Lense-Thirring Precession of Misaligned Discs I

We study Lense-Thirring precession of inviscid and viscous misaligned $\alpha-$discs around a black hole using a gravitomagnetic term in the momentum equation. For weak misalignments, $i \lesssim 10^{\circ}$, the discs behave like rigid bodies, undergoing the full suite of classical harmonic oscillator dynamics including, weak and critically damped motion (due to viscosity), precession (due to Lense-Thirring torque) and nutation (due to apsidal precession). For strong misalignments, $i \gtrsim 30^{\circ}$, we find sufficiently thin, $h/r \lesssim 0.05$ discs break, form a gap and the inner and outer sub-discs evolve quasi independently apart from slow mass transfer. Assuming the sound speed sets the communication speed of warps in the disc, we can estimate the breaking radius by requiring that the inner sub-disc precesses like a rigid body. We explicitly show for the first time using a grid code that an Einstein potential is needed to reproduce the analytic properties of the inner disc edge and find disc breaking. At large inclination angles we find multiple disc breaks, consistent with recent GRMHD simulations of highly inclined discs. Our results suggest that the inclusion of a gravitomagnetic term and appropriate pseudo-Newtonian potential captures the important quantitative features of misaligned discs.