Radial modes of pressure bumps and dips in astrophysical discs

TL;DR

This work develops a topological, microlocal framework to study global oscillations in unstratified astrophysical discs, revealing that pressure extrema act as waveguides for robust topological modes. By deriving a local dispersion relation with a gradient-induced frequency $S$ and computing Chern numbers, the authors show that spectral flow enforces modes that transit between inertial and acoustic bands, with distinctive behavior at pressure maxima and minima. In slender-torus and gap-containing disc models, they obtain analytic solutions where a fundamental mode propagates for all frequencies near a pressure maximum, while a minimum-associated mode travels at fixed $\omega=\kappa$ with arbitrary vertical phase velocity. These results suggest new avenues for discoseismology and offer predictions for how radial structure shapes large-scale disc oscillations detectable via line-kinematic measurements.

Abstract

This study investigates the signatures of pressure extrema on global oscillations in discs. To this end, we use the framework of wave topology to establish a generalised local dispersion relation that includes pressure gradients. We highlight the influence of a previously unrecognized epicyclic-acoustic frequency and derive an analytical criterion for the existence of a branch of modes transiting between the inertial and the pressure bands. We find that pressure extrema consist of wave guides in which such topological modes propagate. The fundamental mode trapped at a pressure bump can propagate at all frequencies, allowing it to resonate with any temporal forcing, while the mode associated with a pressure gap propagates at a fixed frequency, propagates with arbitrary vertical phase velocity. These specific features make them attractive candidates for future discoseismology.

Figures (10)

• Figure 1: The Berry curvature $\boldsymbol{F}$ of the acoustic wave is singular at $(c_\mathrm{s}k_r,S,c_\mathrm{s}k_z)=(0,0,\pm\kappa)$. This obstruction is a topological constraint, characterized by the two charges $\mathcal{C}=\pm 1$. Length and brightness of the arrows indicate the norm of $\boldsymbol{F}$.
• Figure 2: Global modes of a monotonic disc. The one topological branch transits from inertial modes (blue) at low $k_z$ to acoustic modes (yellow) at high $k_z$. It matches the dispersion relation $\omega=c_\mathrm{s}(r_0)k_z$ (grey thin line).
• Figure 3: Dispersion relations of the global modes in a slender torus.
• Figure 4: Dispersion relations of the global modes in a pressure minimum.
• Figure 5: Same as Fig. \ref{['fig:monotonic']}, for a disc with a gap and zoomed on lower frequencies. In a disc where a gap of density is present, one local minimum and one local maximum of density are found. Associated to these extrema, two topological modes are expected and found in the spectrum, one inertial and one acoustic respectively. Grey lines: $\omega = \kappa(r=r_\mathrm{min})$ and $\omega = c_\mathrm{s}(r=r_\mathrm{max})k_z$.