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Many accretion discs have been found to be distorted: either warped due a misalignment in the system, or non-circular as a result of orbital eccentricity or tidal deformation by a binary companion. Warped, eccentric, and tidally distorted discs are not in vertical hydrostatic equilibrium, and thus exhibit vertical oscillations in the direction perpendicular to the disc, a phenomenon that is absent in circular and flat discs. In extreme cases, this vertical motion is manifested as a vertical `bouncing' of the gas, potentially leading to shocks and heating, as observed in recent global numerical simulations. In this paper we isolate the mechanics of vertical disc oscillations by means of quasi-2D and fully 3D hydrodynamic local (shearing-box) models. To determine the numerical and physical dissipation mechanisms at work during an oscillation we start by investigating unforced oscillations, examining the effect of initial oscillation amplitude, as well as resolution, boundary conditions, and vertical box size on the dissipation and energetics of the oscillations. We then drive the oscillations by introducing a time-dependent gravitational potential. A key result is that even a purely vertically oscillating disc is (parametrically) unstable to developing inertial waves, as we confirm through a linear stability analysis. The most important of these has the character of a bending wave, whose radial wavelength depends on the frequency of the vertical oscillation. The nonlinear phase of the instability exhibits shocks, which dampen the oscillations, although energy can also flow from the bending wave back to the vertical oscillation., Comment: 21 pages, 18 figures, 1 table. Submitted to MNRAS

The key difficulty faced by 2D models for planet-disc interaction is in appropriately accounting for the impact of the disc's vertical structure on the dynamics. 3D effects are often mimicked via softening of the planet's potential; however, the planet-induced flow and torques often depend strongly on the choice of softening length. We show that for a linear adiabatic flow perturbing a vertically isothermal disc, there is a particular vertical average of the 3D equations of motion which exactly reproduces 2D fluid equations for arbitrary adiabatic index. There is a strong connection here with the Lubow-Pringle 2D mode of the disc. Correspondingly, we find a simple, general prescription for the consistent treatment of planetary potentials embedded within '2D' discs. The flow induced by a low-mass planet involves large-scale excited spiral density waves which transport angular momentum radially away from the planet, and 'horseshoe streamlines' within the co-orbital region. We derive simple linear equations governing the flow which locally capture both effects faithfully simultaneously. We present an accurate co-orbital flow solution allowing for inexpensive future study of corotation torques, and predict the vertical structure of the co-orbital flow and horseshoe region width for different values of adiabatic index, as well as the vertical dependence of the initial shock location. We find strong agreement with the flow computed in 3D numerical simulations, and with 3D one-sided Lindblad torque estimates, which are a factor of 2 to 3 times lower than values from previous 2D simulations., Comment: Accepted MNRAS, 18 pages, 6 figures

The hydrogen deuteride (HD) molecule is an important deuterium tracer in astrophysical studies. The atmospheres of gas giants are dominated by molecular hydrogen, and simultaneous observation of H$_2$ and HD lines provides reliable information on the D/H ratios on these planets. The reference spectroscopic parameters play a crucial role in such studies. Under thermodynamic conditions encountered in these atmospheres, the spectroscopic studies of HD require not only the knowledge of line intensities and positions but also accurate reference data on pressure-induced line shapes and shifts. Our aim is to provide accurate collision-induced line-shape parameters for HD lines that cover any thermodynamic conditions relevant to the atmospheres of giant planets, i.e., any relevant temperature, pressure, and perturbing gas (the H$_2$/He mixture) composition. We perform quantum-scattering calculations on a new highly accurate ab initio potential energy surface, and we use scattering S-matrices obtained this way to determine the collision-induced line-shape parameters. We use the cavity ring-down spectroscopy for validation of our theoretical methodology. We report accurate collision-induced line-shape parameters for the pure rotational R(0), R(1), and R(2) lines, the most relevant HD lines for the investigations of atmospheres of the giant planets. Besides the basic Voigt-profile collisional parameters (i.e. the broadening and shift parameters), we also report their speed dependences and the complex Dicke parameter, which can influence the effective width and height of the HD lines up to almost a factor of 2 for giant planet conditions. The sub-percent-level accuracy, reached in this work, considerably improves the previously available data. All the reported parameters are consistent with the HITRAN database format, hence allowing for the use of HAPI for generating the beyond-Voigt spectra of HD.

