Gordon I. Ogilvie

We review some of the interesting consequences that tilts, warps, and eccentricities can introduce into the dynamics, thermodynamics, and observational appearance of accreting systems, with an emphasis on disks around black holes and compact stars. We begin with a review of the two types of precession that are associated with eccentric and tilted orbits in general relativity and Newtonian gravity. We then discuss the types of accretion systems that may manifest tilted or eccentric disks. In separate sections we discuss first tilted and then eccentric disks, each section covering relevant and interesting observational, theoretical, and numerical results. Next, we explore potential connections between the phenomenology of quasi-periodic oscillations and either tilted or eccentric disks. Finally, we present some concluding thoughts and discuss future directions this research might take.

We derive a simple, accurate, non-linear, global equation governing spiral density waves in thin, non-self-gravitating, inviscid accretion discs. These discs may have any slowly varying surface density or temperature profile. For specific ‘self-similar’ disc profiles, solutions to our equation match (novel) smooth non-linear exact spiral solutions derived via a separate method, which highlight that non-linear spiral waves need not shock. Indeed, at low amplitudes, we find that dispersion can overcome wave steepening, and may prevent the inner spiral wakes excited by low-mass planets (below roughly 1 per cent of a thermal mass) embedded in protoplanetary discs from shocking. At high amplitudes, we find a simple universal description of non-linear spiral waves with shocks, as well as caps on the possible amplitude and wave action flux of non-linear spirals both with and without shocks, depending on how many arms they have. We further find that highly non-linear spirals are far more loosely wound than their linear counterparts. These developments shed light on why two-armed spirals are prevalent across a range of astrophysical systems which do not necessarily possess an intrinsic twofold symmetry, and why they appear surprisingly loosely wound in observations. These results are supported by very high-resolution numerical simulations.

Context . The vertical shear instability (VSI) is a promising mechanism for generating turbulence and transporting angular momentum in magnetically decoupled regions of protoplanetary discs. While most recent work has focused on adding more complex physics, the saturation properties of the instability in radially extended discs, and its convergence as a function of resolution, are still largely unknown. Aims . We address the question of VSI saturation and associated turbulence using radially extended, fully 3D global disc models with very high resolution in the locally isothermal approximation, to capture both the largest VSI scales and the small-scale turbulent cascade. Methods . We used the GPU-accelerated code Idefix to achieve resolutions of up to 200 points per scale height in the three spatial directions. We chose numerical techniques that minimise numerical diffusion as much as possible: third-order reconstruction schemes, orbital advection, and a third-order time integrator. We modelled the VSI in disc domains extending up to R _out / R _in = 7 in the highest resolution case and R _out / R _in = 25 in our intermediate model (100 points per scale height), with a full 2 π azimuthal extent and a disc aspect ratio H / R = 0.1. Results . We demonstrate that large-scale transport properties converge with 100 points per scale height, leading to a Shakura-Sunyaev α=1.3 × 10 ⁻³ in the bulk of the computational domain. Inner boundary condition artefacts propagate deep inside the computational domain (typically over Δ R / R ∼ 2−3), leading to a reduced α in these regions. The large-scale corrugation wave zones identified in 2D models persist in 3D, albeit with less coherence. Our models show no signs of long-lived zonal flows, pressure bumps, or vortices, in contrast to lower-resolution simulations. Finally, we show that the turbulent cascade resulting from VSI saturation can be interpreted within the framework of critically balanced rotating turbulence. We propose that the small non-axisymmetric scales could be modelled with an effective anisotropic viscosity in 2D simulations, significantly reducing the computational cost of these models while still capturing the important physics. Conclusions . The VSI leads to vigorous turbulence in protoplanetary discs, associated with outward angular momentum transport, but without any significant long-lived features that could enhance planet formation. The innermost regions of VSI simulations are consistently polluted by boundary-condition artefacts influencing the first VSI wave train. Radially extended domains should therefore be used in a more systematic manner, and more realistic inner boundaries should be explored to mimic the radial structure of real protoplanetary discs.

The classical Laplace surface defines the location of circular particle orbits that do not undergo nodal precession around a planet with some obliquity. Close to the planet the surface coincides with the equator of the planet, while far from the planet it coincides with the orbital plane of the planet. We determine the shape of the Laplace surface of a circumplanetary disc that results from accretion of circumstellar gas and experiences the effects of gas pressure, self-gravity, and viscosity, as well as the gravitational effects due to the planetary spin and the star. We apply the linear theory of warped discs in the wave-like regime for a small-obliquity planet such as Jupiter. As a result of dissipation, a disc that begins slightly away from its Laplace surface will evolve to it. Because of pressure effects in typically warm circumplanetary discs, the disc is highly flattened compared with the classical Laplace surface, meaning that it is much less warped but still significantly tilted. For the case of Jupiter, the disc does not align anywhere with the equator of the planet. For such discs, the effects of self-gravity and viscosity on warping are typically small. The disc tilt is intermediate between the planet’s equatorial and orbital planes. Circumplanetary discs that are much cooler than expected can undergo warping and alignment with the planet’s equator at small radii. The results have implications for the orbital evolution of satellites in the Solar system that are observed to be somewhat aligned with their classical Laplace surface.

The classical Laplace surface defines the location of circular particle orbits that do not undergo nodal precession around a planet with some obliquity. Close to the planet the surface coincides with the equator of the planet, while far from the planet it coincides with the orbital plane of the planet. We determine the shape of the Laplace surface of a circumplanetary disc that results from accretion of circumstellar gas and experiences the effects of gas pressure, self-gravity, and viscosity, as well as the gravitational effects due to the planetary spin and the star. We apply the linear theory of warped discs in the wavelike regime for a small-obliquity planet such as Jupiter. As a result of dissipation, a disc that begins slightly away from its Laplace surface will evolve to it. Because of pressure effects in typically warm circumplanetary discs, the disc is highly flattened compared with the classical Laplace surface, meaning that it is much less warped but still significantly tilted. For the case of Jupiter, the disc does not align anywhere with the equator of the planet. For such discs, the effects of self-gravity and viscosity on warping are typically small. The disc tilt is intermediate between the planet's equatorial and orbital planes. Circumplanetary discs that are much cooler than expected can undergo warping and alignment with the planet's equator at small radii. The results have implications for the orbital evolution of satellites in the solar system that are observed to be somewhat aligned with their classical Laplace surface.

We develop several aspects of the theory of gaseous astrophysical discs in which the gravity of the disc makes a significant contribution to its structure and dynamics. We show how the internal gravitational potential can be expanded in powers of the aspect ratio of the disc (or of a structure within it) and separated into near and far contributions. We analyse the hydrostatic vertical structure of a wide family of disc models, both analytically and numerically, and show that the near contribution to the internal gravitational potential energy can be written in an almost universal form in terms of the surface density and scaleheight. We thereby develop an affine model of the dynamics of (generally non-hydrostatic) self-gravitating discs in which this contribution to the energy acts as a gravitational pressure in the plane of the disc. This combines with and significantly reinforces the gas pressure, allowing us to define an enhanced effective sound speed and Toomre stability parameter Q for self-gravitating discs. We confirm that this theory fairly accurately reproduces the onset of axisymmetric gravitational instability in discs with resolved vertical structure. Among other things, this analysis shows that the critical wavelength is on the order of twenty times the scaleheight, helping to justify the validity of the affine model. The weakly nonlinear theory also typically exhibits subcritical behaviour, with equilibrium solutions of finite amplitude being found in the linearly stable regime Q > 1 for adiabatic exponents less than 1.50.