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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

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

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

The nonlinear behaviour of low-viscosity warped discs is poorly understood. We verified a nonlinear bending-wave theory, in which fluid columns undergo affine transformations, with direct 3D hydrodynamical simulations. We employed a second-order Godunov-type scheme, Meshless Finite Mass (MFM), and also the Smoothed Particle Hydrodynamics (SPH) method, with up to 128M particles. For moderate nonlinearity, MFM maintains well the steady nonlinear warp predicted by the affine model for a tilted inviscid disc around a central object with a quadrupole moment. However, numerical dissipation in SPH is so severe that even a low-amplitude nonlinear warp degrades at a resolution where MFM performs well. A low-amplitude arbitrary warp tends to evolve towards a nonlinear steady state. However, no such state exists in our thin disc with an angular semi-thickness H/R = 0.02 when the outer tilt angle is beyond about 14 degrees. The warp breaks tenuously and reconnects in adiabatic simulations, or breaks into distinct annuli in isothermal simulations. The breaking radius lies close to the location with the most extreme nonlinear deformation. Parametric instability is captured only in our highest-resolution simulation, leading to ring structures that may serve as incubators for planets around binaries., Comment: accepted to 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

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

Observations of distorted discs have highlighted the ubiquity of warps in a variety of astrophysical contexts. This has been complemented by theoretical efforts to understand the dynamics of warp evolution. Despite significant efforts to understand the dynamics of warped discs, previous work fails to address arguably the most prevalent regime -- nonlinear warps in Keplerian discs for which there is a resonance between the orbital, epicyclic and vertical oscillation frequencies. In this work, we implement a novel nonlinear ring model, developed recently by Fairbairn and Ogilvie, as a framework for understanding such resonant warp dynamics. Here we uncover two distinct nonlinear regimes as the warp amplitude is increased. Initially we find a smooth modulation theory which describes warp evolution in terms of the averaged Lagrangian of the oscillatory vertical motions of the disc. This hints towards the possibility of connecting previous warp theory under a generalised secular framework. Upon the warp amplitude exceeding a critical value, which scales as the square root of the aspect-ratio of our ring, the disc enters into a bouncing regime with extreme vertical compressions twice per orbit. We develop an impulsive theory which predicts special retrograde and prograde precessing warped solutions, which are identified numerically using our full equation set. Such solutions emphasise the essential activation of nonlinear vertical oscillations within the disc and may have important implications for energy and warp dissipation. Future work should search for this behaviour in detailed numerical studies of the internal flow structure of warped discs., Comment: 21 pages, 13 figures. Accepted for publication 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

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., 16 pages, 15 figures, accepted in MNRAS. Updated with grammatical and typographical corrections

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

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

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., 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

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

We explore the evolution of the eccentricity of an accretion disc perturbed by an embedded planet whose mass is sufficient to open a large gap in the disc. Various methods for representing the orbit-averaged motion of an eccentric disc are discussed. We characterize the linear instability which leads to the growth of eccentricity by means of hydrodynamical simulations. We numerically recover the known result that eccentricity growth in the disc is possible when the planet-to-star mass ratio exceeds 0.003. For mass ratios larger than this threshold, the precession rates and growth rates derived from simulations, as well as the shape of the eccentric mode, compare well with the predictions of a linear theory of eccentric discs. We study mechanisms by which the eccentricity growth eventually saturates into a non-linear regime., 16 pages, 18 figures, accepted for publication in MNRAS

The diurnal tidal forces can excite a normal mode of the Earth's core, the free inner core nutation (FICN), which is characterized by a tilt of the rotation axis of the inner core with respect to the rotation axis of the outer core. The differential rotation between the inner core and the outer core induces fluid motions in the outer core and gives rise to Ohmic dissipation in the presence of the Earth's internal magnetic field. Nutation measurements can reflect such dissipation if it is sufficiently strong and thus can provide insights into the properties and dynamics of the Earth's core. In this study we perform a set of numerical calculations of the linear perturbations in the outer core induced by the FICN at very low Ekman numbers (as small as $10^{-11}$). Our numerical results show that the back-reaction of the magnetic field notably alters the structure and length scale of the perturbations induced by the FICN, and thus influences the Ohmic dissipation resulting from the perturbations. When the Ekman number is sufficiently small, Ohmic dissipation tends to be insensitive to the fluid viscosity and to the magnetic diffusivity, which allows us to estimate the Ohmic dissipation associated with the FICN without relying on an extrapolation. In contrast to the results of Buffett (2010b), the estimated Ohmic dissipation based on our numerical calculations is too weak to account for the observed damping of the FICN mode. This also implies that nutation measurements cannot provide effective constraints on the strength of the magnetic field inside the Earth's outer core., Accepted for publication in EPSL

