Linear Bending Wave Propagation in Laminar and Turbulent Disks

Bending waves are perhaps the most fundamental and analytically tractable phenomena in warped disk dynamics. In this work, we conduct 3D grid-based, numerical experiments of bending waves in laminar, viscous hydrodynamic and turbulent, weakly magnetized disks, capturing their behavior in unprecedented detail. We clearly elucidate the theory from first principles, wherein the general Fourier-Hermite formalism can be simplified to a reduced framework which extends previous results toward locally isothermal disks. We obtain remarkable agreement with our laminar simulations wherein the tilt evolution is well described by the reduced theory, while higher-order vertical modes should be retained for capturing the detailed disk twisting and internal velocity profiles. We then relax this laminar assumption and instead launch bending waves atop a magnetorotationally turbulent disk. Although the turbulence can be quantified with an effective alpha parameter, the bending waves behave distinctly from a classical viscous evolution and are readily disrupted when the turbulent velocity is comparable to the induced warping flows. This may have implications for the inclination damping rates induced by planet-disk interactions, the capture rate of black holes in active galactic nucleus disks or the warped shapes assumed by disks in misaligned systems.