An affine model of the dynamics of astrophysical discs

© 2018 The Author(s). Published by Oxford University Press on behalf of the Royal Astronomical Society. Thin astrophysical discs are very often modelled using the equations of 2D hydrodynamics. We derive an extension of this model that describes more accurately the behaviour of a thin disc in the absence of self-gravity, magnetic fields, and complex internal motions. The ideal fluid theory is derived directly from Hamilton's Principle for a 3D fluid after making a specific approximation to the deformation gradient tensor. We express the equations in Eulerian form after projection on to a reference plane. The disc is thought of as a set of fluid columns, each of which is capable of a time-dependent affine transformation, consisting of a translation together with a linear transformation in three dimensions. Therefore, in addition to the usual 2D hydrodynamics in the reference plane, the theory allows for a deformation of the mid-plane (as occurs in warped discs) and for the internal shearing motions that accompany such deformations. It also allows for the vertical expansions driven in non-circular discs by a variation of the vertical gravitational field around the horizontal streamlines, or by a divergence of the horizontal velocity. The equations of the affine model embody conservation laws for energy and potential vorticity, even for non-planar discs.We verify that they reproduce exactly the linear theories of 3D warped and eccentric discs in a secular approximation. However, the affine model does not rely on any secular or small-amplitude assumptions and should be useful in more general circumstances

Non-linear hydrodynamic instability and turbulence in eccentric astrophysical discs with vertical structure

Non-linear evolution of the parametric instability of inertial waves inherent to eccentric discs is studied by way of a new local numerical model. Mode coupling of tidal deformation with the disc eccentricity is known to produce exponentially growing eccentricities at certain mean-motion resonances. However, the details of an efficient saturation mechanism balancing this growth still are not fully understood. This paper develops a local numerical model for an eccentric quasi-axisymmetric shearing box which generalizes the often-used Cartesian shearing box model. The numerical method is an overall second-order well-balanced finite volume method which maintains the stratified and oscillatory steady-state solution by construction. This implementation is employed to study the non-linear outcome of the parametric instability in eccentric discs with vertical structure. Stratification is found to constrain the perturbation energy near the mid-plane and localize the effective region of inertial wave breaking that sources turbulence. A saturated marginally sonic turbulent state results from the non-linear breaking of inertial waves and is subsequently unstable to large-scale axisymmetric zonal flow structures. This resulting limit-cycle behaviour reduces access to the eccentric energy source and prevents substantial transport of angular momentum radially through the disc. Still, the saturation of this parametric instability of inertial waves is shown to damp eccentricity on a time-scale of a thousand orbital periods. It may thus be a promising mechanism for intermittently regaining balance with the exponential growth of eccentricity from the eccentric Lindblad resonances and may also help explain the occurrence of ’bursty’ dynamics such as the superhump phenomenon

Non-axisymmetric instabilities in discs with imposed zonal flows

We conduct a linear stability calculation of an ideal Keplerian flow on which a sinusoidal zonal flow is imposed. The analysis uses the shearing sheet model and is carried out both in isothermal and adiabatic conditions, with and without self-gravity (SG). In the non-SG regime, a structure in the potential vorticity (PV) leads to a non-axisymmetric Kelvin–Helmholtz (KH) instability; in the short-wavelength limit its growth rate agrees with the incompressible calculation by Lithwick, which only considers perturbations elongated in the streamwise direction. The instability’s strength is analysed as a function of the structure’s properties, and zonal flows are found to be stable if their wavelength is $\gtrsim$ 8 $\small{H}$, where $\small{H}$ is the disc’s scaleheight, regardless of the value of the adiabatic index $\gamma$. The non-axisymmetric KH instability can operate in Rayleigh-stable conditions, and it therefore represents the limiting factor to the structure’s properties. Introducing SG triggers a second non-axisymmetric instability, which is found to be located around a PV maximum, while the KH instability is linked to a PV minimum, as expected. In the adiabatic regime, the same gravitational instability is detected even when the structure is present only in the entropy (not in the PV) and the instability spreads to weaker SG conditions as the entropy structure’s amplitude is increased. This eventually yields a non-axisymmetric instability in the non-SG regime, albeit of weak strength, localized around an entropy maximum.Science and Technology Facilities CouncilThis is the final version of the article. It first appeared from Oxford University Press via http://dx.doi.org/10.1093/mnras/stw223