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., 18 pages, 6 figures, accepted manuscript to appear in MNRAS

We revisit the global modes and instabilities of homogeneous rotating ellipsoidal fluid masses, which are the simplest global models of rotationally and tidally deformed gaseous planets or stars. The tidal flow in a short-period planet may be unstable to the elliptical instability, a hydrodynamic instability that can drive tidal evolution. We perform a global (and local WKB) analysis to study this instability using the elegant formalism of Lebovitz & Lifschitz. We survey the parameter space of global instabilities with harmonic orders $\ell\leq 5$, for planets with spins that are purely aligned (prograde) or anti-aligned (retrograde) with their orbits. In general, the instability has a much larger growth rate if the planetary spin and orbit are anti-aligned rather than aligned. We have identified a violent instability for anti-aligned spins outside of the usual frequency range for the elliptical instability (when $\frac{n}{\Omega}\lesssim -1$, where $n$ and $\Omega$ are the orbital and spin angular frequencies, respectively) if the tidal amplitude is sufficiently large. We also explore the instability in a rigid ellipsoidal container, which is found to be quantitatively similar to that with a realistic free surface. Finally, we study the effect of rotation and tidal deformation on mode frequencies. We find that larger rotation rates and larger tidal deformations both decrease the frequencies of the prograde sectoral surface gravity modes. This increases the prospect of their tidal excitation, potentially enhancing the tidal response over expectations from linear theory. In a companion paper, we use our results to interpret global simulations of the elliptical instability., Comment: 16 pages, 10 figures, accepted for publication in MNRAS (2016 March 22)

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

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., Comment: 17 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

We study the interaction between stellar irradiation and tidal heating in gaseous planets with short orbital periods. The intentionally simplified atmospheric model we employ makes the problem analytically tractable and permits the derivation of useful scaling relations. We show that many tidal models provide thermal feedback, producing interior radiative zones and leading to enhanced g-mode dissipation with a wide spectrum of resonances. These resonances are dynamically tuned by the thermal feedback, and so represent a novel form of thermomechanical feedback, coupling vibrational modes to the very slow thermal evolution of the planet. We then show that stellar irradiation allows the heat produced by these modes to be trapped at depth with high efficiency, leading to entropy increase in the central convective region, as well as expansion of the planet's radius sufficient to match observed swelling. We find that thermally driven winds play an essential role in this process by making the thermal structure of the atmosphere spherically symmetric within a few scale heights of the photosphere. We characterise the relationship between the swelling factor, the orbital period and the host star and determine the timescale for swelling. We show that these g-modes suffice to produce bloating on the order of the radius of the planet over $\mathrm{Gyr}$ timescales when combined with significant insolation and we provide analytic relations for the relative magnitudes of tidal heating and insolation., 15 pages, 5 figures, accepted to MNRAS

Spin-orbit misalignments have been detected in exoplanetary systems and binary star systems. Tidal interactions may have played an important role in the evolution of the spin-orbit angle. In this study, we investigate the tidal interactions in spin-orbit misaligned systems. In particular, we focus on the tidal response of a rotating fluid body to the obliquity tide, which may be important for the evolution of the spin-orbit angle but hardly affects the orbital evolution. The obliquity tide also provides a torque for the mutual precession of the spin and orbital axes around the total angular momentum vector, which has not yet been considered in previous studies on the tidal interactions. In this paper, we first formulate a set of linearized equations describing the tidal response in spin-orbit misaligned systems, taking into account the precessional motion. Numerical solutions in a homogeneous fluid and in a polytrope of index 1 show that dissipative inertial waves can be excited on top of precession by the obliquity tide in the presence of a rigid core. The tidal quality factor associated with the obliquity tide $Q'_{210}$ can be several orders of magnitude smaller than those associated with other tidal components if their frequencies fall outside the frequency range of inertial waves. Therefore, it is possible that the spin-orbit misalignment undergoes much more rapid decay than the orbital decay in hot Jupiter systems owing to the enhanced dissipation of the obliquity tide., 11 pages, 8 figures, accepted for publication in MNRAS