Attempts to understand the dynamics of warped astrophysical discs have garnered significant attention, largely motivated by the growing catalogue of observed distorted systems. Previous studies have shown that the evolution of the warp is crucially regulated by the internal flow fields established by the undulating geometry. These are typically modelled as laminar horizontal, shearing flows which oscillate back and forth at approximately the orbital frequency. However this shearing motion is known to be susceptible to a hydrodynamic, parametric instability of inertial waves which might modify the warped dynamics. Whilst the linear growth phase is well understood, the subsequent nonlinear saturation combined with the self-consistent feedback onto the warp has not been studied. In this work, we implement a novel numerical setup using the recent ring model framework of Fairbairn and Ogilvie, within the Lagrangian code GIZMO. We formally identify several locally growing modes in the simulation, as predicted by a three-mode coupling analysis of the instability, and find decent agreement with the theoretical growth rates. We understand the saturation mechanism as a wave breaking process which suppresses the growth of shorter wavelength parametric couplings first, whilst allowing the longest mode to dominate the final quasi-steady, wavelike turbulence. The Reynolds stresses, transporting energy from the warp to the small scales, can be effectively modelled using a time-dependent, anisotropic viscous alpha model which closely captures the amplitude and phase evolution of the warp. Consequently, this model might help inform future global studies which are commonplace but typically don't resolve the parametric instability., Comment: Accepted MNRAS, 22 pages, 17 figures

We simulate the propagation and dissipation of tidally induced nonlinear gravity waves in the cores of solar-type stars. We perform hydrodynamical simulations of a previously developed Boussinesq model using a spectral-element code to study the stellar core as a wave cavity that is periodically forced at the outer boundary with a given azimuthal wavenumber and an adjustable frequency. For low-amplitude forcing, the system exhibits resonances with standing g-modes at particular frequencies, corresponding to a situation in which the tidal torque is highly frequency-dependent. For high-amplitude forcing, the excited waves break promptly near the centre and spin up the core so that subsequent waves are absorbed in an expanding critical layer, as found in previous work, leading to a tidal torque with a smooth frequency-dependence. For intermediate-amplitude forcing, we find that linear damping of the waves gradually spins up the core such that the resonance condition can be altered drastically. The system can evolve towards or away from g-mode resonances, depending on the difference between the forcing frequency and the closest eigenfrequency. Eventually, a critical layer forms and absorbs the incoming waves, leading to a situation similar to the high-amplitude case in which the waves break promptly. We study the dependence of this process on the forcing amplitude and frequency, as well as on the diffusion coefficients. We emphasize that the small Prandtl number in the centre of solar-like stars facilitates the development of a differentially rotating core owing to the nonlinear feedback of waves. Our simulations and analysis reveal that this important mechanism may drastically change the phase of gravity waves and thus the classical picture of resonance locking in solar-type stars needs to be revised., Comment: 22 pages, 26 figures, MNRAS accepted, comments welcome

Astrophysical disks that are sufficiently cold and dense are linearly unstable to the formation of axisymmetric rings as a result of the disk's gravity. In practice, spiral structures are formed, which may in turn produce bound fragments. We study a nonlinear dynamical path that can explain the development of spirals in a local model of a gaseous disk on the subcritical side of the gravitational instability bifurcation. Axisymmetric equilibria can be radially periodic or localized, in the form of standing solitary waves. The solitary solutions have an energy slightly larger than a smooth disk. They are further unstable to non-axisymmetric perturbations with a wide range of azimuthal wavenumbers. The solitary waves may act as a pathway to spirals and fragmentation., Comment: 6 pages, 4 figures, ApJL accepted

