On the ideal magnetohydrodynamic stability of the sheared-flow Z pinch

Sheared-flow Z-pinch stability has previously been studied within ideal magnetohydrodynamic (MHD) primarily through growth rate calculations, which find that even trans-Alfvénic sheared flows apparently fail to suppress the kink instability. Here, trans-Alfvénic sheared flow is shown to stabilize the MHD kink but also to excite shear-driven instabilities characteristic of high-Reynolds-number supersonic flow. This distinction is evident from the dispersion relations underlying the growth rates, computed as the analytic dispersion function in the complex-frequency plane. Regularization splits this function into adiabatic and resonant parts describing how discrete modes emerge from and interact with the continuous spectrum. The Doppler-shifted flow continuum is then seen to interact with the interchange and kink instabilities in distinct ways. For interchange, the continuum overlaps the instability branch at all wavenumbers, so even sub-Alfvénic sheared flow stabilizes profiles modestly beyond the interchange threshold. The kink, by contrast, is shielded from the continuum by a frequency gap, and trans-Alfvénic flow is required to Doppler-shift the continuum into resonance with it, giving a geometric picture of the stabilization threshold. However, shear-driven instabilities arise at this same threshold, including reflection modes and an acoustic kink. It is these shear-driven modes, not the original MHD instabilities, that dominate the ideal-MHD spectrum in trans-Alfvénic conditions. The ideal analysis thus describes the stabilization mechanism while showing that the stability of the sheared-flow Z pinch ultimately rests on non-ideal physics, such as finite orbit width and dissipation.