This chapter explores the physics shared by planetary rings and the various disks that populate the Universe. It begins with an observational overview, ranging from protoplanetary disks to spiral galaxies, and then compares and contrasts these astrophysical disks with the rings of the Solar System. Emphasis is placed on fundamental physics and dynamics, and how research into the two classes of object connects. Topics covered include disk formation, accretion, collisional processes, waves, instabilities, and satellite-disk interactions., Comment: To appear in: Tiscareno & Murray Eds (2017), Planetary Ring Systems, Cambridge University Press

Eccentric Keplerian discs are believed to be unstable to three-dimensional hydrodynamical instabilities driven by the time-dependence of fluid properties around an orbit. These instabilities could lead to small-scale turbulence, and ultimately modify the global disc properties. We use a local model of an eccentric disc, derived in a companion paper, to compute the nonlinear vertical ("breathing mode") oscillations of the disc. We then analyse their linear stability to locally axisymmetric disturbances for any disc eccentricity and eccentricity gradient using a numerical Floquet method. In the limit of small departures from a circular reference orbit, the instability of an isothermal disc is explained analytically. We also study analytically the small-scale instability of an eccentric neutrally stratified polytropic disc with any polytropic index using a WKB approximation. We find that eccentric discs are generically unstable to the parametric excitation of small-scale inertial waves. The nonlinear evolution of these instabilities should be studied in numerical simulations, where we expect them to lead to a decay of the disc eccentricity and eccentricity gradient as well as to induce additional transport and mixing. Our results highlight that it is essential to consider the three-dimensional structure of eccentric discs, and their resulting vertical oscillatory flows, in order to correctly capture their evolution., Comment: 18 pages, 6 figures, to be published in MNRAS

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 (2007), 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 $\gtrsim 8H$, where $H$ is the disc's scale height, 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, localised around an entropy maximum., Comment: 12 pages, 16 figures // v2: corrected mistakes in adiabatic terms in Eq. 24-26. These changes impacted only on Figs. 11-13, which were also updated

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., Comment: 14 pages, 12 figures, 1 table, accepted for publication in MNRAS (2016 March 08)

These lecture notes and example problems are based on a course given at the University of Cambridge in Part III of the Mathematical Tripos. Fluid dynamics is involved in a very wide range of astrophysical phenomena, such as the formation and internal dynamics of stars and giant planets, the workings of jets and accretion discs around stars and black holes, and the dynamics of the expanding Universe. Effects that can be important in astrophysical fluids include compressibility, self-gravitation and the dynamical influence of the magnetic field that is 'frozen in' to a highly conducting plasma. The basic models introduced and applied in this course are Newtonian gas dynamics and magnetohydrodynamics (MHD) for an ideal compressible fluid. The mathematical structure of the governing equations and the associated conservation laws are explored in some detail because of their importance for both analytical and numerical methods of solution, as well as for physical interpretation. Linear and nonlinear waves, including shocks and other discontinuities, are discussed. The spherical blast wave resulting from a supernova, and involving a strong shock, is a classic problem that can be solved analytically. Steady solutions with spherical or axial symmetry reveal the physics of winds and jets from stars and discs. The linearized equations determine the oscillation modes of astrophysical bodies, as well as determining their stability and their response to tidal forcing., 94 pages, 23 figures, to be published in the Journal of Plasma Physics (JPP Lecture Notes)