We show how the local approximation of astrophysical discs, which is the basis for the well known model of the shearing box, can be used to study many aspects of the dynamics of warped discs. In the local model, inclination of the orbit of a test particle with respect to the reference orbit corresponds to a vertical oscillation of the particle at the orbital frequency. Warping of a disc corresponds to a locally axisymmetric corrugation of the midplane of the disc that oscillates vertically at the orbital frequency, while evolution of the warp corresponds to a modulation of the complex amplitude of the vertical oscillation. We derive a conservation law for this amplitude that is the local equivalent of the conservation of angular momentum and therefore governs the evolution of the warp. For lengthscales that are long compared to the vertical scaleheight of the disc, the known non-resonant and resonant regimes of warp dynamics, including the diffusive and wavelike regimes of Keplerian discs, occur in the local model in the same way as in a global view of warped discs., Comment: 18 pages, 2 figures, accepted version to be published in MNRAS

We revisit the Tidally Excited Oscillations (TEOs) in the A-type, main-sequence, eccentric binary KOI-54, the prototype of heartbeat stars. Although the linear tidal response of the star is a series of orbital-harmonic frequencies which are not stellar eigenfrequencies, we show that the non-linearly excited non-orbital-harmonic TEOs are eigenmodes. By carefully choosing the modes which satisfy the mode-coupling selection rules, a period spacing ($\Delta P$) pattern of quadrupole gravity modes ($\Delta P \approx 2520-2535$ sec) can be discerned in the Fourier spectrum, with a detection significance level of $99.9\%$. The inferred period spacing value agrees remarkably well with the theoretical $l=2,m=0$ g modes from a stellar model with the measured mass, radius and effective temperature. We also find that the two largest-amplitude TEOs at $N=90, 91$ harmonics are very close to resonance with $l=2,m=0$ eigenmodes, and likely come from different stars. Previous works on tidal oscillations primarily focus on the modeling of TEO amplitudes and phases, the high sensitivity of TEO amplitude to the frequency detuning (tidal forcing frequency minus the closest stellar eigenfrequency) requires extremely dense grids of stellar models and prevents us from constraining the stellar physical parameters easily. This work, however, opens the window of real tidal asteroseismology by using the eigenfrequencies of the star inferred from the non-linear TEOs and possibly very-close-to-resonance linear TEOs. Our seismic modeling of these identified eigen g-modes shows that the best-matching stellar models have ($M \approx 2.20, 2.35 M_{\odot}$) and super-solar metallicity, in good agreement with previous measurements., Comment: MNRAS Accepted, comments welcome

In rotating stars and planets, excitation of inertial waves in convective envelopes provides an important channel for tidal dissipation, but the dissipation rate due to inertial waves depends erratically on the tidal frequency. Tidal dissipation is significantly enhanced at some frequencies, suggesting possible resonances between the tidal forcing and some eigenmodes. However, the nature of these resonances remains enigmatic owing to the singularity of the eigenvalue problem of inertial waves, and the resonances are often mistakenly attributed to wave attractors in the literature. In this letter, we reveal that resonant tidal responses correspond to inertial modes with large-scale flows hidden beneath localized wave beams. Strong couplings between the tidal forcing and the hidden large-scale flows intensify the localized wave beams emanating from the critical latitudes, leading to enhanced tidal dissipation. This study resolves a long-standing puzzle regarding the frequency-dependence of tidal dissipation due to inertial waves in convective envelopes., Comment: 11 pages, 5 figures, Accepted for publication in ApJL

Understanding oscillations and waves in astrophysical fluid bodies helps to elucidate their observed variability and the underlying physical mechanisms. Indeed, global oscillations and bending modes of accretion discs or tori may be relevant to quasi-periodicity and warped structures around compact objects. While most studies rely on linear theory, observationally significant, nonlinear dynamics is still poorly understood, especially in Keplerian discs for which resonances typically demand a separate treatment. In this work we introduce a novel analytical model which exactly solves the ideal, compressible fluid equations for a non-self-gravitating elliptical cylinder within a local shearing sheet. The aspect ratio of the ring is an adjustable parameter, allowing a continuum of models ranging from a torus of circular cross-section to a thin ring. We restrict attention to flow fields which are a linear function of the coordinates, capturing the lowest order global motions and reducing the dynamics to a set of coupled ordinary differential equations (ODEs). This system acts as a framework for exploring a rich range of hydrodynamic phenomena in both the large amplitude and Keplerian regimes. We demonstrate the connection between tilting tori and warped discs within this model, showing that the linear modes of the ring correspond to oppositely precessing global bending modes. These are further confirmed within a numerical grid based simulation. Crucially, the ODE system developed here allows for a more tractable investigation of nonlinear dynamics. This will be demonstrated in a subsequent paper which evidences mode coupling between warping and vertical motions in thin tilted rings., Comment: 14 pages, 7 figures, accepted version for publication in 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

