2021ApJ...922..243D

Accretion disks around black holes can become warped by Lense-Thirring precession. When the disk viscosity is sufficiently small, such that the warp propagates as a wave, then steady-state solutions to the linearized fluid equations exhibit an oscillatory radial profile of the disk tilt angle. Here we show, for the first time, that these solutions are in good agreement with three-dimensional hydrodynamical simulations, in which the viscosity is isotropic and measured to be small compared to the disk angular semi-thickness, and in the case that the disk tilt-and thus the warp amplitude-remains small. We show, using both the linearized fluid equations and hydrodynamical simulations, that the inner disk tilt can be more than several times larger than the original disk tilt, and we provide physical reasoning for this effect. We explore the transition in disk behavior as the misalignment angle is increased, finding increased dissipation associated with regions of strong warping. For large enough misalignments the disk becomes unstable to disk tearing and breaks into discrete planes. For the simulations we present here, we show that the total (physical and numerical) viscosity at the time the disk breaks is small enough that the disk tearing occurs in the wave-like regime, substantiating that disk tearing is possible in this region of parameter space. Our simulations demonstrate that high spatial resolution, and thus low numerical viscosity, is required to accurately model the warp dynamics in this regime. Finally, we discuss the observational implications of our results.