In Keplerian accretion disks, turbulence and magnetic fields may be jointly excited through a subcritical dynamo process involving the magnetorotational instability (MRI). High-resolution simulations exhibit a tendency towards statistical self-organization of MRI dynamo turbulence into large-scale cyclic dynamics. Understanding the physical origin of these structures, and whether they can be sustained and transport angular momentum efficiently in astrophysical conditions, represents a significant theoretical challenge. The discovery of simple periodic nonlinear MRI dynamo solutions has recently proven useful in this respect, and has notably served to highlight the role of turbulent magnetic diffusion in the seeming decay of the dynamics at low magnetic Prandtl number Pm (magnetic diffusivity larger than viscosity), a common regime in accretion disks. The connection between these simple structures and the statistical organization reported in turbulent simulations remained elusive, though. Here, we report the numerical discovery in moderate aspect ratio Keplerian shearing boxes of new periodic, incompressible, three-dimensional nonlinear MRI dynamo solutions with a larger dynamical complexity reminiscent of such simulations. These "chimera" cycles are characterized by multiple MRI-unstable dynamical stages, but their basic physical principles of self-sustainment are nevertheless identical to those of simpler cycles found in azimuthally elongated boxes. In particular, we find that they are not sustained at low Pm either due to subcritical turbulent magnetic diffusion. These solutions offer a new perspective into the transition from laminar to turbulent instability-driven dynamos, and may prove useful to devise improved statistical models of turbulent accretion disk dynamos., Comment: 12 pages, 8 figures, submitted to A&A

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., Comment: 27 pages, 14 figures, accepted for publication in MNRAS

Jets, from the protostellar to the AGN context, have been extensively studied but their connection to the turbulent dynamics of the underlying accretion disc is poorly understood. Following a similar approach to Lesur et al. (2013), we examine the role of the magnetorotational instability (MRI) in the production and acceleration of outflows from discs. Via a suite of one-dimensional shearing-box simulations of stratified discs we show that magneto-centrifugal winds exhibit cyclic activity with a period of 10-20 Omega^{-1}, a few times the orbital period. The cycle seems to be more vigorous for strong vertical field; it is robust to the variation of relevant parameters and independent of numerical details. The convergence of these solutions (in particular the mass loss rate) with vertical box size is also studied. By considering a sequence of magnetohydrostatic equilibria and their stability, the periodic activity may be understood as the succession of the following phases: (a) a dominant MRI channel mode, (b) strong magnetic field generation, (c) consequent wind launching, and ultimately (d) vertical expulsion of the excess magnetic field by the expanding and accelerating gas associated with the wind. We discuss potential connections between this behaviour and observed time-variability in disk-jet systems., Comment: 19 pages, 16 figures, accepted in MNRAS (August, 30th 2016)

Tidal friction is thought to be important in determining the long-term spin-orbit evolution of short-period extrasolar planetary systems. Using a simple model of the orbit-averaged effects of tidal friction, we study the evolution of close-in planets on inclined orbits, due to tides. We analyse the effects of the inclusion of stellar magnetic braking by performing a phase-plane analysis of a simplified system of equations, including the braking torque. The inclusion of magnetic braking is found to be important, and its neglect can result in a very different system history. We then present the results of numerical integrations of the tidal evolution equations, where we find that it is essential to consider coupled evolution of the orbital and rotational elements, including dissipation in both the star and planet, to accurately model the evolution. The main result of our integrations is that for typical Hot Jupiters, tidal friction aligns the stellar spin with the orbit on a similar time as it causes the orbit to decay. This means that if a planet is observed to be aligned, then it probably formed coplanar. This reinforces the importance of Rossiter-McLaughlin effect observations in determining the degree of spin-orbit alignment in transiting systems. We apply these results to the XO-3 system, and constrain the tidal quality factors Q' in both the star and planet in this system. Using a model in which inertial waves are excited by tidal forcing in the outer convective envelope and dissipated by turbulent viscosity, we calculate Q' for a range of F-star models, and find it to vary considerably within this class of stars. This means that assuming a single Q' applies to all stars is probably incorrect. We propose an explanation for the survival of WASP-12 b & OGLE-TR-56 b, in terms of weak dissipation in the star., 19 pages, 8 figures, accepted in MNRAS