This article is the second in a pair of articles on the topic of the generation of a two-color artificial star (which we term a "laser photometric ratio star," or LPRS) of de-excitation light from neutral sodium atoms in the mesosphere, for use in precision telescopic measurements in astronomy and atmospheric physics, and more specifically for the calibration of measurements of dark energy using type Ia supernovae. The two techniques respectively described in both this and the previous article would each generate an LPRS with a precisely 1:1 ratio of yellow (589/590 nm) photons to near-infrared (819/820 nm) photons produced in the mesosphere. Both techniques would provide novel mechanisms for establishing a spectrophotometric calibration ratio of unprecedented precision, from above most of Earth's atmosphere, for upcoming telescopic observations across astronomy and atmospheric physics. The technique described in this article has the advantage of producing a much brighter (specifically, brighter by approximately a factor of 1000) LPRS, using lower-power (<30 W average power) lasers, than the technique using a single 500 W average power laser described in the first article of this pair. However, the technique described here would require polarization filters to be installed into the telescope camera in order to sufficiently remove laser atmospheric Rayleigh backscatter from telescope images, whereas the technique described in the first article would only require more typical wavelength filters in order to sufficiently remove laser Rayleigh backscatter., Comment: The first article in this pair of articles is at arXiv:2001.10958 . This article has 15 pages and 3 figures. Both articles are published in MNRAS

Warped accretion discs of low viscosity are prone to hydrodynamic instability due to parametric resonance of inertial waves as confirmed by local simulations. Global simulations of warped discs, using either smoothed particle hydrodynamics (SPH) or grid-based codes, are ubiquitous but no such instability has been seen. Here we utilize a hybrid Godunov-type Lagrangian method to study parametric instability in global simulations of warped Keplerian discs at unprecedentedly high resolution (up to 120 million particles). In the global simulations, the propagation of the warp is well described by the linear bending-wave equations before the instability sets in. The ensuing turbulence, captured for the first time in a global simulation, damps relative orbital inclinations and leads to a decrease in the angular momentum deficit. As a result, the warp undergoes significant damping within one bending-wave crossing time. Observed protoplanetary disc warps are likely maintained by companions or aftermath of disc breaking., Comment: Accepted for publication in MNRAS

After the Tidal Disruption Event (TDE) of a star around a SuperMassive Black Hole (SMBH), if the stellar debris stream rapidly circularizes and forms a compact disk, the TDE emission is expected to peak in the soft X-ray or far Ultra-Violet (UV). The fact that many TDE candidates are observed to peak in the near UV and optical has challenged conventional TDE emission models. By idealizing a disk as a nested sequence of elliptical orbits which communicate adiabatically via pressure forces, and are heated by energy dissipated during the circularization of the nearly parabolic debris streams, we investigate the dynamics and thermal emission of highly eccentric TDE disks, including the effect of General-Relativistic apsidal precession from the SMBH. We calculate the properties of uniformly precessing, apsidally aligned, and highly eccentric TDE disks, and find highly eccentric disk solutions exist for realistic TDE properties (SMBH and stellar mass, periapsis distance, etc.). Taking into account compressional heating (cooling) near periapsis (apoapsis), we find our idealized eccentric disk model can produce emission consistent with the X-ray and UV/Optical luminosities of many optically bright TDE candidates. Our work attempts to quantify the thermal emission expected from the shock-heating model for TDE emission, and finds stream-stream collisions are a promising way to power optically bright TDEs., Comment: 16 pages, 11 figures, accepted to MNRAS

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., Comment: MNRAS accepted. 25 pages

