torques describe the interaction of the tearing modes with static magnetic perturbations located outside the plasma region. The interior torques describe the nonlinear interaction of three tearing modes that satisfy a wave number resonance condition. For conditions typical of MST, the internal torques dominate the external torques, which suggest the nonlinear interaction of tearing instabilities play a prominent role in the momentum degradation and mode locking.