Identifying generic physical mechanisms responsible for the generation of magnetic fields and turbulence in differentially rotating flows is fundamental to understand the dynamics of astrophysical objects such as accretion disks and stars. In this paper, we discuss the concept of subcritical dynamo action and its hydrodynamic analogue exemplified by the process of nonlinear transition to turbulence in non-rotating wall-bounded shear flows. To illustrate this idea, we describe some recent results on nonlinear hydrodynamic transition to turbulence and nonlinear dynamo action in rotating shear flows pertaining to the problem of turbulent angular momentum transport in accretion disks. We argue that this concept is very generic and should be applicable to many astrophysical problems involving a shear flow and non-axisymmetric instabilities of shear-induced axisymmetric toroidal velocity or magnetic fields, such as Kelvin-Helmholtz, magnetorotational, Tayler or global magnetoshear instabilities. In the light of several recent numerical results, we finally suggest that, similarly to a standard linear instability, subcritical MHD dynamo processes in high-Reynolds number shear flows could act as a large-scale driving mechanism of turbulent flows that would in turn generate an independent small-scale dynamo., Comment: 12 pages. Submitted to Astronomische Nachrichten (2007 Catania MHD Workshop special issue)

We study the development of finite eccentricity in accretion disks in close binary systems using a two-dimensional grid-based numerical scheme. We perform detailed parameter studies to explore the dependence on viscosity, disk aspect ratio, the inclusion of a mass-transfer stream and the role of the boundary conditions. We consider mass ratios 0.05, Comment: 19 pages, 27 Figures, accepted by A&A

We calculate the excitation and dissipation of low-frequency tidal oscillations in uniformly rotating solar-type stars. For tidal frequencies smaller than twice the spin frequency, inertial waves are excited in the convective envelope and are dissipated by turbulent viscosity. Enhanced dissipation occurs over the entire frequency range rather than in a series of very narrow resonant peaks, and is relatively insensitive to the effective viscosity. Hough waves are excited at the base of the convective zone and propagate into the radiative interior. We calculate the associated dissipation rate under the assumption that they do not reflect coherently from the center of the star. Tidal dissipation in a model based on the present Sun is significantly enhanced through the inclusion of the Coriolis force but may still fall short of that required to explain the circularization of close binary stars. However, the dependence of the results on the spin frequency, tidal frequency, and stellar model indicate that a more detailed evolutionary study including inertial and Hough waves is required. We also discuss the case of higher tidal frequencies appropriate to stars with very close planetary companions. The survival of even the closest hot Jupiters can be plausibly explained provided that the Hough waves they generate are not damped at the center of the star. We argue that this is the case because the tide excited by a hot Jupiter in the present Sun would marginally fail to achieve nonlinearity. As conditions at the center of the star evolve, nonlinearity may set in at a critical age, resulting in a relatively rapid inspiral of the hot Jupiter., Comment: 12 pages, 7 figures, to be published in Astrophys. J

Subcritical transition to turbulence in Keplerian accretion disks is still a controversial issue and some theoretical progress is required in order to determine whether or not this scenario provides a plausible explanation for the origin of angular momentum transport in non-magnetized accretion disks. Motivated by the recent discoveries of exact nonlinear steady self-sustaining solutions in linearly stable non-rotating shear flows, we attempt to compute similar solutions in Rayleigh-stable rotating plane Couette flows and to identify transition mechanisms in such flows by combining nonlinear continuation methods and asymptotic theory. We obtain exact nonlinear solutions for Rayleigh-stable cyclonic regimes but show that it is not possible to compute solutions for Rayleigh-stable anticyclonic regimes, including Keplerian flow, using similar techniques. We also present asymptotic descriptions of these various problems at large Reynolds numbers that provide some insight into the differences between the non-rotating and Rayleigh-stable anticyclonic regimes and derive some necessary conditions for mechanisms analogous to the non-rotating self-sustaining process to be present in flows on the Rayleigh line. Our results demonstrate that subcritical transition mechanisms cannot be identified in wall-bounded Rayleigh-stable anticyclonic shear flows by transposing directly the phenomenology of subcritical transition in cyclonic and non-rotating wall-bounded shear flows. Asymptotic developments, however, leave open the possibility that nonlinear self-sustaining solutions may exist in unbounded or periodic flows on the Rayleigh line. These could serve as a starting point to discover solutions in Rayleigh-stable flows, but the nonlinear stability of Keplerian accretion disks remains to be determined., Comment: 16 pages, 12 figures. Accepted for publication in A&A