Trapped inertial oscillations (r-modes) provide a promising explanation for high-frequency quasi-periodic oscillations (HFQPOs) observed in the emission from black hole X-ray binary systems. An eccentricity (or warp) can excite r-modes to large amplitudes, but concurrently the oscillations are likely damped by magnetohydrodynamic (MHD) turbulence driven by the magnetorotational instability (MRI). We force eccentricity in global, unstratified, zero-net flux MHD simulations of relativistic accretion discs, and find that a sufficiently strong disc distortion generates trapped inertial waves despite this damping. In our simulations, eccentricities above ~ 0.03 in the inner disc excite trapped waves. In addition to the competition between r-mode damping and driving, we observe that larger amplitude eccentric structures modify and in some cases suppress MRI turbulence. Given the variety of distortions (warps as well as eccentricities) capable of amplifying r-modes, the robustness of trapped inertial wave excitation in the face of MRI turbulence in our simulations provides support for a discoseismic explanation for HFQPOs., Comment: 16 pages, 14 figures, accepted in MNRAS. Updated with grammatical, typographical, and superficial image corrections

The largest uncertainty on measurements of dark energy using type Ia supernovae is presently due to systematics from photometry; specifically to the relative uncertainty on photometry as a function of wavelength in the optical spectrum. We show that a precise constraint on relative photometry between the visible and near-infrared can be achieved in upcoming surveys (such as in LSST at the Vera C. Rubin Observatory) via a mountaintop-located laser source tuned to the 342.78 nm vacuum excitation wavelength of neutral sodium atoms. Using a high-power (500 W) laser modified from laser guide star studies, this excitation will produce an artificial star (which we term a "laser photometric ratio star," or LPRS) of de-excitation light in the mesosphere that is observable from the ground at approximately 20 magnitude (i.e., well within the expected single-image magnitude limit of LSST) at wavelengths in vacuum of 589.16 nm, 589.76 nm, 818.55 nm, and 819.70 nm, with the sum of the numbers of 589.16 nm and 589.76 nm photons produced by this process equal to the sum of the numbers of 818.55 nm and 819.70 nm photons, establishing a precise calibration ratio between, for example, the LSST r and z filters. This technique can thus provide a novel mechanism for establishing a spectrophotometric calibration ratio of unprecedented precision, from above most of the Earth's atmosphere, for upcoming telescopic observations across astronomy and atmospheric physics. This article is the first in a pair of articles on this topic. The second article of the pair describes an alternative technique to achieve a similar, but brighter, LPRS than the technique described in this paper, by using two mountaintop-located lasers, at optical frequencies approximately 4 GHz away from resonances at wavelengths in vacuum of 589.16 nm and 819.71 nm respectively., Comment: The second article in this pair of articles is at arXiv:2010.08683 . This article has 14 pages & 7 figures. Both articles are published in MNRAS. This most recent arXiv version fixes a minor error in the caption of Figure 4

High-frequency quasi-periodic oscillations (HFQPOs) observed in the emission of black-hole X-ray binary systems promise insight into strongly curved spacetime. `Discoseismic' oscillations with frequencies set by the intrinsic properties of the central black hole, in particular `trapped inertial waves' (r-modes), offer an attractive explanation for HFQPOs. To produce an observable signature, however, such oscillations must be excited to sufficiently large amplitudes. Turbulence driven by the magnetorotational instability (MRI) fails to provide the necessary amplification, but r-modes may still be excited via interaction with accretion disc warps or eccentricities. We present 3D global hydrodynamic simulations of relativistic accretion discs, which demonstrate for the first time the excitation of trapped inertial waves by an imposed eccentricity in the flow. While the r-modes' saturated state depends on the vertical boundary conditions used in our unstratified, cylindrical framework, their excitation is unambiguous in all runs with eccentricity >0.005 near the ISCO. These simulations provide a proof of concept, demonstrating the robustness of trapped inertial wave excitation in a non-magnetized context. In a companion paper, we explore the competition between this excitation, and damping by magnetohydrodynamic turbulence., Comment: 16 pages, 15 figures, accepted in MNRAS. Updated with grammatical and typographical corrections

