Instability and vertical eccentricity variation in global hydrodynamic disk simulations

TL;DR

This work tackles how strong disk eccentricities induce dynamical instabilities in hydrodynamic disks by coupling global 3D nonlinear simulations with 2+1D linear theory. It confirms that parametric excitation of inertial waves saturates via vertical motions and transfers energy away from radial eccentricity, while also revealing persistent slow global modes that introduce vertical variations in eccentricity. The study shows that boundary conditions and vertical domain extent crucially shape the nonlinear evolution, including elevator flows and the prominence of global modes, suggesting that vertical gravity and stratification will be important for realistic disks. Overall, the findings demonstrate that even in purely hydrodynamic, Newtonian disks, eccentric distortions drive rich, multi-scale dynamics with potential implications for variability in X-ray binaries, Be stars, and protoplanetary systems, and motivate future vertically stratified investigations.

Abstract

Many dynamical interactions can induce eccentricities in astrophysical accretion disks. Disk eccentricities in turn seed a variety of instabilities, even in ideal hydrodynamics. We use 3D nonlinear simulations and 2+1D linear calculations to characterize local and global instabilities in strongly distorted disks. On local scales, our simulations show the growth of parametrically excited inertial waves, which drive wave turbulence. The inertial waves' growth rates and localizations agree with the predictions of local theory. On global scales, we observe the growth of a separate family of low-frequency, vertically structured modes that compare favorably with eigenmodes computed from the linear theory of an eccentric background state. These low-frequency modes interact nonlinearly with the inertial wave turbulence driven by parametric instability, and they induce variation in eccentricity profiles that are initially uniform in the vertical direction. Extrapolating from our vertically local framework, we postulate that these secondary distortions may correspond to the corrugation of an initially planar eccentric disk. Our simulations demonstrate that strong disk eccentricities drive numerous dynamical phenomena even in a purely hydrodynamic, Newtonian framework.

Figures (18)

• Figure 1: Top: unfolded plot showing the radial velocity field associated with an eccentric disk solution to a non-linear secular theory, calculated to achieve a maximum eccentricity of $0.25$ in the middle of the radial domain ($r\in[r_0,3r_0],$$c_s=0.05r_0\Omega_0$). The streamlines superimposed as black lines illustrate the disk's eccentricity (circular streamlines would be vertical on the unfolded polar plot). Bottom: corresponding eccentricity profile (blue) and maximum growth rate (in units of $\Omega_0$) for the local parametric instability (orange), both plotted as a function of semilatus rectum $\lambda$.
• Figure 2: Spacetime diagram showing radial velocity (evaluated at a fixed $\phi=0$) as a function of radius (y-axis) and time (x-axis) for a 2D simulation (A25r32D) initialized with a free eccentric mode with a maximum eccentricity of $0.25$ (and a radial domain with $r_1/r_0=3$).
• Figure 3: Left panels: radial kinetic energy (top) and maximum eccentricity values (bottom) for 2D simulations initialized with free eccentric modes (left) and driven eccentric distortions (right). Right colormaps: snapshots showing radial velocity for a simulation with driven eccentricity. The $\phi$-profile of density enforced in the ghost cells at the outer boundary is taken from a non-linear eccentric eigenmode with maximum eccentricity $\max[e]=0.25$.
• Figure 4: Spacetime diagrams analogous to those shown in \ref{['fig:A25r32D_spct']}, but showing mid-plane radial profiles for the 3D simulation A25r3D. The additional spacetime diagram showing midplane vertical velocity (bottom) illustrates the growth of small-scale inertial oscillations close to $r_0$, and their breakdown into subsonic turbulence.
• Figure 5: Time evolution of integrated quantities for our 3D simulations. Column titles indicate the source of eccentricity (free mode vs. driven distortion), while line colors indicate different eccentricity values or driving amplitudes. From top to bottom, the each row plots the volume-integrated radial kinetic energy, the vertical kinetic energy, the maximum eccentricity, and the (relative) difference in maximum eccentricity between each 3D simulation and its 2D counterpart. The $y$-axes in the final row transition from log to linear scales at $10^{-3}.$ A combination of local and global instabilities intrinsic to the disk distortions drive the growth of vertical oscillations, which in turn modify the background disk eccentricity.