Time-dependent quasi-spherical accretion

Differentially rotating, "advection-dominated" accretion flows are considered in which the heat generated by viscous dissipation is retained in the fluid. The equations of time-dependent quasi-spherical accretion are solved in a simplified one-dimensional model that neglects the latitudinal dependence of the flow. A self-similar solution is presented that has finite size, mass, angular momentum and energy. This may be expected to be an attractor for the initial-value problem in which a cool and narrow ring of fluid orbiting around a central mass heats up, spreads radially and is accreted. The solution provides some insight into the dynamics of quasi-spherical accretion and avoids many of the strictures of the steady self-similar solution of Narayan & Yi. Special attention is given to the astrophysically important case in which the adiabatic exponent gamma=5/3; even in this case, the flow is found to be differentially rotating and bound to the central object, and accretion can occur without the need for powerful outflows.Comment: 6 pages, 3 figures, to be published in MNRA

Nonlinear bending waves in Keplerian accretion discs

The nonlinear dynamics of a warped accretion disc is investigated in the important case of a thin Keplerian disc with negligible viscosity and self-gravity. A one-dimensional evolutionary equation is formally derived that describes the primary nonlinear and dispersive effects on propagating bending waves other than parametric instabilities. It has the form of a derivative nonlinear Schroedinger equation with coefficients that are obtained explicitly for a particular model of a disc. The properties of this equation are analysed in some detail and illustrative numerical solutions are presented. The nonlinear and dispersive effects both depend on the compressibility of the gas through its adiabatic index Gamma. In the physically realistic case Gamma<3, nonlinearity does not lead to the steepening of bending waves but instead enhances their linear dispersion. In the opposite case Gamma>3, nonlinearity leads to wave steepening and solitary waves are supported. The effects of a small effective viscosity, which may suppress parametric instabilities, are also considered. This analysis may provide a useful point of comparison between theory and numerical simulations of warped accretion discs.Comment: 15 pages, 2 figures, to be published in MNRA

The non-linear fluid dynamics of a warped accretion disc

The dynamics of a viscous accretion disc subject to a slowly varying warp of large amplitude is considered. Attention is restricted to discs in which self-gravitation is negligible, and to the generic case in which the resonant wave propagation found in inviscid Keplerian discs does not occur. The equations of fluid dynamics are derived in a coordinate system that follows the principal warping motion of the disc. They are reduced using asymptotic methods for thin discs, and solved to extract the equation governing the warp. In general, this is a wave equation of parabolic type with non-linear dispersion and diffusion, which describes fully non-linear bending waves. This method generalizes the linear theory of Papaloizou & Pringle (1983) to allow for an arbitrary rotation law, and extends it into the non-linear domain, where it connects with a generalized version of the theory of Pringle (1992). The astrophysical implications of this analysis are discussed briefly.Comment: 23 pages, 5 figures, to be published in MNRA

The effect of planetary migration on the corotation resonance

The migration of a planet through a gaseous disc causes the locations of their resonant interactions to drift and can alter the torques exerted between the planet and the disc. We analyse the time-dependent dynamics of a non-coorbital corotation resonance under these circumstances. The ratio of the resonant torque in a steady state to the value given by Goldreich & Tremaine (1979) depends essentially on two dimensionless quantities: a dimensionless turbulent diffusion time-scale and a dimensionless radial drift speed. When the drift speed is comparable to the libration speed and the viscosity is small, the torque can become much larger than the unsaturated value in the absence of migration, but is still proportional to the large-scale vortensity gradient in the disc. Fluid that is trapped in the resonance and drifts with it acquires a vortensity anomaly relative to its surroundings. If the anomaly is limited by viscous diffusion in a steady state, the resulting torque is inversely proportional to the viscosity, although a long time may be required to achieve this state. A further, viscosity-independent, contribution to the torque comes from fluid that streams through the resonant region. In other cases, torque oscillations occur before the steady value is achieved. We discuss the significance of these results for the evolution of eccentricity in protoplanetary systems. We also describe the possible application of these findings to the coorbital region and the concept of runaway (or type III) migration. [Abridged]Comment: 15 pages, 6 figures, to be published in MNRA