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

The evolution of a large-scale poloidal magnetic field in an accretion disc is an important problem because it determines the launching of winds and the feasibility of the magnetorotational instability to generate turbulence or channel flows. Recent studies, both semi-analytical calculations and numerical simulations, have highlighted the crucial role non-ideal MHD effects (Ohmic resistivity, Hall drift and ambipolar diffusion), relevant in the protoplanetary disc context, might play in magnetic flux evolution in the disc. We investigate the flux transport in discs through the use of two one-dimensional semi-analytic models in the vertical direction, exploring regimes where different physical source terms and effects dominate. The governing equations for both models are derived by performing an asymptotic expansion in the limit of a thin disc, with the different regimes isolated through setting the relative order of the leading terms between variables. Flux transport rates and vertical structure profiles are calculated for a range of diffusivities and disc magnetisations. We found that Ohmic and ambipolar diffusivities drive radially outward flux transport with an outwardly inclined field. A wind outflow drives inward flux transport, which is significantly enhanced in the presence of Hall drift in the positive polarity case, $\eta_H (\boldsymbol{B}_z \cdot \boldsymbol{\Omega}) > 0$, an effect which has only been briefly noted before. Coupled only with outward inclination, the Hall effect reduces the flux transport given by a background Ohmic and/or ambipolar diffusivity, but drives no flux transport when it is the only non-ideal effect present., Comment: In press. Accepted by 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

The excitation of trapped inertial waves (r-modes) by warps and eccentricities in the inner regions of a black hole accretion disc may explain the high-frequency quasi-periodic oscillations (HFQPOs) observed in the emission of Galactic X-ray binaries. However, it has been suggested that strong vertical magnetic fields push the oscillations' trapping region toward the innermost stable circular orbit (ISCO), where conditions could be unfavourable for their excitation. This paper explores the effects of large-scale magnetic fields that exhibit \textit{both} toroidal and vertical components, through local and global linear analyses. We find that a strong toroidal magnetic field can reduce the detrimental effects of a vertical field: in fact, the isolation of the trapping region from the ISCO may be restored by toroidal magnetic fields approaching thermal strengths. The toroidal field couples the r-modes to the disc's magneto-acoustic response and inflates the effective pressure within the oscillations. As a consequence, the restoring force associated with the vertical magnetic field's tension is reduced. Given the analytical and numerical evidence that accretion discs threaded by poloidal magnetic field lines develop a strong toroidal component, our result provides further evidence that the detrimental effects of magnetic fields on trapped inertial modes are not as great as previously thought., Comment: 16 pages, 6 figures, MNRAS accepted

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 generalises 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 localise 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., Comment: 20 pages, 17 figures, accepted for publication in MNRAS

We derive expressions for the local ideal magnetohydrodynamic (MHD) equations for a warped 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 magnetised warped discs in the absence of 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 Alfv\'{e}nic-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 magnetised discs., Comment: 15 pages, 9 figures, 1 table, MNRAS accepted

Thin astrophysical discs are very often modelled using the equations of two-dimensional 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 three-dimensional 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 two-dimensional hydrodynamics in the reference plane, the theory allows for a deformation of the midplane (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 three-dimensional 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., Comment: 18 pages, 4 figures, to be published in MNRAS

The high-frequency quasi-periodic oscillations (HFQPOs) that punctuate the light curves of X-ray binary systems present a window onto the intrinsic properties of stellar-mass black holes and hence a testbed for general relativity. One explanation for these features is that relativistic distortion of the accretion disc's differential rotation creates a trapping region in which inertial waves (r-modes) might grow to observable amplitudes. Local analyses, however, predict that large-scale magnetic fields push this trapping region to the inner disc edge, where conditions may be unfavorable for r-mode growth. We revisit this problem from a pseudo-Newtonian but fully global perspective, deriving linearized equations describing a relativistic, magnetized accretion flow, and calculating normal modes with and without vertical density stratification. In an unstratified model, the choice of vertical wavenumber determines the extent to which vertical magnetic fields drive the r-modes toward the inner edge. In a global model fully incorporating density stratification, we confirm that this susceptibility to magnetic fields depends on disc thickness. Our calculations suggest that in thin discs, r-modes may remain independent of the inner disc edge for vertical magnetic fields with plasma betas as low as $\beta\approx 100-300$. We posit that the appearance of r-modes in observations may be more determined by a competition between excitation and damping mechanisms near the ISCO than the modification of the trapping region by magnetic fields., Comment: 20 pages, 17 figures, MNRAS accepted