Global evolution of the magnetic field in a thin disc and its consequences for protoplanetary systems

The strength and structure of the large-scale magnetic field in protoplanetary discs are still unknown, although they could have important consequences for the dynamics and evolution of the disc. Using a mean-field approach in which we model the effects of turbulence through enhanced diffusion coefficients, we study the time-evolution of the large-scale poloidal magnetic field in a global model of a thin accretion disc, with particular attention to protoplanetary discs. With the transport coefficients usually assumed, the magnetic field strength does not significantly increase radially inwards, leading to a relatively weak magnetic field in the inner part of the disc. We show that with more realistic transport coefficients that take into account the vertical structure of the disc and the back-reaction of the magnetic field on the flow as obtained by Guilet & Ogilvie (2012), the magnetic field can significantly increase radially inwards. The magnetic-field profile adjusts to reach an equilibrium value of the plasma $\beta$ parameter (the ratio of midplane thermal pressure to magnetic pressure) in the inner part of the disc. This value of $\beta$ depends strongly on the aspect ratio of the disc and on the turbulent magnetic Prandtl number, and lies in the range $10^4-10^7$ for protoplanetary discs. Such a magnetic field is expected to affect significantly the dynamics of protoplanetary discs by increasing the strength of MHD turbulence and launching an outflow. We discuss the implications of our results for the evolution of protoplanetary discs and for the formation of powerful jets as observed in T-Tauri star systems.Comment: 19 pages, 12 figures, accepted for publication in MNRA

The linear stability of dilute particulate rings

Irregular structure in planetary rings is often attributed to the intrinsic instabilities of a homogeneous state undergoing Keplerian shear. Previously these have been analysed with simple hydrodynamic models. We instead employ a kinetic theory, in which we solve the linearised moment equations derived in Shu and Stewart 1985 for a dilute ring. This facilitates an examination of velocity anisotropy and non-Newtonian stress, and their effects on the viscous and viscous/gravitational instabilities thought to occur in Saturn's rings. Because we adopt a dilute gas model, the applicability of our results to the actual dense rings of Saturn are significantly curtailled. Nevertheless this study is a necessary preliminary before an attack on the difficult problem of dense ring dynamics. We find the Shu and Stewart formalism admits analytic stability criteria for the viscous overstability, viscous instability, and thermal instability. These criteria are compared with those of a hydrodynamic model incorporating the effective viscosity and cooling function computed from the kinetic steady state. We find the two agree in the `hydrodynamic limit' (i.e. many collisions per orbit) but disagree when collisions are less frequent, when we expect the viscous stress to be increasingly non-Newtonian and the velocity distribution increasingly anisotropic. In particular, hydrodynamics predicts viscous overstability for a larger portion of parameter space. We also numerically solve the linearised equations of the more accurate Goldreich and Tremaine 1978 kinetic model and discover its linear stability to be qualitatively the same as that of Shu and Stewart's. Thus the simple collision operator adopted in the latter would appear to be an adequate approximation for dilute rings, at least in the linear regime

Hydrodynamic instability in warped astrophysical discs

Warped astrophysical discs are usually treated as laminar viscous flows, which have anomalous properties when the disc is nearly Keplerian and the viscosity is small: fast horizontal shearing motions and large torques are generated, which cause the warp to evolve rapidly, in some cases at a rate that is inversely proportional to the viscosity. However, these flows are often subject to a linear hydrodynamic instability, which may produce small-scale turbulence and modify the large-scale dynamics of the disc. We use a warped shearing sheet to compute the oscillatory laminar flows in a warped disc and to analyse their linear stability by the Floquet method. We find widespread hydrodynamic instability deriving from the parametric resonance of inertial waves. Even very small, unobservable warps in nearly Keplerian discs of low viscosity can be expected to generate hydrodynamic turbulence, or at least wave activity, by this mechanism.Comment: 17 pages, 7 figures, revised version, to be published in MNRA

Tidal interactions of a Maclaurin spheroid. I: Properties of free oscillation modes

We review the work of Bryan (1889) on the normal modes of a Maclaurin spheroid, carrying out numerical calculations of the frequencies and spatial forms of these modes that have not been previously published. We study all modes of degree $l \le 4$, which includes both inertial modes and surface gravity modes, with the aim of better understanding the effect of rapid rotation on tidal interactions. The inclusion of these higher degree modes greatly increases the number of frequencies at which tidal resonances may occur. We derive an expression for the decay rates of these modes to first order in viscosity and explicitly plot these for modes. We see that the equatorial bulge of the spheroid has a significant effect on the decay rates (changing some of these by a factor of 2 between an eccentricity of $e=0$ and $0.5$), and a more modest effect on the mode frequencies. This suggests that models of tidal interaction between rapidly rotating stars and giant planets that model the Coriolis force while neglecting the centrifugal distortion of the body may be in error by an order unity factor. In a subsequent paper we shall examine the case of a forced flow in this spheroid, and complete the model by considering how the tides raised by the orbiting companion change the orbital elements.Comment: 27 pages, 39 figures, 1 table, accepted for publication in MNRA

Local and global dynamics of warped astrophysical discs

Astrophysical discs are warped whenever a misalignment is present in the system, or when a flat disc is made unstable by external forces. The evolution of the shape and mass distribution of a warped disc is driven not only by external influences but also by an internal torque, which transports angular momentum through the disc. This torque depends on internal flows driven by the oscillating pressure gradient associated with the warp, and on physical processes operating on smaller scales, which may include instability and turbulence. We introduce a local model for the detailed study of warped discs. Starting from the shearing sheet of Goldreich & Lynden-Bell, we impose the oscillating geometry of the orbital plane by means of a coordinate transformation. This warped shearing sheet (or box) is suitable for analytical and computational treatments of fluid dynamics, magnetohydrodynamics, etc., and it can be used to compute the internal torque that drives the large-scale evolution of the disc. The simplest hydrodynamic states in the local model are horizontally uniform laminar flows that oscillate at the orbital frequency. These correspond to the nonlinear solutions for warped discs found in previous work by Ogilvie, and we present an alternative derivation and generalization of that theory. In a companion paper we show that these laminar flows are often linearly unstable, especially if the disc is nearly Keplerian and of low viscosity. The local model can be used in future work to determine the nonlinear outcome of the hydrodynamic instability of warped discs, and its interaction with others such as the magnetorotational instability.Comment: 17 pages, 10 figures, revised version, to be published in MNRA

Viscous overstability and eccentricity evolution in three-dimensional gaseous discs

We investigate the growth or decay rate of the fundamental mode of even symmetry in a viscous accretion disc. This mode occurs in eccentric discs and is known to be potentially overstable. We determine the vertical structure of the disc and its modes, treating radiative energy transport in the diffusion approximation. In the limit of very long radial wavelength, an analytical criterion for viscous overstability is obtained, which involves the effective shear and bulk viscosity, the adiabatic exponent and the opacity law of the disc. This differs from the prediction of a two-dimensional model. On shorter wavelengths (a few times the disc thickness), the criterion for overstability is more difficult to satisfy because of the different vertical structure of the mode. In a low-viscosity disc a third regime of intermediate wavelengths appears, in which the overstability is suppressed as the horizontal velocity perturbations develop significant vertical shear. We suggest that this effect determines the damping rate of eccentricity in protoplanetary discs, for which the long-wavelength analysis is inapplicable and overstability is unlikely to occur on any scale. In thinner accretion discs and in decretion discs around Be stars overstability may occur only on the longest wavelengths, leading to the preferential excitation of global eccentric modes.Comment: 11 pages, 8 figure