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In the astrophysics community it is common practice to model collisionless dust, entrained in a gas flow, as a pressureless fluid. However a pressureless fluid is fundamentally different from a collisionless fluid - the latter of which generically possess a non-zero anisotropic pressure or stress tensor. In this paper we derive a fluid model for collisionless dust, entrained in a turbulent gas, starting from the equations describing the motion of individual dust grains. We adopt a covariant formulation of our model to allow for the geometry and coordinate systems prevalent in astrophysics, and provide a closure valid for the accretion disc context. We show that the continuum mechanics properties of a dust fluid corresponds to a higher-dimensional anisotropic Maxwell fluid, after the extra dimensions are averaged out, with a dynamically important rheological stress tensor. This higher-dimensional treatment has the advantage of keeping the dust velocity and velocity of the fluid seen, and their respective moments, on the same footing. This results in a simplification of the constitutive relation describing the evolution of the dust Rheological stress., Comment: 51 pages, 7 figures, accepted for publication in JfM

We implement a local model for a spherical collapsing/expanding gas cloud into the Athena++ magnetohydrodynamic code. This local model consists of a Cartesian periodic box with time-dependent geometry. We present a series of benchmark test problems, including non-linear solutions and linear perturbations of the local model, confirming the code's desired performance. During a spherical collapse, a horizontal shear flow is amplified, corresponding to angular momentum conservation of zonal flows in the global problem; wave speed and amplitude of sound waves increase in the local frame, due to the reduction in the characteristic length scale of the box, which can lead to an anisotropic effective sound speed in the local box. Our code conserves both mass and momentum to machine precision. This numerical implementation of the local model has potential applications to the study of local physics and hydrodynamic instabilities during protostellar collapse, providing a powerful framework for better understanding the earliest stages of star and planet formation., Comment: 14 pages, 9 figures, plus appendix (3 pages, 1 figure). Accepted for publication in Astronomy & Astrophysics

There is a common assumption in the particulate disc community that the pressure in particulate discs is essentially zero and that the disc streamlines follow Keplerian orbits, in the absence of self-gravity or external perturbations. It is also often assumed that the fluid description of particulate discs is not valid in the presence of crossing orbits (e.g. from nonzero free eccentricities). These stem from the misconception that fluid pressure arises due to the (typically rare) collisions between particles and that the velocity of particles in fluids are single-valued in space. In reality, pressure is a statistical property of the particle distribution function which arises precisely because there is a distribution of velocities at a given position. In this letter we demonstrate, with simple examples, that pressure in particulate discs is non-zero and is related to the inclination and free eccentricity distributions of the constituent particles in the discs. This means many common models of debris discs implicitly assume a nonzero, and potentially quite significant, dust pressure. We shall also demonstrate that the bulk motion of the dust is not the same as the particle motion and that the presence of pressure gradients can lead to strong departures from Keplerian motion., Comment: 5 pages, 3 figures, accepted for publication in MNRAS letters

In this paper we develop a framework for studying unstratified, magnetised eccentric discs and compute uniformly precessing eccentric modes in a cylindrical annulus which provide convenient initial conditions for numerical simulations. The presence of a magnetic field in an eccentric disc can be described by an effective gas with a modified equation of state. At magnetic field strengths relevant to the magneto-rotational instability the magnetic field has negligible influence on the evolution of the eccentric disc, however the eccentric disc can significantly enhance the magnetic field strength over that in the a circular disc. We verify the suitability of these eccentric disc solutions by carrying out 2D simulations in RAMSES. Our simulated modes (in 2D) follow a similar evolution to the purely hydrodynamical modes, matching theoretical expectations, provided they are adequately resolved. Such solutions will provide equilibrium states for studies of the eccentric magneto-rotational instability and magnetised parametric instability in unstratified discs and are useful for exploring the response of disc turbulence on top of a fluid flow varying on the orbital timescale., Comment: 16 pages, 12 figures, accepted for publication in MNRAS

Spherical flows are a classic problem in astrophysics which are typically studied from a global perspective. However, much like with accretion discs, there are likely many instabilities and small scale phenomena which would be easier to study from a local perspective. For this purpose, we develop a local model for a spherically contracting/expanding gas cloud, in the spirit of the shearing box, $\beta$-plane and expanding box models which have had extensive use in studies of accretion discs, planets and stellar winds respectively. The local model consists of a, spatially homogeneous, periodic box with a time varying aspect ratio, along with a scale factor (analogous to that in FRW/Newtonian cosmology) relating the box coordinates to the physical coordinates of the global problem. We derive a number of symmetries and conservation laws exhibited by the local model. Some of these reflect symmetries of the periodic box, modified by the time dependant geometry, while others are local analogues for symmetries of the global problem. The energy, density and vorticity in the box also generically increase(/decrease) as a consequence of the collapse(/expansion). We derive a number of nonlinear solutions, including a local analogue of uniform density zonal flows, which grow as a consequence of angular momentum conservation. Our model is closely related to the accelerated expanding box model of Tenerani \& Velli and is an extension of the isotropic model considered by Robertson \& Goldreich., Comment: 20 pages, 6 figures, accepted for publication in MNRAS