Nonlinear hydrodynamical evolution of eccentric Keplerian discs in two dimensions: Validation of secular theory

We perform global two-dimensional hydrodynamical simulations of Keplerian discs with free eccentricity over thousands of orbital periods. Our aim is to determine the validity of secular theory in describing the evolution of eccentric discs, and to explore their nonlinear evolution for moderate eccentricities. Linear secular theory is found to correctly predict the structure and precession rates of discs with small eccentricities. However, discs with larger eccentricities (and eccentricity gradients) are observed to precess faster (retrograde relative to the orbital motion), at a rate that depends on their eccentricities (and eccentricity gradients). We derive analytically a nonlinear secular theory for eccentric gas discs, which explains this result as a modification of the pressure forces whenever eccentric orbits in a disc nearly intersect. This effect could be particularly important for highly eccentric discs produced in tidal disruption events, or for narrow gaseous rings; it might also play a role in causing some of the variability in superhump binary systems. In two dimensions, the eccentricity of a moderately eccentric disc is long-lived and persists throughout the duration of our simulations. Eccentric modes are however weakly damped by their interaction with non-axisymmetric spiral density waves (driven by the Papaloizou-Pringle instability, which occurs in our idealised setup with solid walls), as well as numerical diffusion.AJB is supported by the Leverhulme Trust and Isaac Newton Trust through the award of an Early Career Fellowship. The early stages of this research were supported by STFC through grants ST/J001570/1 and ST/L000636/1.This is the author accepted manuscript. The final version is available from Oxford University Press via http://dx.doi.org/10.1093/mnras/stw58

Focusing of non-linear eccentric waves in astrophysical discs

ABSTRACT We develop a fully non-linear approximation to the short-wavelength limit of eccentric waves in astrophysical discs, based on the averaged Lagrangian method of Whitham. In this limit there is a separation of scales between the rapidly varying eccentric wave and the background disc. Despite having small eccentricities, such rapidly varying waves can be highly non-linear, potentially approaching orbital intersection, and this can result in strong pressure gradients in the disc. We derive conditions for the steepening of non-linearity and eccentricity as the waves propagate in a radially structured disc in this short-wavelength limit and show that the behaviour of the solution can be bounded by the behaviour of the WKB solution to the linearized equations.E. Lynch would like to thank the Science and Technologies Facilities Council (STFC) for funding this work through a STFC studentship

Wind-MRI interactions in local models of protoplanetary discs - I. Ohmic resistivity

A magnetic disc wind is an important mechanism that may be responsible for driving accretion and structure formation in protoplanetary discs. Recent numerical simulations have shown that these winds can take either the traditional `hourglass' symmetry about the mid-plane, or a `slanted' symmetry dominated by a mid-plane toroidal field of a single sign. The formation of this slanted symmetry state has not previously been explained. We use radially local 1D vertical shearing box simulations to assess the importance of large-scale MRI channel modes in influencing the formation and morphologies of these wind solutions. We consider only Ohmic resistivity and explore the effect of different magnetisations, with the mid-plane $\beta$ parameter ranging from $10^5$ to $10^2$. We find that our magnetic winds go through three stages of development: cyclic, transitive and steady, with the steady wind taking a slanted symmetry profile similar to those observed in local and global simulations. We show that the cycles are driven by periodic excitation of the $n=2$ or $3$ MRI channel mode coupled with advective eviction, and that the transition to the steady wind is caused by a much more slowly growing $n=1$ mode altering the wind structure. Saturation is achieved through a combination of advective damping from the strong wind, and suppression of the instability due to a strong toroidal field. A higher disc magnetisation leads to a greater tendency towards, and more rapid settling into the slanted symmetry steady wind, which may have important implications for mass and flux transport processes in protoplanetary discs

