Ripples spreading across the Galactic disc. Interplay of direct and indirect effects of the Sagittarius dwarf impact

Gaia data have revealed vertically asymmetric phase-space structures in the Milky Way (MW) disc, such as phase spirals, indicating vertical oscillations. These oscillations exhibit two distinct modes: the bending mode and the breathing mode, associated with one-arm and two-arm phase spirals, respectively. This study aims to explore the excitation mechanisms of the bending and breathing modes and their subsequent evolution in the MW disc, focusing on the interplay between direct perturbations from the Sagittarius dwarf galaxy and indirect contributions from tidally induced spiral arms. We perform high-resolution $N$-body simulations to model the interaction between an MW-like disc galaxy and a Sagittarius dwarf-like satellite. These simulations resolve fine phase-space structures, enabling analysis of the bending and breathing modes at both macroscopic (global bending and breathing waves) and microscopic (local phase spirals) scales. Our simulations demonstrate that the satellite's perturbation directly excites the bending mode and induces spiral arms in the galactic disc. These spiral arms excite the breathing mode, making it an indirect consequence of the satellite interaction. Initially, the bending mode dominates, but it rapidly decays due to horizontal mixing. In contrast, the breathing mode persists for a longer duration, sustained by the spiral arms, leading to a transition from a bending-dominated to a breathing-dominated state. This transition progresses faster in the inner galaxy than in the outer regions. The simulations reproduce the one-arm phase spiral observed in the solar neighbourhood and reveal two-arm phase spirals, particularly in the inner galaxy, associated with spiral arm-induced breathing modes. Our findings highlight the combined effects of direct satellite perturbations and indirect spiral arm dynamics in shaping the vertical structure of the MW disc.