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 (θ_c) between perturbed stream function of velocity (ф˜) and magnetic field (φ˜ ), i.e. θ_c = θ_ф − θ_ψ. A dimensionless parameter, analogous to the Richardson number, R_i = 16γ_kink²/ω_E² (γ_kink: the normalized growth rate of the pure kink mode; ωˆ_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 R_i> 1, θ_c is locked to a fixed value, corresponding to the conventional eigenmode solution. When R_i ≤1, θ_c 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 ynamical behavior of the kink mode in the presence of E × B shear flow.