A local model of warped magnetized accretion discs

We derive expressions for the local ideal magnetohydrodynamic (MHD) equations for awarped astrophysical disc using a warped shearing box formalism. A perturbation expansion of these equations to first order in the warping amplitude leads to a linear theory for the internal local structure of magnetized warped discs in the absence of magnetorotational instability (MRI) turbulence. In the special case of an external magnetic field oriented normal to the disc surface, these equations are solved semi-analytically via a spectral method. The relatively rapid warp propagation of low-viscosity Keplerian hydrodynamic warped discs is diminished by the presence of a magnetic field. The magnetic tension adds a stiffness to the epicyclic oscillations, detuning the natural frequency from the orbital frequency and thereby removing the resonant forcing of epicyclic modes characteristic of hydrodynamic warped discs. In contrast to a single hydrodynamic resonance, we find a series of Alfvénic-epicyclic modes which may be resonantly forced by the warped geometry at critical values of the orbital shear rate q and magnetic field strength. At these critical points large internal torques are generated and anomalously rapid warp propagation occurs. As our treatment omits MRI turbulence, these results are of greatest applicability to strongly magnetized discs.This work was funded by the Science and Technology Facilities Council (STFC) through a studentship to J. Paris and through grant ST/L000636/1

Local numerical simulations of warped discs

We study the hydrodynamical stability of the laminar flows associated with warped astrophysical discs using numerical simulations of warped shearing boxes. We recover linear growth rates reported previously due to a parametric resonance of inertial waves, and show that the nonlinear saturated state can significantly reduce the laminar flows, meaning that the warp would evolve on much longer time scales than would be concluded from the internal torques due to these laminar flows. Towards larger warp amplitudes, we find first of all a reversal of angular momentum flux, indicating that the mass distribution would evolve in an anti-diffusive manner, and second that the linear growth rates disappear, possibly because of the very strong shear in the laminar flows in this regime. For discs with small enough viscosity, a nonlinear state can still be found when linear growth rates are absent by introducing a large enough perturbation, either by starting from a nonlinear state obtained at smaller warp amplitude, or by starting from a state with no laminar flows

Growth of eccentric modes in disc-planet interactions

We formulate a set of linear equations that describe the behaviour of small eccentricities in a protoplanetary system consisting of a gaseous disc and a planet. Eccentricity propagates through the disc by means of pressure and self-gravity, and is exchanged with the planet via secular interactions. Excitation and damping of eccentricity can occur through Lindblad and corotation resonances, as well as viscosity. We compute normal modes of the coupled disc–planet system in the case of short-period giant planets orbiting inside an inner cavity, possibly carved by the stellar magnetosphere. Three-dimensional effects allow for a mode to be trapped in the inner parts of the disc. This mode can easily grow within the disc’s lifetime. An eccentric mode dominated by the planet can also grow, although less rapidly. We compute the structure and growth rates of these modes and their dependence on the assumed properties of the disc.We acknowledge support from STFC through grant ST/L000636/1.This is the author accepted manuscript. The final version is available from Oxford University Press via http://dx.doi.org/10.1093/mnras/stw52

Dynamical structure of highly eccentric discs with applications to tidal disruption events

ABSTRACT Whether tidal disruption events circularize or accrete directly as highly eccentric discs is the subject of current research and appears to depend sensitively on the disc thermodynamics. One aspect of this problem that has not received much attention is that a highly eccentric disc must have a strong, non-hydrostatic variation of the disc scale height around each orbit. As a complement to numerical simulations carried out by other groups, we investigate the dynamical structure of TDE discs using the non-linear theory of eccentric accretion discs. In particular, we study the variation of physical quantities around each elliptical orbit, taking into account the dynamical vertical structure, as well as viscous dissipation and radiative cooling. The solutions include a structure similar to the nozzle-like structure seen in simulations. We find evidence for the existence of the thermal instability in highly eccentric discs dominated by radiation pressure. For thermally stable solutions many of our models indicate a failure of the α-prescription for turbulent stresses. We discuss the consequences of our results for the structure of eccentric TDE discs.</jats:p