We present Paper II of the Eccentric Debris Disc Morphologies series to explore the effects that significant free and forced eccentricities have on high-resolution millimetre-wavelength observations of debris discs, motivated by recent ALMA images of HD53143's disc. In this work, we explore the effects of free eccentricity, and by varying disc fractional widths and observational resolutions, show for a range of narrow eccentric discs, orbital overlaps result in dust emission distributions that have either one or two radial peaks at apocentre and/or pericentre. The narrowest discs contain two radial peaks, whereas the broadest discs contain just one radial peak. For fixed eccentricities, as fractional disc widths are increased, we show that these peaks merge first at apocentre (producing apocentre glow), and then at pericentre (producing pericentre glow). Our work thus demonstrates that apocentre/pericentre glows in models with constant free and forced eccentricities can be both width and resolution dependent at millimetre wavelengths, challenging the classical assertion that apocentre/pericentre glows are purely wavelength dependent. We discuss future high-resolution observations that can distinguish between competing interpretations of underlying debris disc eccentricity distributions., Comment: 7 pages, 5 figures. Accepted for publication in MNRAS Letters

In this paper I develop a nonlinear theory of tightly-wound (highly twisted) eccentric waves in astrophysical discs, based on the averaged Lagrangian method of Whitham. Viscous dissipation is included in the theory by use of a pseudo-Lagrangian. This work is an extension of the theory developed by Lee \& Goodman to 3D discs, with the addition of viscosity. I confirm that linear tightly-wound eccentric waves are overstable and are excited by the presence of a shear viscosity and show this persists for weakly nonlinear waves. I find the waves are damped by shear viscosity when the wave become sufficiently nonlinear, a result previously found in particulate discs. Additionally I compare the results of this model to recent simulations of eccentric waves propagating in the inner regions of black hole discs and show that an ingoing eccentric wave can be strongly damped near the marginally stable orbit, resulting in a nearly circular disc with a strong azimuthal variation in the disc density., Comment: 21 pages, 7 figures, Accepted 2021 November 19. Received 2021 November 19; in original form 2021 October 20

The location of surface brightness maxima (e.g. apocentre and pericentre glow) in eccentric debris discs are often used to infer the underlying orbits of the dust and planetesimals that comprise the disc. However, there is a misconception that eccentric discs have higher surface densities at apocentre and thus necessarily exhibit apocentre glow at long wavelengths. This arises from the expectation that the slower velocities at apocentre lead to a "pile up'" of dust, which fails to account for the greater area over which dust is spread at apocentre. Instead we show with theory and by modelling three different regimes that the morphology and surface brightness distributions of face-on debris discs are strongly dependent on their eccentricity profile (i.e. whether this is constant, rising or falling with distance). We demonstrate that at shorter wavelengths the classical pericentre glow effect remains true, whereas at longer wavelengths discs can either demonstrate apocentre glow or pericentre glow. We additionally show that at long wavelengths the same disc morphology can produce either apocentre glow or pericentre glow depending on the observational resolution. Finally, we show that the classical approach of interpreting eccentric debris discs using line densities is only valid under an extremely limited set of circumstances, which are unlikely to be met as debris disc observations become increasingly better resolved., Comment: 14 pages, 11 figures, Accepted 2021 December 1. Received 2021 September 9; in original form 2021 July 26 - MNRAS

Whether tidal disruption events (TDEs) circularise or accrete directly as a highly eccentric disc is the subject of current research and appears to depend sensitively on the disc thermodynamics. In a previous paper we applied the theory of eccentric discs to TDE discs using an $\alpha-$prescription for the disc stress, which leads to solutions that exhibit extreme, potentially unphysical, behaviour. In this paper we further explore the dynamical vertical structure of highly eccentric discs using alternative stress models that are better motivated by the behaviour of magnetic fields in eccentric discs. We find that the presence of a coherent magnetic field has a stabilising effect on the dynamics and can significantly alter the behaviour of highly eccentric radiation dominated discs. We conclude that magnetic fields are important for the evolution of TDE discs., Comment: 20 pages, 20 figure, accepted in MNRAS

Whether tidal disruption events circularise 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 nonlinear 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 $\alpha-$prescription for turbulent stresses. We discuss the consequences of our results for the structure of eccentric TDE discs., Comment: 19 pages, 20 figures, accepted in MNRAS

We develop a fully nonlinear approximation to the short-wavelength limit of eccentric waves in astrophysical discs, based on the averaged Lagrangian method of Whitham (1965). 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 nonlinear, potentially approaching orbital intersection, and this can result in strong pressure gradients in the disc. We derive conditions for the steepening of nonlinearity 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 linearised equations., Comment: 15 pages, 8 figures, accepted manuscript to appear in MNRAS

We show that the ideal hydrodynamics of an eccentric astrophysical disc can be derived from a variational principle. The nonlinear secular theory describes the slow evolution of a continuous set of nested elliptical orbits as a result of the pressure in a thin disc. In the artificial but widely considered case of a 2D disc, the hydrodynamic Hamiltonian is just the orbit-averaged internal energy of the disc, which can be determined from its eccentricity distribution using the geometry of the elliptical orbits. In the realistic case of a 3D disc, the Hamiltonian needs to be modified to take into account the dynamical vertical structure of the disc. The simplest solutions of the theory are uniformly precessing nonlinear eccentric modes, which make the energy stationary subject to the angular momentum being fixed. We present numerical examples of nonlinear eccentric modes up to their limiting amplitudes. Although it lacks dissipation, which is important in many astrophysical contexts, this formalism allows a simpler theoretical approach to the nonlinear dynamics of eccentric discs than that derived from stress integrals, and also connects better with established methods of celestial mechanics for cases in which the disc interacts gravitationally with one or more orbital companion., Comment: 18 pages, 6 figures, accepted manuscript to appear in MNRAS