Nonlinear phase dynamics of ideal kink mode in the presence of shear flow

We investigate nonlinear phase dynamics of an ideal kink mode,induced by E × B flow.Here the phase is the cross phase (0c) between perturbed stream function of velocity ((φ)) and magnetic field ((ψ)),i.e.θc =θφ-θψ.A dimensionless parameter,analogous to the Richardson number,Ri =16γkink2/(ω)E2 (γkink: the normalized growth rate of the pure kink mode;(ω)E: normalized E × B shearing rate) is defined to measure the competition between phase pinning by the current density and phase detuning by the flow shear.When Ri ＞ 1,θc is locked to a fixed value,corresponding to the conventional eigenmode solution.When Ri ≤ 1,θc enters a phase slipping or oscillating state,corresponding to a nonmodal solution.The nonlinear phase dynamics method provides a more intuitive explanation of the complex dynamical behavior of the kink mode in the presence of E × B shear flow.