We have applied an eccentric accretion disc theory in simplified form to the case of an accretion disc in a binary system, where the disc contains the 3:1 Lindblad resonance. This is relevant to the case of superhumps in SU Ursae Majoris cataclysmic variables and other systems, where it is thought that this resonance leads to growth of eccentricity and a modulation in the light curve due to the interaction of a precessing eccentric disc with tidal stresses. A single differential equation is formulated which describes the propagation, resonant excitation and viscous damping of eccentricity. The theory is first worked out in the simple case of a narrow ring and leads to the conclusion that the eccentricity distribution is locally suppressed by the presence of the resonance, creating a dip in the eccentricity at the resonant radius. Application of this theory to the superhump case confirms this conclusion and produces a more accurate expression for the precession rate of the disc than has been previously accomplished with simple dynamical estimates., 10 pages, 11 figures. Accepted for publication in MNRAS

[Abridged] Tides may play an important role in determining the observed distributions of mass, orbital period, and eccentricity of the extrasolar planets. In addition, tidal interactions between giant planets in the solar system and their moons are thought to be responsible for the orbital migration of the satellites, leading to their capture into resonant configurations. We treat the underlying fluid dynamical problem with the aim of determining the efficiency of tidal dissipation in gaseous giant planets. In cases of interest, the tidal forcing frequencies are comparable to the spin frequency of the planet but small compared to its dynamical frequency. We therefore study the linearized response of a slowly and possibly differentially rotating planet to low-frequency tidal forcing. Convective regions of the planet support inertial waves, while any radiative regions support generalized Hough waves. We present illustrative numerical calculations of the tidal dissipation rate and argue that inertial waves provide a natural avenue for efficient tidal dissipation in most cases of interest. The resulting value of Q depends in a highly erratic way on the forcing frequency, but we provide evidence that the relevant frequency-averaged dissipation rate may be asymptotically independent of the viscosity in the limit of small Ekman number. In short-period extrasolar planets, if the stellar irradiation of the planet leads to the formation of a radiative outer layer that supports generalized Hough modes, the tidal dissipation rate can be enhanced through the excitation and damping of these waves. These dissipative mechanisms offer a promising explanation of the historical evolution and current state of the Galilean satellites as well as the observed circularization of the orbits of short-period extrasolar planets., 74 pages, 12 figures, submitted to The Astrophysical Journal

We analyse the non-linear, three-dimensional response of a gaseous, viscous protoplanetary disc to the presence of a planet of mass ranging from one Earth mass (1 M$_e$) to one Jupiter mass (1 M$_J$) by using the ZEUS hydrodynamics code. We determine the gas flow pattern, and the accretion and migration rates of the planet. The planet is assumed to be in a fixed circular orbit about the central star. It is also assumed to be able to accrete gas without expansion on the scale of its Roche radius. Only planets with masses $M \gsim 0.1$ M$_J$ produce significant perturbations in the disc's surface density. The flow within the Roche lobe of the planet is fully three-dimensional. Gas streams generally enter the Roche lobe close to the disc midplane, but produce much weaker shocks than the streams in two-dimensional models. The streams supply material to a circumplanetary disc that rotates in the same sense as the planet's orbit. Much of the mass supply to the circumplanetary disc comes from non-coplanar flow. The accretion rate peaks with a planet mass of approximately 0.1 M$_J$ and is highly efficient, occurring at the local viscous rate. The migration timescales for planets of mass less than 0.1 M$_J$, based on torques from disc material outside the planets' Roche lobes, are in excellent agreement with the linear theory of Type I (non-gap) migration for three-dimensional discs. The transition from Type I to Type II (gap) migration is smooth, with changes in migration times of about a factor of 2. Starting with a core which can undergo runaway growth, a planet can gain up to a few M$_J$ with little migration. Planets with final masses of order 10 M$_J$ would undergo large migration, which makes formation and survival difficult., Comment: Accepted by MNRAS, 18 pages, 13 figures (6 degraded resolution). Paper with high-resolution figures available at http://www.astro.ex.ac.uk/people/mbate/

