On the ideal stability of the sheared-flow Z pinch
Sheared-flow Z-pinch stability has been studied within ideal MHD primarily through growth rate calculations, which find that even trans-Alfv\'{e}nic sheared flows apparently fail to suppress the kink instability. Trans-Alfv\'{e}nic sheared flow does stabilize the MHD kink but also excites shear-driven instabilities characteristic of high-Reynolds-number supersonic flow. This distinction is evident from the dispersion relations underlying the growth rates, computed here 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 interacts with the interchange and kink instabilities in distinct ways. For interchange, the continuum overlaps the instability branch at all wavenumbers, so even sub-Alfv\'{e}nic sheared flow stabilizes profiles modestly beyond the interchange threshold. The kink, by contrast, is shielded from the continuum by a frequency gap, and trans-Alfv\'{e}nic flow is required to Doppler-shift the continuum into resonance with it, giving a geometric picture of the stabilization threshold. But 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-Alfv\'{e}nic 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, including finite orbit width and dissipation.