Effects of Resolution and Local Stability on Galactic Disks: I. Multiple Spiral Mode Formation via Swing Amplification

We investigate the formation of multiple spiral modes in Milky Way–like disk–halo systems without explicitly exciting perturbations. We explore how numerical resolution, the level of local disk stability, and the presence of a live halo influence both the initial appearance and the subsequent evolution of these modes. To characterize spiral structure, we compute Fourier amplitudes for modes \(m=1\)–\(6\). In marginally unstable, lower-resolution disks (\(N_\star = 5\times10^6\), \(N_{\rm DM}=1.14\times10^7\)), faint features appear within the first \(0.5\)~Gyr due to numerical noise, in contrast to high-resolution (\(N_\star = 5\times10^7\), \(N_{\rm DM}=1.14\times10^8\)) models where perturbations emerge later. We find that the delay in the spiral and bar formation epoch due to the enhanced resolution is more significant in the more stable models. Across all sufficiently resolved, live-halo models with \(m_{\rm DM}/m_\star \le 10\), the spirals exhibit a cascading sequence in both mode number and radius: higher-\(m\) modes form and decay first, followed by the delayed emergence of lower-\(m\) modes, with an inward drift of the activity’s epicenter. This behavior reflects a combination of local swing amplification, which explains the initial growth of short-wavelength modes, and interference between coexisting coherent spiral components with different pattern speeds, which accounts for the recurrent short-timescale amplitude modulations. In contrast, models with a fixed halo potential or coarse halo resolution (\(N_{\rm DM}=1.14\times10^6\) and \(m_{\rm DM}/m_\star=100\)) show strong early spirals but lack this coherent cascading behavior, owing to excessive shot noise and insufficient halo responsiveness. The \(m=3\) mode plays a transitional role, marking the onset of angular-momentum transport in the inner disk that precedes bar formation, a process absent in fixed-potential models. Our results show that a live halo with appropriate mass resolution provides the gravitational response needed to sustain and regenerate multi-mode spiral structure, even though the total angular-momentum exchange remains small.