[Abridged] The turbulent stresses that lead to angular momentum transport in accretion discs have often been treated as resulting from an isotropic effective viscosity, related to the pressure through the alpha parametrization of Shakura & Sunyaev. This simple approach may be adequate for the simplest aspects of accretion disc theory, and was necessitated historically by an incomplete understanding of the origin of the turbulence. More recently, Balbus & Hawley have shown that the magnetorotational instability provides a robust mechanism of generating turbulent Reynolds and Maxwell stresses in sufficiently ionized discs. The alpha viscosity model fails to describe numerous aspects of this process. This paper introduces a new analytical model that aims to represent more faithfully the dynamics of magnetorotational turbulent stresses and bridge the gap between analytical studies and numerical simulations. Covariant evolutionary equations for the mean Reynolds and Maxwell tensors are presented, which correctly include the linear interaction with the mean flow. Non-linear and dissipative effects, in the absence of an imposed magnetic flux and in the limit of large Reynolds number and magnetic Reynolds number, are modelled through five non-linear terms that represent known physical processes and are strongly constrained by symmetry properties and dimensional considerations. The resulting model should be particularly useful in understanding the dynamics of warped, eccentric and tidally distorted discs, non-Keplerian accretion flows close to black holes, and a variety of time-dependent accretion phenomena., 15 pages, 7 figures, to be published in MNRAS

We determine the torque exerted in a steady state by an external potential on a three-dimensional gaseous disk at a non-coorbital corotation resonance. Our model accounts for the feedback of the torque on the surface density and vorticity in the corotation region, and assumes that the disk has a barotropic equation of state and a nonzero effective viscosity. The ratio of the torque to the value given by the formula of Goldreich & Tremaine depends essentially on a single dimensionless parameter, which quantifies the extent to which the resonance is saturated. We discuss the implications for the eccentricity evolution of young planets., 23 pages, 1 figure, revised version, to be published in the Astrophysical Journal

We demonstrate a close analogy between a viscoelastic medium and an electrically conducting fluid containing a magnetic field. Specifically, the dynamics of the Oldroyd-B fluid in the limit of large Deborah number corresponds to that of a magnetohydrodynamic (MHD) fluid in the limit of large magnetic Reynolds number. As a definite example of this analogy, we compare the stability properties of differentially rotating viscoelastic and MHD flows. We show that there is an instability of the Oldroyd-B fluid that is physically distinct from both the inertial and elastic instabilities described previously in the literature, but is directly equivalent to the magnetorotational instability in MHD. It occurs even when the specific angular momentum increases outwards, provided that the angular velocity decreases outwards; it derives from the kinetic energy of the shear flow and does not depend on the curvature of the streamlines. However, we argue that the elastic instability of viscoelastic Couette flow has no direct equivalent in MHD., 21 pages, 3 figures, to be published in J. Fluid Mech

We study the viscous effects on the interaction between the coplanar Be-star disc and the neutron star in Be/X-ray binaries, using a three-dimensional, smoothed particle hydrodynamics code. For simplicity, we assume the Be disc to be isothermal at the temperature of half the stellar effective temperature. In order to mimic the gas ejection process from the Be star, we inject particles with the Keplerian rotation velocity at a radius just outside the star. Both Be star and neutron star are treated as point masses. We find that the Be-star disc is effectively truncated if the Shakura-Sunyaev viscosity parameter alpha_SS >> 1, which confirms the previous semi-analytical result. In the truncated disc, the material decreted from the Be star accumulates, so that the disc becomes denser more rapidly than if around an isolated Be star. The resonant truncation of the Be disc results in a significant reduction of the amount of gas captured by the neutron star and a strong dependence of the mass capture rate on the orbital phase. We also find that an eccentric mode is excited in the Be disc through direct driving due to a one-armed bar potential of the binary. The strength of the mode becomes greater in the case of a smaller viscosity. In a high-resolution simulation with alpha_SS=0.1, the eccentric mode is found to precess in a prograde sense. The mass capture rate by the neutron star modulates as the mode precesses., 15 pages, including 20 figures and 1 table, accepted for publication in MNRAS

