Extreme Scale Height Variations and Nozzle Shocks in Warped Disks

Abstract Accretion disks around both stellar-mass and supermassive black holes (BHs) are likely often warped. Whenever a disk is warped, its scale height varies with azimuth. Sufficiently strong warps cause extreme compressions of the scale height, which fluid parcels “bounce” off of twice per orbit to high latitudes. We study the dynamics of strong warps using (i) the nearly analytic “ring theory” of Fairbairn &amp; Ogilvie, which we generalize to the Kerr metric, and (ii) three-dimensional general-relativistic hydrodynamic simulations of tori (“rings”) around BHs, using the H-AMR code. We initialize a ring with a warp and study its evolution on tens of orbital periods. The simulations agree excellently with the ring theory until the warp amplitude, ψ , reaches a critical value ψ c . When ψ &gt; ψ c , the rings enter the bouncing regime. We analytically derive (and numerically validate) that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>ψ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≈</mml:mo> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">g</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> in the non-Keplerian regime, where r g = GM / c 2 is the gravitational radius, and M is the mass of the central object. Whenever the scale height bounces, the vertical velocity becomes supersonic, leading to “nozzle shocks” as gas collides at the scale height minima. Nozzle shocks damp the warp within ≈10–20 orbits, which is not captured by the ring theory. Nozzle shock dissipation leads to inflow timescales 1–2 orders of magnitude shorter than unwarped α disks, which may result in rapid variability, such as in changing-look active galactic nuclei or in the soft state of X-ray binaries. We propose that steady disks with strong warps may self-regulate to have amplitudes near ψ c .