An important and widely neglected aspect of the interaction between an accretion disc and a massive companion with a coplanar orbit is the vertical component of the tidal force. As shown by Lubow, the response of the disc to vertical forcing is resonant at certain radii, at which a localized torque is exerted, and from which a compressive wave (p mode) may be emitted. Although these vertical resonances are weaker than the corresponding Lindblad resonances, the m=2 inner vertical resonance in a binary star is typically located within the tidal truncation radius of a circumstellar disc. In this paper I develop a general theory of vertical resonances, allowing for non-linearity of the response, and dissipation by radiative damping and turbulent viscosity. The problem is reduced to a universal, non-linear ordinary differential equation with two real parameters. Solutions of the complex non-linear Airy equation are presented to illustrate the non-linear saturation of the resonance and the effects of dissipation. It is argued that the m=2 inner vertical resonance is unlikely to truncate the disc in cataclysmic variable stars, but contributes to angular momentum transport and produces a potentially observable non-axisymmetric structure., Comment: 17 pages, 3 figures, to be published in MNRAS

The magnetorotational (MRI) dynamo has long been considered one of the possible drivers of turbulent angular momentum transport in astrophysical accretion disks. However, various numerical results suggest that this dynamo may be difficult to excite in the astrophysically relevant regime of magnetic Prandtl number (Pm) significantly smaller than unity, for reasons currently not well understood. The aim of this article is to present the first results of an ongoing numerical investigation of the role of both linear and nonlinear dissipative effects in this problem. Combining a parametric exploration and an energy analysis of incompressible nonlinear MRI dynamo cycles representative of the transitional dynamics in large aspect ratio shearing boxes, we find that turbulent magnetic diffusion makes the excitation and sustainment of this dynamo at moderate magnetic Reynolds number (Rm) increasingly difficult for decreasing Pm. This results in an increase in the critical Rm of the dynamo for increasing kinematic Reynolds number (Re), in agreement with earlier numerical results. Given its very generic nature, we argue that turbulent magnetic diffusion could be an important determinant of MRI dynamo excitation in disks, and may also limit the efficiency of angular momentum transport by MRI turbulence in low Pm regimes., 7 pages, 6 figures

Saturn's inner B-ring and its C-ring support wavetrains of contrasting amplitudes but with similar length scales, 100-1000 km. In addition, the inner B-ring is punctuated by two intriguing `flat' regions between radii 93,000 km and 98,000 km in which the waves die out, whereas the C-ring waves coexist with a forest of plateaus, narrow ringlets, and gaps. In both regions the waves are probably generated by a large-scale linear instability whose origin lies in the meteoritic bombardment of the rings: the ballistic transport instability. In this paper, the third in a series, we numerically simulate the long-term nonlinear evolution of this instability in a convenient local model. Our C-ring simulations confirm that the unstable system forms low-amplitude wavetrains possessing a preferred band of wavelengths. B-ring simulations, on the other hand, exhibit localised nonlinear wave `packets' separated by linearly stable flat zones. Wave packets travel slowly while spreading in time, a result that suggests the observed flat regions in Saturn's B-ring are shrinking. Finally, we present exploratory runs of the inner B-ring edge which reproduce earlier numerical results: ballistic transport can maintain the sharpness of a spreading edge while building a `ramp' structure at its base. Moreover, the ballistic transport instability can afflict the ramp region, but only in low-viscosity runs., 11 pages, 9 figures, MNRAS accepted

The ejecta discharged by impacting meteorites can redistribute a planetary ring's mass and angular momentum. This `ballistic transport' of ring properties instigates a linear instability that could generate the 100--1000-km undulations observed in Saturn's inner B-ring and in its C-ring. We present semi-analytic results demonstrating how the instability sustains steadily travelling nonlinear wavetrains. At low optical depths, the instability produces approximately sinusoidal waves of low amplitude, which we identify with those observed between radii 77,000 and 86,000 km in the C-ring. On the other hand, optical depths of 1 or more exhibit hysteresis, whereby the ring falls into multiple stable states: the homogeneous background equilibrium or large-amplitude wave states. Possibly the `flat zones' and `wave zones' between radii 93,000 and 98,000 km in the B-ring correspond to the stable homogeneous and wave states, respectively. In addition, we test the linear stability of the wavetrains and show that only a small subset are stable. In particular, stable solutions all possess wavelengths greater than the lengthscale of fastest linear growth. We supplement our calculations with a weakly nonlinear analysis that suggests the C-ring reproduces some of the dynamics of the complex Ginzburg--Landau equation. In the third paper in the series, these results will be tested and extended with numerical simulations., 14 pages, 13 figures, MNRAS accepted